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psdlag-agn/psdlag/src/interpolation.cpp
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/*************************************************************************
Copyright (c) Sergey Bochkanov (ALGLIB project).
>>> SOURCE LICENSE >>>
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation (www.fsf.org); either version 2 of the
License, or (at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
A copy of the GNU General Public License is available at
http://www.fsf.org/licensing/licenses
>>> END OF LICENSE >>>
*************************************************************************/
#include "stdafx.h"
#include "interpolation.h"
// disable some irrelevant warnings
#if (AE_COMPILER==AE_MSVC)
#pragma warning(disable:4100)
#pragma warning(disable:4127)
#pragma warning(disable:4702)
#pragma warning(disable:4996)
#endif
using namespace std;
/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS IMPLEMENTATION OF C++ INTERFACE
//
/////////////////////////////////////////////////////////////////////////
namespace alglib
{
/*************************************************************************
IDW interpolant.
*************************************************************************/
_idwinterpolant_owner::_idwinterpolant_owner()
{
p_struct = (alglib_impl::idwinterpolant*)alglib_impl::ae_malloc(sizeof(alglib_impl::idwinterpolant), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_idwinterpolant_init(p_struct, NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_idwinterpolant_owner::_idwinterpolant_owner(const _idwinterpolant_owner &rhs)
{
p_struct = (alglib_impl::idwinterpolant*)alglib_impl::ae_malloc(sizeof(alglib_impl::idwinterpolant), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_idwinterpolant_init_copy(p_struct, const_cast<alglib_impl::idwinterpolant*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_idwinterpolant_owner& _idwinterpolant_owner::operator=(const _idwinterpolant_owner &rhs)
{
if( this==&rhs )
return *this;
alglib_impl::_idwinterpolant_clear(p_struct);
if( !alglib_impl::_idwinterpolant_init_copy(p_struct, const_cast<alglib_impl::idwinterpolant*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
return *this;
}
_idwinterpolant_owner::~_idwinterpolant_owner()
{
alglib_impl::_idwinterpolant_clear(p_struct);
ae_free(p_struct);
}
alglib_impl::idwinterpolant* _idwinterpolant_owner::c_ptr()
{
return p_struct;
}
alglib_impl::idwinterpolant* _idwinterpolant_owner::c_ptr() const
{
return const_cast<alglib_impl::idwinterpolant*>(p_struct);
}
idwinterpolant::idwinterpolant() : _idwinterpolant_owner()
{
}
idwinterpolant::idwinterpolant(const idwinterpolant &rhs):_idwinterpolant_owner(rhs)
{
}
idwinterpolant& idwinterpolant::operator=(const idwinterpolant &rhs)
{
if( this==&rhs )
return *this;
_idwinterpolant_owner::operator=(rhs);
return *this;
}
idwinterpolant::~idwinterpolant()
{
}
/*************************************************************************
IDW interpolation
INPUT PARAMETERS:
Z - IDW interpolant built with one of model building
subroutines.
X - array[0..NX-1], interpolation point
Result:
IDW interpolant Z(X)
-- ALGLIB --
Copyright 02.03.2010 by Bochkanov Sergey
*************************************************************************/
double idwcalc(const idwinterpolant &z, const real_1d_array &x)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
double result = alglib_impl::idwcalc(const_cast<alglib_impl::idwinterpolant*>(z.c_ptr()), const_cast<alglib_impl::ae_vector*>(x.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return *(reinterpret_cast<double*>(&result));
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
IDW interpolant using modified Shepard method for uniform point
distributions.
INPUT PARAMETERS:
XY - X and Y values, array[0..N-1,0..NX].
First NX columns contain X-values, last column contain
Y-values.
N - number of nodes, N>0.
NX - space dimension, NX>=1.
D - nodal function type, either:
* 0 constant model. Just for demonstration only, worst
model ever.
* 1 linear model, least squares fitting. Simpe model for
datasets too small for quadratic models
* 2 quadratic model, least squares fitting. Best model
available (if your dataset is large enough).
* -1 "fast" linear model, use with caution!!! It is
significantly faster than linear/quadratic and better
than constant model. But it is less robust (especially
in the presence of noise).
NQ - number of points used to calculate nodal functions (ignored
for constant models). NQ should be LARGER than:
* max(1.5*(1+NX),2^NX+1) for linear model,
* max(3/4*(NX+2)*(NX+1),2^NX+1) for quadratic model.
Values less than this threshold will be silently increased.
NW - number of points used to calculate weights and to interpolate.
Required: >=2^NX+1, values less than this threshold will be
silently increased.
Recommended value: about 2*NQ
OUTPUT PARAMETERS:
Z - IDW interpolant.
NOTES:
* best results are obtained with quadratic models, worst - with constant
models
* when N is large, NQ and NW must be significantly smaller than N both
to obtain optimal performance and to obtain optimal accuracy. In 2 or
3-dimensional tasks NQ=15 and NW=25 are good values to start with.
* NQ and NW may be greater than N. In such cases they will be
automatically decreased.
* this subroutine is always succeeds (as long as correct parameters are
passed).
* see 'Multivariate Interpolation of Large Sets of Scattered Data' by
Robert J. Renka for more information on this algorithm.
* this subroutine assumes that point distribution is uniform at the small
scales. If it isn't - for example, points are concentrated along
"lines", but "lines" distribution is uniform at the larger scale - then
you should use IDWBuildModifiedShepardR()
-- ALGLIB PROJECT --
Copyright 02.03.2010 by Bochkanov Sergey
*************************************************************************/
void idwbuildmodifiedshepard(const real_2d_array &xy, const ae_int_t n, const ae_int_t nx, const ae_int_t d, const ae_int_t nq, const ae_int_t nw, idwinterpolant &z)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::idwbuildmodifiedshepard(const_cast<alglib_impl::ae_matrix*>(xy.c_ptr()), n, nx, d, nq, nw, const_cast<alglib_impl::idwinterpolant*>(z.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
IDW interpolant using modified Shepard method for non-uniform datasets.
This type of model uses constant nodal functions and interpolates using
all nodes which are closer than user-specified radius R. It may be used
when points distribution is non-uniform at the small scale, but it is at
the distances as large as R.
INPUT PARAMETERS:
XY - X and Y values, array[0..N-1,0..NX].
First NX columns contain X-values, last column contain
Y-values.
N - number of nodes, N>0.
NX - space dimension, NX>=1.
R - radius, R>0
OUTPUT PARAMETERS:
Z - IDW interpolant.
NOTES:
* if there is less than IDWKMin points within R-ball, algorithm selects
IDWKMin closest ones, so that continuity properties of interpolant are
preserved even far from points.
-- ALGLIB PROJECT --
Copyright 11.04.2010 by Bochkanov Sergey
*************************************************************************/
void idwbuildmodifiedshepardr(const real_2d_array &xy, const ae_int_t n, const ae_int_t nx, const double r, idwinterpolant &z)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::idwbuildmodifiedshepardr(const_cast<alglib_impl::ae_matrix*>(xy.c_ptr()), n, nx, r, const_cast<alglib_impl::idwinterpolant*>(z.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
IDW model for noisy data.
This subroutine may be used to handle noisy data, i.e. data with noise in
OUTPUT values. It differs from IDWBuildModifiedShepard() in the following
aspects:
* nodal functions are not constrained to pass through nodes: Qi(xi)<>yi,
i.e. we have fitting instead of interpolation.
* weights which are used during least squares fitting stage are all equal
to 1.0 (independently of distance)
* "fast"-linear or constant nodal functions are not supported (either not
robust enough or too rigid)
This problem require far more complex tuning than interpolation problems.
Below you can find some recommendations regarding this problem:
* focus on tuning NQ; it controls noise reduction. As for NW, you can just
make it equal to 2*NQ.
* you can use cross-validation to determine optimal NQ.
* optimal NQ is a result of complex tradeoff between noise level (more
noise = larger NQ required) and underlying function complexity (given
fixed N, larger NQ means smoothing of compex features in the data). For
example, NQ=N will reduce noise to the minimum level possible, but you
will end up with just constant/linear/quadratic (depending on D) least
squares model for the whole dataset.
INPUT PARAMETERS:
XY - X and Y values, array[0..N-1,0..NX].
First NX columns contain X-values, last column contain
Y-values.
N - number of nodes, N>0.
NX - space dimension, NX>=1.
D - nodal function degree, either:
* 1 linear model, least squares fitting. Simpe model for
datasets too small for quadratic models (or for very
noisy problems).
* 2 quadratic model, least squares fitting. Best model
available (if your dataset is large enough).
NQ - number of points used to calculate nodal functions. NQ should
be significantly larger than 1.5 times the number of
coefficients in a nodal function to overcome effects of noise:
* larger than 1.5*(1+NX) for linear model,
* larger than 3/4*(NX+2)*(NX+1) for quadratic model.
Values less than this threshold will be silently increased.
NW - number of points used to calculate weights and to interpolate.
Required: >=2^NX+1, values less than this threshold will be
silently increased.
Recommended value: about 2*NQ or larger
OUTPUT PARAMETERS:
Z - IDW interpolant.
NOTES:
* best results are obtained with quadratic models, linear models are not
recommended to use unless you are pretty sure that it is what you want
* this subroutine is always succeeds (as long as correct parameters are
passed).
* see 'Multivariate Interpolation of Large Sets of Scattered Data' by
Robert J. Renka for more information on this algorithm.
-- ALGLIB PROJECT --
Copyright 02.03.2010 by Bochkanov Sergey
*************************************************************************/
void idwbuildnoisy(const real_2d_array &xy, const ae_int_t n, const ae_int_t nx, const ae_int_t d, const ae_int_t nq, const ae_int_t nw, idwinterpolant &z)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::idwbuildnoisy(const_cast<alglib_impl::ae_matrix*>(xy.c_ptr()), n, nx, d, nq, nw, const_cast<alglib_impl::idwinterpolant*>(z.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Barycentric interpolant.
*************************************************************************/
_barycentricinterpolant_owner::_barycentricinterpolant_owner()
{
p_struct = (alglib_impl::barycentricinterpolant*)alglib_impl::ae_malloc(sizeof(alglib_impl::barycentricinterpolant), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_barycentricinterpolant_init(p_struct, NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_barycentricinterpolant_owner::_barycentricinterpolant_owner(const _barycentricinterpolant_owner &rhs)
{
p_struct = (alglib_impl::barycentricinterpolant*)alglib_impl::ae_malloc(sizeof(alglib_impl::barycentricinterpolant), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_barycentricinterpolant_init_copy(p_struct, const_cast<alglib_impl::barycentricinterpolant*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_barycentricinterpolant_owner& _barycentricinterpolant_owner::operator=(const _barycentricinterpolant_owner &rhs)
{
if( this==&rhs )
return *this;
alglib_impl::_barycentricinterpolant_clear(p_struct);
if( !alglib_impl::_barycentricinterpolant_init_copy(p_struct, const_cast<alglib_impl::barycentricinterpolant*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
return *this;
}
_barycentricinterpolant_owner::~_barycentricinterpolant_owner()
{
alglib_impl::_barycentricinterpolant_clear(p_struct);
ae_free(p_struct);
}
alglib_impl::barycentricinterpolant* _barycentricinterpolant_owner::c_ptr()
{
return p_struct;
}
alglib_impl::barycentricinterpolant* _barycentricinterpolant_owner::c_ptr() const
{
return const_cast<alglib_impl::barycentricinterpolant*>(p_struct);
}
barycentricinterpolant::barycentricinterpolant() : _barycentricinterpolant_owner()
{
}
barycentricinterpolant::barycentricinterpolant(const barycentricinterpolant &rhs):_barycentricinterpolant_owner(rhs)
{
}
barycentricinterpolant& barycentricinterpolant::operator=(const barycentricinterpolant &rhs)
{
if( this==&rhs )
return *this;
_barycentricinterpolant_owner::operator=(rhs);
return *this;
}
barycentricinterpolant::~barycentricinterpolant()
{
}
/*************************************************************************
Rational interpolation using barycentric formula
F(t) = SUM(i=0,n-1,w[i]*f[i]/(t-x[i])) / SUM(i=0,n-1,w[i]/(t-x[i]))
Input parameters:
B - barycentric interpolant built with one of model building
subroutines.
T - interpolation point
Result:
barycentric interpolant F(t)
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
double barycentriccalc(const barycentricinterpolant &b, const double t)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
double result = alglib_impl::barycentriccalc(const_cast<alglib_impl::barycentricinterpolant*>(b.c_ptr()), t, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return *(reinterpret_cast<double*>(&result));
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Differentiation of barycentric interpolant: first derivative.
Algorithm used in this subroutine is very robust and should not fail until
provided with values too close to MaxRealNumber (usually MaxRealNumber/N
or greater will overflow).
INPUT PARAMETERS:
B - barycentric interpolant built with one of model building
subroutines.
T - interpolation point
OUTPUT PARAMETERS:
F - barycentric interpolant at T
DF - first derivative
NOTE
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentricdiff1(const barycentricinterpolant &b, const double t, double &f, double &df)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::barycentricdiff1(const_cast<alglib_impl::barycentricinterpolant*>(b.c_ptr()), t, &f, &df, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Differentiation of barycentric interpolant: first/second derivatives.
INPUT PARAMETERS:
B - barycentric interpolant built with one of model building
subroutines.
T - interpolation point
OUTPUT PARAMETERS:
F - barycentric interpolant at T
DF - first derivative
D2F - second derivative
NOTE: this algorithm may fail due to overflow/underflor if used on data
whose values are close to MaxRealNumber or MinRealNumber. Use more robust
BarycentricDiff1() subroutine in such cases.
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentricdiff2(const barycentricinterpolant &b, const double t, double &f, double &df, double &d2f)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::barycentricdiff2(const_cast<alglib_impl::barycentricinterpolant*>(b.c_ptr()), t, &f, &df, &d2f, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine performs linear transformation of the argument.
INPUT PARAMETERS:
B - rational interpolant in barycentric form
CA, CB - transformation coefficients: x = CA*t + CB
OUTPUT PARAMETERS:
B - transformed interpolant with X replaced by T
-- ALGLIB PROJECT --
Copyright 19.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentriclintransx(const barycentricinterpolant &b, const double ca, const double cb)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::barycentriclintransx(const_cast<alglib_impl::barycentricinterpolant*>(b.c_ptr()), ca, cb, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine performs linear transformation of the barycentric
interpolant.
INPUT PARAMETERS:
B - rational interpolant in barycentric form
CA, CB - transformation coefficients: B2(x) = CA*B(x) + CB
OUTPUT PARAMETERS:
B - transformed interpolant
-- ALGLIB PROJECT --
Copyright 19.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentriclintransy(const barycentricinterpolant &b, const double ca, const double cb)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::barycentriclintransy(const_cast<alglib_impl::barycentricinterpolant*>(b.c_ptr()), ca, cb, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Extracts X/Y/W arrays from rational interpolant
INPUT PARAMETERS:
B - barycentric interpolant
OUTPUT PARAMETERS:
N - nodes count, N>0
X - interpolation nodes, array[0..N-1]
F - function values, array[0..N-1]
W - barycentric weights, array[0..N-1]
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentricunpack(const barycentricinterpolant &b, ae_int_t &n, real_1d_array &x, real_1d_array &y, real_1d_array &w)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::barycentricunpack(const_cast<alglib_impl::barycentricinterpolant*>(b.c_ptr()), &n, const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Rational interpolant from X/Y/W arrays
F(t) = SUM(i=0,n-1,w[i]*f[i]/(t-x[i])) / SUM(i=0,n-1,w[i]/(t-x[i]))
INPUT PARAMETERS:
X - interpolation nodes, array[0..N-1]
F - function values, array[0..N-1]
W - barycentric weights, array[0..N-1]
N - nodes count, N>0
OUTPUT PARAMETERS:
B - barycentric interpolant built from (X, Y, W)
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentricbuildxyw(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const ae_int_t n, barycentricinterpolant &b)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::barycentricbuildxyw(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), n, const_cast<alglib_impl::barycentricinterpolant*>(b.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Rational interpolant without poles
The subroutine constructs the rational interpolating function without real
poles (see 'Barycentric rational interpolation with no poles and high
rates of approximation', Michael S. Floater. and Kai Hormann, for more
information on this subject).
Input parameters:
X - interpolation nodes, array[0..N-1].
Y - function values, array[0..N-1].
N - number of nodes, N>0.
D - order of the interpolation scheme, 0 <= D <= N-1.
D<0 will cause an error.
D>=N it will be replaced with D=N-1.
if you don't know what D to choose, use small value about 3-5.
Output parameters:
B - barycentric interpolant.
Note:
this algorithm always succeeds and calculates the weights with close
to machine precision.
-- ALGLIB PROJECT --
Copyright 17.06.2007 by Bochkanov Sergey
*************************************************************************/
void barycentricbuildfloaterhormann(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t d, barycentricinterpolant &b)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::barycentricbuildfloaterhormann(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, d, const_cast<alglib_impl::barycentricinterpolant*>(b.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Conversion from barycentric representation to Chebyshev basis.
This function has O(N^2) complexity.
INPUT PARAMETERS:
P - polynomial in barycentric form
A,B - base interval for Chebyshev polynomials (see below)
A<>B
OUTPUT PARAMETERS
T - coefficients of Chebyshev representation;
P(x) = sum { T[i]*Ti(2*(x-A)/(B-A)-1), i=0..N-1 },
where Ti - I-th Chebyshev polynomial.
NOTES:
barycentric interpolant passed as P may be either polynomial obtained
from polynomial interpolation/ fitting or rational function which is
NOT polynomial. We can't distinguish between these two cases, and this
algorithm just tries to work assuming that P IS a polynomial. If not,
algorithm will return results, but they won't have any meaning.
-- ALGLIB --
Copyright 30.09.2010 by Bochkanov Sergey
*************************************************************************/
void polynomialbar2cheb(const barycentricinterpolant &p, const double a, const double b, real_1d_array &t)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::polynomialbar2cheb(const_cast<alglib_impl::barycentricinterpolant*>(p.c_ptr()), a, b, const_cast<alglib_impl::ae_vector*>(t.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Conversion from Chebyshev basis to barycentric representation.
This function has O(N^2) complexity.
INPUT PARAMETERS:
T - coefficients of Chebyshev representation;
P(x) = sum { T[i]*Ti(2*(x-A)/(B-A)-1), i=0..N },
where Ti - I-th Chebyshev polynomial.
N - number of coefficients:
* if given, only leading N elements of T are used
* if not given, automatically determined from size of T
A,B - base interval for Chebyshev polynomials (see above)
A<B
OUTPUT PARAMETERS
P - polynomial in barycentric form
-- ALGLIB --
Copyright 30.09.2010 by Bochkanov Sergey
*************************************************************************/
void polynomialcheb2bar(const real_1d_array &t, const ae_int_t n, const double a, const double b, barycentricinterpolant &p)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::polynomialcheb2bar(const_cast<alglib_impl::ae_vector*>(t.c_ptr()), n, a, b, const_cast<alglib_impl::barycentricinterpolant*>(p.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Conversion from Chebyshev basis to barycentric representation.
This function has O(N^2) complexity.
INPUT PARAMETERS:
T - coefficients of Chebyshev representation;
P(x) = sum { T[i]*Ti(2*(x-A)/(B-A)-1), i=0..N },
where Ti - I-th Chebyshev polynomial.
N - number of coefficients:
* if given, only leading N elements of T are used
* if not given, automatically determined from size of T
A,B - base interval for Chebyshev polynomials (see above)
A<B
OUTPUT PARAMETERS
P - polynomial in barycentric form
-- ALGLIB --
Copyright 30.09.2010 by Bochkanov Sergey
*************************************************************************/
void polynomialcheb2bar(const real_1d_array &t, const double a, const double b, barycentricinterpolant &p)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
n = t.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::polynomialcheb2bar(const_cast<alglib_impl::ae_vector*>(t.c_ptr()), n, a, b, const_cast<alglib_impl::barycentricinterpolant*>(p.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Conversion from barycentric representation to power basis.
This function has O(N^2) complexity.
INPUT PARAMETERS:
P - polynomial in barycentric form
C - offset (see below); 0.0 is used as default value.
S - scale (see below); 1.0 is used as default value. S<>0.
OUTPUT PARAMETERS
A - coefficients, P(x) = sum { A[i]*((X-C)/S)^i, i=0..N-1 }
N - number of coefficients (polynomial degree plus 1)
NOTES:
1. this function accepts offset and scale, which can be set to improve
numerical properties of polynomial. For example, if P was obtained as
result of interpolation on [-1,+1], you can set C=0 and S=1 and
represent P as sum of 1, x, x^2, x^3 and so on. In most cases you it
is exactly what you need.
However, if your interpolation model was built on [999,1001], you will
see significant growth of numerical errors when using {1, x, x^2, x^3}
as basis. Representing P as sum of 1, (x-1000), (x-1000)^2, (x-1000)^3
will be better option. Such representation can be obtained by using
1000.0 as offset C and 1.0 as scale S.
2. power basis is ill-conditioned and tricks described above can't solve
this problem completely. This function will return coefficients in
any case, but for N>8 they will become unreliable. However, N's
less than 5 are pretty safe.
3. barycentric interpolant passed as P may be either polynomial obtained
from polynomial interpolation/ fitting or rational function which is
NOT polynomial. We can't distinguish between these two cases, and this
algorithm just tries to work assuming that P IS a polynomial. If not,
algorithm will return results, but they won't have any meaning.
-- ALGLIB --
Copyright 30.09.2010 by Bochkanov Sergey
*************************************************************************/
void polynomialbar2pow(const barycentricinterpolant &p, const double c, const double s, real_1d_array &a)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::polynomialbar2pow(const_cast<alglib_impl::barycentricinterpolant*>(p.c_ptr()), c, s, const_cast<alglib_impl::ae_vector*>(a.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Conversion from barycentric representation to power basis.
This function has O(N^2) complexity.
INPUT PARAMETERS:
P - polynomial in barycentric form
C - offset (see below); 0.0 is used as default value.
S - scale (see below); 1.0 is used as default value. S<>0.
OUTPUT PARAMETERS
A - coefficients, P(x) = sum { A[i]*((X-C)/S)^i, i=0..N-1 }
N - number of coefficients (polynomial degree plus 1)
NOTES:
1. this function accepts offset and scale, which can be set to improve
numerical properties of polynomial. For example, if P was obtained as
result of interpolation on [-1,+1], you can set C=0 and S=1 and
represent P as sum of 1, x, x^2, x^3 and so on. In most cases you it
is exactly what you need.
However, if your interpolation model was built on [999,1001], you will
see significant growth of numerical errors when using {1, x, x^2, x^3}
as basis. Representing P as sum of 1, (x-1000), (x-1000)^2, (x-1000)^3
will be better option. Such representation can be obtained by using
1000.0 as offset C and 1.0 as scale S.
2. power basis is ill-conditioned and tricks described above can't solve
this problem completely. This function will return coefficients in
any case, but for N>8 they will become unreliable. However, N's
less than 5 are pretty safe.
3. barycentric interpolant passed as P may be either polynomial obtained
from polynomial interpolation/ fitting or rational function which is
NOT polynomial. We can't distinguish between these two cases, and this
algorithm just tries to work assuming that P IS a polynomial. If not,
algorithm will return results, but they won't have any meaning.
-- ALGLIB --
Copyright 30.09.2010 by Bochkanov Sergey
*************************************************************************/
void polynomialbar2pow(const barycentricinterpolant &p, real_1d_array &a)
{
alglib_impl::ae_state _alglib_env_state;
double c;
double s;
c = 0;
s = 1;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::polynomialbar2pow(const_cast<alglib_impl::barycentricinterpolant*>(p.c_ptr()), c, s, const_cast<alglib_impl::ae_vector*>(a.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Conversion from power basis to barycentric representation.
This function has O(N^2) complexity.
INPUT PARAMETERS:
A - coefficients, P(x) = sum { A[i]*((X-C)/S)^i, i=0..N-1 }
N - number of coefficients (polynomial degree plus 1)
* if given, only leading N elements of A are used
* if not given, automatically determined from size of A
C - offset (see below); 0.0 is used as default value.
S - scale (see below); 1.0 is used as default value. S<>0.
OUTPUT PARAMETERS
P - polynomial in barycentric form
NOTES:
1. this function accepts offset and scale, which can be set to improve
numerical properties of polynomial. For example, if you interpolate on
[-1,+1], you can set C=0 and S=1 and convert from sum of 1, x, x^2,
x^3 and so on. In most cases you it is exactly what you need.
However, if your interpolation model was built on [999,1001], you will
see significant growth of numerical errors when using {1, x, x^2, x^3}
as input basis. Converting from sum of 1, (x-1000), (x-1000)^2,
(x-1000)^3 will be better option (you have to specify 1000.0 as offset
C and 1.0 as scale S).
2. power basis is ill-conditioned and tricks described above can't solve
this problem completely. This function will return barycentric model
in any case, but for N>8 accuracy well degrade. However, N's less than
5 are pretty safe.
-- ALGLIB --
Copyright 30.09.2010 by Bochkanov Sergey
*************************************************************************/
void polynomialpow2bar(const real_1d_array &a, const ae_int_t n, const double c, const double s, barycentricinterpolant &p)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::polynomialpow2bar(const_cast<alglib_impl::ae_vector*>(a.c_ptr()), n, c, s, const_cast<alglib_impl::barycentricinterpolant*>(p.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Conversion from power basis to barycentric representation.
This function has O(N^2) complexity.
INPUT PARAMETERS:
A - coefficients, P(x) = sum { A[i]*((X-C)/S)^i, i=0..N-1 }
N - number of coefficients (polynomial degree plus 1)
* if given, only leading N elements of A are used
* if not given, automatically determined from size of A
C - offset (see below); 0.0 is used as default value.
S - scale (see below); 1.0 is used as default value. S<>0.
OUTPUT PARAMETERS
P - polynomial in barycentric form
NOTES:
1. this function accepts offset and scale, which can be set to improve
numerical properties of polynomial. For example, if you interpolate on
[-1,+1], you can set C=0 and S=1 and convert from sum of 1, x, x^2,
x^3 and so on. In most cases you it is exactly what you need.
However, if your interpolation model was built on [999,1001], you will
see significant growth of numerical errors when using {1, x, x^2, x^3}
as input basis. Converting from sum of 1, (x-1000), (x-1000)^2,
(x-1000)^3 will be better option (you have to specify 1000.0 as offset
C and 1.0 as scale S).
2. power basis is ill-conditioned and tricks described above can't solve
this problem completely. This function will return barycentric model
in any case, but for N>8 accuracy well degrade. However, N's less than
5 are pretty safe.
-- ALGLIB --
Copyright 30.09.2010 by Bochkanov Sergey
*************************************************************************/
void polynomialpow2bar(const real_1d_array &a, barycentricinterpolant &p)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
double c;
double s;
n = a.length();
c = 0;
s = 1;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::polynomialpow2bar(const_cast<alglib_impl::ae_vector*>(a.c_ptr()), n, c, s, const_cast<alglib_impl::barycentricinterpolant*>(p.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Lagrange intepolant: generation of the model on the general grid.
This function has O(N^2) complexity.
INPUT PARAMETERS:
X - abscissas, array[0..N-1]
Y - function values, array[0..N-1]
N - number of points, N>=1
OUTPUT PARAMETERS
P - barycentric model which represents Lagrange interpolant
(see ratint unit info and BarycentricCalc() description for
more information).
-- ALGLIB --
Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialbuild(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, barycentricinterpolant &p)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::polynomialbuild(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, const_cast<alglib_impl::barycentricinterpolant*>(p.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Lagrange intepolant: generation of the model on the general grid.
This function has O(N^2) complexity.
INPUT PARAMETERS:
X - abscissas, array[0..N-1]
Y - function values, array[0..N-1]
N - number of points, N>=1
OUTPUT PARAMETERS
P - barycentric model which represents Lagrange interpolant
(see ratint unit info and BarycentricCalc() description for
more information).
-- ALGLIB --
Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialbuild(const real_1d_array &x, const real_1d_array &y, barycentricinterpolant &p)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
if( (x.length()!=y.length()))
throw ap_error("Error while calling 'polynomialbuild': looks like one of arguments has wrong size");
n = x.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::polynomialbuild(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, const_cast<alglib_impl::barycentricinterpolant*>(p.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Lagrange intepolant: generation of the model on equidistant grid.
This function has O(N) complexity.
INPUT PARAMETERS:
A - left boundary of [A,B]
B - right boundary of [A,B]
Y - function values at the nodes, array[0..N-1]
N - number of points, N>=1
for N=1 a constant model is constructed.
OUTPUT PARAMETERS
P - barycentric model which represents Lagrange interpolant
(see ratint unit info and BarycentricCalc() description for
more information).
-- ALGLIB --
Copyright 03.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialbuildeqdist(const double a, const double b, const real_1d_array &y, const ae_int_t n, barycentricinterpolant &p)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::polynomialbuildeqdist(a, b, const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, const_cast<alglib_impl::barycentricinterpolant*>(p.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Lagrange intepolant: generation of the model on equidistant grid.
This function has O(N) complexity.
INPUT PARAMETERS:
A - left boundary of [A,B]
B - right boundary of [A,B]
Y - function values at the nodes, array[0..N-1]
N - number of points, N>=1
for N=1 a constant model is constructed.
OUTPUT PARAMETERS
P - barycentric model which represents Lagrange interpolant
(see ratint unit info and BarycentricCalc() description for
more information).
-- ALGLIB --
Copyright 03.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialbuildeqdist(const double a, const double b, const real_1d_array &y, barycentricinterpolant &p)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
n = y.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::polynomialbuildeqdist(a, b, const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, const_cast<alglib_impl::barycentricinterpolant*>(p.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Lagrange intepolant on Chebyshev grid (first kind).
This function has O(N) complexity.
INPUT PARAMETERS:
A - left boundary of [A,B]
B - right boundary of [A,B]
Y - function values at the nodes, array[0..N-1],
Y[I] = Y(0.5*(B+A) + 0.5*(B-A)*Cos(PI*(2*i+1)/(2*n)))
N - number of points, N>=1
for N=1 a constant model is constructed.
OUTPUT PARAMETERS
P - barycentric model which represents Lagrange interpolant
(see ratint unit info and BarycentricCalc() description for
more information).
-- ALGLIB --
Copyright 03.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialbuildcheb1(const double a, const double b, const real_1d_array &y, const ae_int_t n, barycentricinterpolant &p)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::polynomialbuildcheb1(a, b, const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, const_cast<alglib_impl::barycentricinterpolant*>(p.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Lagrange intepolant on Chebyshev grid (first kind).
This function has O(N) complexity.
INPUT PARAMETERS:
A - left boundary of [A,B]
B - right boundary of [A,B]
Y - function values at the nodes, array[0..N-1],
Y[I] = Y(0.5*(B+A) + 0.5*(B-A)*Cos(PI*(2*i+1)/(2*n)))
N - number of points, N>=1
for N=1 a constant model is constructed.
OUTPUT PARAMETERS
P - barycentric model which represents Lagrange interpolant
(see ratint unit info and BarycentricCalc() description for
more information).
-- ALGLIB --
Copyright 03.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialbuildcheb1(const double a, const double b, const real_1d_array &y, barycentricinterpolant &p)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
n = y.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::polynomialbuildcheb1(a, b, const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, const_cast<alglib_impl::barycentricinterpolant*>(p.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Lagrange intepolant on Chebyshev grid (second kind).
This function has O(N) complexity.
INPUT PARAMETERS:
A - left boundary of [A,B]
B - right boundary of [A,B]
Y - function values at the nodes, array[0..N-1],
Y[I] = Y(0.5*(B+A) + 0.5*(B-A)*Cos(PI*i/(n-1)))
N - number of points, N>=1
for N=1 a constant model is constructed.
OUTPUT PARAMETERS
P - barycentric model which represents Lagrange interpolant
(see ratint unit info and BarycentricCalc() description for
more information).
-- ALGLIB --
Copyright 03.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialbuildcheb2(const double a, const double b, const real_1d_array &y, const ae_int_t n, barycentricinterpolant &p)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::polynomialbuildcheb2(a, b, const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, const_cast<alglib_impl::barycentricinterpolant*>(p.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Lagrange intepolant on Chebyshev grid (second kind).
This function has O(N) complexity.
INPUT PARAMETERS:
A - left boundary of [A,B]
B - right boundary of [A,B]
Y - function values at the nodes, array[0..N-1],
Y[I] = Y(0.5*(B+A) + 0.5*(B-A)*Cos(PI*i/(n-1)))
N - number of points, N>=1
for N=1 a constant model is constructed.
OUTPUT PARAMETERS
P - barycentric model which represents Lagrange interpolant
(see ratint unit info and BarycentricCalc() description for
more information).
-- ALGLIB --
Copyright 03.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialbuildcheb2(const double a, const double b, const real_1d_array &y, barycentricinterpolant &p)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
n = y.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::polynomialbuildcheb2(a, b, const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, const_cast<alglib_impl::barycentricinterpolant*>(p.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Fast equidistant polynomial interpolation function with O(N) complexity
INPUT PARAMETERS:
A - left boundary of [A,B]
B - right boundary of [A,B]
F - function values, array[0..N-1]
N - number of points on equidistant grid, N>=1
for N=1 a constant model is constructed.
T - position where P(x) is calculated
RESULT
value of the Lagrange interpolant at T
IMPORTANT
this function provides fast interface which is not overflow-safe
nor it is very precise.
the best option is to use PolynomialBuildEqDist()/BarycentricCalc()
subroutines unless you are pretty sure that your data will not result
in overflow.
-- ALGLIB --
Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
double polynomialcalceqdist(const double a, const double b, const real_1d_array &f, const ae_int_t n, const double t)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
double result = alglib_impl::polynomialcalceqdist(a, b, const_cast<alglib_impl::ae_vector*>(f.c_ptr()), n, t, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return *(reinterpret_cast<double*>(&result));
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Fast equidistant polynomial interpolation function with O(N) complexity
INPUT PARAMETERS:
A - left boundary of [A,B]
B - right boundary of [A,B]
F - function values, array[0..N-1]
N - number of points on equidistant grid, N>=1
for N=1 a constant model is constructed.
T - position where P(x) is calculated
RESULT
value of the Lagrange interpolant at T
IMPORTANT
this function provides fast interface which is not overflow-safe
nor it is very precise.
the best option is to use PolynomialBuildEqDist()/BarycentricCalc()
subroutines unless you are pretty sure that your data will not result
in overflow.
-- ALGLIB --
Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
double polynomialcalceqdist(const double a, const double b, const real_1d_array &f, const double t)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
n = f.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
double result = alglib_impl::polynomialcalceqdist(a, b, const_cast<alglib_impl::ae_vector*>(f.c_ptr()), n, t, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return *(reinterpret_cast<double*>(&result));
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Fast polynomial interpolation function on Chebyshev points (first kind)
with O(N) complexity.
INPUT PARAMETERS:
A - left boundary of [A,B]
B - right boundary of [A,B]
F - function values, array[0..N-1]
N - number of points on Chebyshev grid (first kind),
X[i] = 0.5*(B+A) + 0.5*(B-A)*Cos(PI*(2*i+1)/(2*n))
for N=1 a constant model is constructed.
T - position where P(x) is calculated
RESULT
value of the Lagrange interpolant at T
IMPORTANT
this function provides fast interface which is not overflow-safe
nor it is very precise.
the best option is to use PolIntBuildCheb1()/BarycentricCalc()
subroutines unless you are pretty sure that your data will not result
in overflow.
-- ALGLIB --
Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
double polynomialcalccheb1(const double a, const double b, const real_1d_array &f, const ae_int_t n, const double t)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
double result = alglib_impl::polynomialcalccheb1(a, b, const_cast<alglib_impl::ae_vector*>(f.c_ptr()), n, t, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return *(reinterpret_cast<double*>(&result));
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Fast polynomial interpolation function on Chebyshev points (first kind)
with O(N) complexity.
INPUT PARAMETERS:
A - left boundary of [A,B]
B - right boundary of [A,B]
F - function values, array[0..N-1]
N - number of points on Chebyshev grid (first kind),
X[i] = 0.5*(B+A) + 0.5*(B-A)*Cos(PI*(2*i+1)/(2*n))
for N=1 a constant model is constructed.
T - position where P(x) is calculated
RESULT
value of the Lagrange interpolant at T
IMPORTANT
this function provides fast interface which is not overflow-safe
nor it is very precise.
the best option is to use PolIntBuildCheb1()/BarycentricCalc()
subroutines unless you are pretty sure that your data will not result
in overflow.
-- ALGLIB --
Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
double polynomialcalccheb1(const double a, const double b, const real_1d_array &f, const double t)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
n = f.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
double result = alglib_impl::polynomialcalccheb1(a, b, const_cast<alglib_impl::ae_vector*>(f.c_ptr()), n, t, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return *(reinterpret_cast<double*>(&result));
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Fast polynomial interpolation function on Chebyshev points (second kind)
with O(N) complexity.
INPUT PARAMETERS:
A - left boundary of [A,B]
B - right boundary of [A,B]
F - function values, array[0..N-1]
N - number of points on Chebyshev grid (second kind),
X[i] = 0.5*(B+A) + 0.5*(B-A)*Cos(PI*i/(n-1))
for N=1 a constant model is constructed.
T - position where P(x) is calculated
RESULT
value of the Lagrange interpolant at T
IMPORTANT
this function provides fast interface which is not overflow-safe
nor it is very precise.
the best option is to use PolIntBuildCheb2()/BarycentricCalc()
subroutines unless you are pretty sure that your data will not result
in overflow.
-- ALGLIB --
Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
double polynomialcalccheb2(const double a, const double b, const real_1d_array &f, const ae_int_t n, const double t)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
double result = alglib_impl::polynomialcalccheb2(a, b, const_cast<alglib_impl::ae_vector*>(f.c_ptr()), n, t, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return *(reinterpret_cast<double*>(&result));
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Fast polynomial interpolation function on Chebyshev points (second kind)
with O(N) complexity.
INPUT PARAMETERS:
A - left boundary of [A,B]
B - right boundary of [A,B]
F - function values, array[0..N-1]
N - number of points on Chebyshev grid (second kind),
X[i] = 0.5*(B+A) + 0.5*(B-A)*Cos(PI*i/(n-1))
for N=1 a constant model is constructed.
T - position where P(x) is calculated
RESULT
value of the Lagrange interpolant at T
IMPORTANT
this function provides fast interface which is not overflow-safe
nor it is very precise.
the best option is to use PolIntBuildCheb2()/BarycentricCalc()
subroutines unless you are pretty sure that your data will not result
in overflow.
-- ALGLIB --
Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
double polynomialcalccheb2(const double a, const double b, const real_1d_array &f, const double t)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
n = f.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
double result = alglib_impl::polynomialcalccheb2(a, b, const_cast<alglib_impl::ae_vector*>(f.c_ptr()), n, t, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return *(reinterpret_cast<double*>(&result));
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
1-dimensional spline interpolant
*************************************************************************/
_spline1dinterpolant_owner::_spline1dinterpolant_owner()
{
p_struct = (alglib_impl::spline1dinterpolant*)alglib_impl::ae_malloc(sizeof(alglib_impl::spline1dinterpolant), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_spline1dinterpolant_init(p_struct, NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_spline1dinterpolant_owner::_spline1dinterpolant_owner(const _spline1dinterpolant_owner &rhs)
{
p_struct = (alglib_impl::spline1dinterpolant*)alglib_impl::ae_malloc(sizeof(alglib_impl::spline1dinterpolant), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_spline1dinterpolant_init_copy(p_struct, const_cast<alglib_impl::spline1dinterpolant*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_spline1dinterpolant_owner& _spline1dinterpolant_owner::operator=(const _spline1dinterpolant_owner &rhs)
{
if( this==&rhs )
return *this;
alglib_impl::_spline1dinterpolant_clear(p_struct);
if( !alglib_impl::_spline1dinterpolant_init_copy(p_struct, const_cast<alglib_impl::spline1dinterpolant*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
return *this;
}
_spline1dinterpolant_owner::~_spline1dinterpolant_owner()
{
alglib_impl::_spline1dinterpolant_clear(p_struct);
ae_free(p_struct);
}
alglib_impl::spline1dinterpolant* _spline1dinterpolant_owner::c_ptr()
{
return p_struct;
}
alglib_impl::spline1dinterpolant* _spline1dinterpolant_owner::c_ptr() const
{
return const_cast<alglib_impl::spline1dinterpolant*>(p_struct);
}
spline1dinterpolant::spline1dinterpolant() : _spline1dinterpolant_owner()
{
}
spline1dinterpolant::spline1dinterpolant(const spline1dinterpolant &rhs):_spline1dinterpolant_owner(rhs)
{
}
spline1dinterpolant& spline1dinterpolant::operator=(const spline1dinterpolant &rhs)
{
if( this==&rhs )
return *this;
_spline1dinterpolant_owner::operator=(rhs);
return *this;
}
spline1dinterpolant::~spline1dinterpolant()
{
}
/*************************************************************************
This subroutine builds linear spline interpolant
INPUT PARAMETERS:
X - spline nodes, array[0..N-1]
Y - function values, array[0..N-1]
N - points count (optional):
* N>=2
* if given, only first N points are used to build spline
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
OUTPUT PARAMETERS:
C - spline interpolant
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
-- ALGLIB PROJECT --
Copyright 24.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildlinear(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, spline1dinterpolant &c)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dbuildlinear(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, const_cast<alglib_impl::spline1dinterpolant*>(c.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine builds linear spline interpolant
INPUT PARAMETERS:
X - spline nodes, array[0..N-1]
Y - function values, array[0..N-1]
N - points count (optional):
* N>=2
* if given, only first N points are used to build spline
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
OUTPUT PARAMETERS:
C - spline interpolant
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
-- ALGLIB PROJECT --
Copyright 24.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildlinear(const real_1d_array &x, const real_1d_array &y, spline1dinterpolant &c)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
if( (x.length()!=y.length()))
throw ap_error("Error while calling 'spline1dbuildlinear': looks like one of arguments has wrong size");
n = x.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dbuildlinear(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, const_cast<alglib_impl::spline1dinterpolant*>(c.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine builds cubic spline interpolant.
INPUT PARAMETERS:
X - spline nodes, array[0..N-1].
Y - function values, array[0..N-1].
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points are used to build spline
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundLType - boundary condition type for the left boundary
BoundL - left boundary condition (first or second derivative,
depending on the BoundLType)
BoundRType - boundary condition type for the right boundary
BoundR - right boundary condition (first or second derivative,
depending on the BoundRType)
OUTPUT PARAMETERS:
C - spline interpolant
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
SETTING BOUNDARY VALUES:
The BoundLType/BoundRType parameters can have the following values:
* -1, which corresonds to the periodic (cyclic) boundary conditions.
In this case:
* both BoundLType and BoundRType must be equal to -1.
* BoundL/BoundR are ignored
* Y[last] is ignored (it is assumed to be equal to Y[first]).
* 0, which corresponds to the parabolically terminated spline
(BoundL and/or BoundR are ignored).
* 1, which corresponds to the first derivative boundary condition
* 2, which corresponds to the second derivative boundary condition
* by default, BoundType=0 is used
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildcubic(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t boundltype, const double boundl, const ae_int_t boundrtype, const double boundr, spline1dinterpolant &c)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dbuildcubic(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, boundltype, boundl, boundrtype, boundr, const_cast<alglib_impl::spline1dinterpolant*>(c.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine builds cubic spline interpolant.
INPUT PARAMETERS:
X - spline nodes, array[0..N-1].
Y - function values, array[0..N-1].
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points are used to build spline
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundLType - boundary condition type for the left boundary
BoundL - left boundary condition (first or second derivative,
depending on the BoundLType)
BoundRType - boundary condition type for the right boundary
BoundR - right boundary condition (first or second derivative,
depending on the BoundRType)
OUTPUT PARAMETERS:
C - spline interpolant
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
SETTING BOUNDARY VALUES:
The BoundLType/BoundRType parameters can have the following values:
* -1, which corresonds to the periodic (cyclic) boundary conditions.
In this case:
* both BoundLType and BoundRType must be equal to -1.
* BoundL/BoundR are ignored
* Y[last] is ignored (it is assumed to be equal to Y[first]).
* 0, which corresponds to the parabolically terminated spline
(BoundL and/or BoundR are ignored).
* 1, which corresponds to the first derivative boundary condition
* 2, which corresponds to the second derivative boundary condition
* by default, BoundType=0 is used
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildcubic(const real_1d_array &x, const real_1d_array &y, spline1dinterpolant &c)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t boundltype;
double boundl;
ae_int_t boundrtype;
double boundr;
if( (x.length()!=y.length()))
throw ap_error("Error while calling 'spline1dbuildcubic': looks like one of arguments has wrong size");
n = x.length();
boundltype = 0;
boundl = 0;
boundrtype = 0;
boundr = 0;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dbuildcubic(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, boundltype, boundl, boundrtype, boundr, const_cast<alglib_impl::spline1dinterpolant*>(c.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function solves following problem: given table y[] of function values
at nodes x[], it calculates and returns table of function derivatives d[]
(calculated at the same nodes x[]).
This function yields same result as Spline1DBuildCubic() call followed by
sequence of Spline1DDiff() calls, but it can be several times faster when
called for ordered X[] and X2[].
INPUT PARAMETERS:
X - spline nodes
Y - function values
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points are used
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundLType - boundary condition type for the left boundary
BoundL - left boundary condition (first or second derivative,
depending on the BoundLType)
BoundRType - boundary condition type for the right boundary
BoundR - right boundary condition (first or second derivative,
depending on the BoundRType)
OUTPUT PARAMETERS:
D - derivative values at X[]
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
Derivative values are correctly reordered on return, so D[I] is always
equal to S'(X[I]) independently of points order.
SETTING BOUNDARY VALUES:
The BoundLType/BoundRType parameters can have the following values:
* -1, which corresonds to the periodic (cyclic) boundary conditions.
In this case:
* both BoundLType and BoundRType must be equal to -1.
* BoundL/BoundR are ignored
* Y[last] is ignored (it is assumed to be equal to Y[first]).
* 0, which corresponds to the parabolically terminated spline
(BoundL and/or BoundR are ignored).
* 1, which corresponds to the first derivative boundary condition
* 2, which corresponds to the second derivative boundary condition
* by default, BoundType=0 is used
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dgriddiffcubic(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t boundltype, const double boundl, const ae_int_t boundrtype, const double boundr, real_1d_array &d)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dgriddiffcubic(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, boundltype, boundl, boundrtype, boundr, const_cast<alglib_impl::ae_vector*>(d.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function solves following problem: given table y[] of function values
at nodes x[], it calculates and returns table of function derivatives d[]
(calculated at the same nodes x[]).
This function yields same result as Spline1DBuildCubic() call followed by
sequence of Spline1DDiff() calls, but it can be several times faster when
called for ordered X[] and X2[].
INPUT PARAMETERS:
X - spline nodes
Y - function values
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points are used
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundLType - boundary condition type for the left boundary
BoundL - left boundary condition (first or second derivative,
depending on the BoundLType)
BoundRType - boundary condition type for the right boundary
BoundR - right boundary condition (first or second derivative,
depending on the BoundRType)
OUTPUT PARAMETERS:
D - derivative values at X[]
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
Derivative values are correctly reordered on return, so D[I] is always
equal to S'(X[I]) independently of points order.
SETTING BOUNDARY VALUES:
The BoundLType/BoundRType parameters can have the following values:
* -1, which corresonds to the periodic (cyclic) boundary conditions.
In this case:
* both BoundLType and BoundRType must be equal to -1.
* BoundL/BoundR are ignored
* Y[last] is ignored (it is assumed to be equal to Y[first]).
* 0, which corresponds to the parabolically terminated spline
(BoundL and/or BoundR are ignored).
* 1, which corresponds to the first derivative boundary condition
* 2, which corresponds to the second derivative boundary condition
* by default, BoundType=0 is used
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dgriddiffcubic(const real_1d_array &x, const real_1d_array &y, real_1d_array &d)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t boundltype;
double boundl;
ae_int_t boundrtype;
double boundr;
if( (x.length()!=y.length()))
throw ap_error("Error while calling 'spline1dgriddiffcubic': looks like one of arguments has wrong size");
n = x.length();
boundltype = 0;
boundl = 0;
boundrtype = 0;
boundr = 0;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dgriddiffcubic(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, boundltype, boundl, boundrtype, boundr, const_cast<alglib_impl::ae_vector*>(d.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function solves following problem: given table y[] of function values
at nodes x[], it calculates and returns tables of first and second
function derivatives d1[] and d2[] (calculated at the same nodes x[]).
This function yields same result as Spline1DBuildCubic() call followed by
sequence of Spline1DDiff() calls, but it can be several times faster when
called for ordered X[] and X2[].
INPUT PARAMETERS:
X - spline nodes
Y - function values
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points are used
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundLType - boundary condition type for the left boundary
BoundL - left boundary condition (first or second derivative,
depending on the BoundLType)
BoundRType - boundary condition type for the right boundary
BoundR - right boundary condition (first or second derivative,
depending on the BoundRType)
OUTPUT PARAMETERS:
D1 - S' values at X[]
D2 - S'' values at X[]
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
Derivative values are correctly reordered on return, so D[I] is always
equal to S'(X[I]) independently of points order.
SETTING BOUNDARY VALUES:
The BoundLType/BoundRType parameters can have the following values:
* -1, which corresonds to the periodic (cyclic) boundary conditions.
In this case:
* both BoundLType and BoundRType must be equal to -1.
* BoundL/BoundR are ignored
* Y[last] is ignored (it is assumed to be equal to Y[first]).
* 0, which corresponds to the parabolically terminated spline
(BoundL and/or BoundR are ignored).
* 1, which corresponds to the first derivative boundary condition
* 2, which corresponds to the second derivative boundary condition
* by default, BoundType=0 is used
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dgriddiff2cubic(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t boundltype, const double boundl, const ae_int_t boundrtype, const double boundr, real_1d_array &d1, real_1d_array &d2)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dgriddiff2cubic(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, boundltype, boundl, boundrtype, boundr, const_cast<alglib_impl::ae_vector*>(d1.c_ptr()), const_cast<alglib_impl::ae_vector*>(d2.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function solves following problem: given table y[] of function values
at nodes x[], it calculates and returns tables of first and second
function derivatives d1[] and d2[] (calculated at the same nodes x[]).
This function yields same result as Spline1DBuildCubic() call followed by
sequence of Spline1DDiff() calls, but it can be several times faster when
called for ordered X[] and X2[].
INPUT PARAMETERS:
X - spline nodes
Y - function values
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points are used
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundLType - boundary condition type for the left boundary
BoundL - left boundary condition (first or second derivative,
depending on the BoundLType)
BoundRType - boundary condition type for the right boundary
BoundR - right boundary condition (first or second derivative,
depending on the BoundRType)
OUTPUT PARAMETERS:
D1 - S' values at X[]
D2 - S'' values at X[]
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
Derivative values are correctly reordered on return, so D[I] is always
equal to S'(X[I]) independently of points order.
SETTING BOUNDARY VALUES:
The BoundLType/BoundRType parameters can have the following values:
* -1, which corresonds to the periodic (cyclic) boundary conditions.
In this case:
* both BoundLType and BoundRType must be equal to -1.
* BoundL/BoundR are ignored
* Y[last] is ignored (it is assumed to be equal to Y[first]).
* 0, which corresponds to the parabolically terminated spline
(BoundL and/or BoundR are ignored).
* 1, which corresponds to the first derivative boundary condition
* 2, which corresponds to the second derivative boundary condition
* by default, BoundType=0 is used
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dgriddiff2cubic(const real_1d_array &x, const real_1d_array &y, real_1d_array &d1, real_1d_array &d2)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t boundltype;
double boundl;
ae_int_t boundrtype;
double boundr;
if( (x.length()!=y.length()))
throw ap_error("Error while calling 'spline1dgriddiff2cubic': looks like one of arguments has wrong size");
n = x.length();
boundltype = 0;
boundl = 0;
boundrtype = 0;
boundr = 0;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dgriddiff2cubic(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, boundltype, boundl, boundrtype, boundr, const_cast<alglib_impl::ae_vector*>(d1.c_ptr()), const_cast<alglib_impl::ae_vector*>(d2.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function solves following problem: given table y[] of function values
at old nodes x[] and new nodes x2[], it calculates and returns table of
function values y2[] (calculated at x2[]).
This function yields same result as Spline1DBuildCubic() call followed by
sequence of Spline1DDiff() calls, but it can be several times faster when
called for ordered X[] and X2[].
INPUT PARAMETERS:
X - old spline nodes
Y - function values
X2 - new spline nodes
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points from X/Y are used
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundLType - boundary condition type for the left boundary
BoundL - left boundary condition (first or second derivative,
depending on the BoundLType)
BoundRType - boundary condition type for the right boundary
BoundR - right boundary condition (first or second derivative,
depending on the BoundRType)
N2 - new points count:
* N2>=2
* if given, only first N2 points from X2 are used
* if not given, automatically detected from X2 size
OUTPUT PARAMETERS:
F2 - function values at X2[]
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
Function values are correctly reordered on return, so F2[I] is always
equal to S(X2[I]) independently of points order.
SETTING BOUNDARY VALUES:
The BoundLType/BoundRType parameters can have the following values:
* -1, which corresonds to the periodic (cyclic) boundary conditions.
In this case:
* both BoundLType and BoundRType must be equal to -1.
* BoundL/BoundR are ignored
* Y[last] is ignored (it is assumed to be equal to Y[first]).
* 0, which corresponds to the parabolically terminated spline
(BoundL and/or BoundR are ignored).
* 1, which corresponds to the first derivative boundary condition
* 2, which corresponds to the second derivative boundary condition
* by default, BoundType=0 is used
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dconvcubic(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t boundltype, const double boundl, const ae_int_t boundrtype, const double boundr, const real_1d_array &x2, const ae_int_t n2, real_1d_array &y2)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dconvcubic(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, boundltype, boundl, boundrtype, boundr, const_cast<alglib_impl::ae_vector*>(x2.c_ptr()), n2, const_cast<alglib_impl::ae_vector*>(y2.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function solves following problem: given table y[] of function values
at old nodes x[] and new nodes x2[], it calculates and returns table of
function values y2[] (calculated at x2[]).
This function yields same result as Spline1DBuildCubic() call followed by
sequence of Spline1DDiff() calls, but it can be several times faster when
called for ordered X[] and X2[].
INPUT PARAMETERS:
X - old spline nodes
Y - function values
X2 - new spline nodes
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points from X/Y are used
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundLType - boundary condition type for the left boundary
BoundL - left boundary condition (first or second derivative,
depending on the BoundLType)
BoundRType - boundary condition type for the right boundary
BoundR - right boundary condition (first or second derivative,
depending on the BoundRType)
N2 - new points count:
* N2>=2
* if given, only first N2 points from X2 are used
* if not given, automatically detected from X2 size
OUTPUT PARAMETERS:
F2 - function values at X2[]
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
Function values are correctly reordered on return, so F2[I] is always
equal to S(X2[I]) independently of points order.
SETTING BOUNDARY VALUES:
The BoundLType/BoundRType parameters can have the following values:
* -1, which corresonds to the periodic (cyclic) boundary conditions.
In this case:
* both BoundLType and BoundRType must be equal to -1.
* BoundL/BoundR are ignored
* Y[last] is ignored (it is assumed to be equal to Y[first]).
* 0, which corresponds to the parabolically terminated spline
(BoundL and/or BoundR are ignored).
* 1, which corresponds to the first derivative boundary condition
* 2, which corresponds to the second derivative boundary condition
* by default, BoundType=0 is used
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dconvcubic(const real_1d_array &x, const real_1d_array &y, const real_1d_array &x2, real_1d_array &y2)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t boundltype;
double boundl;
ae_int_t boundrtype;
double boundr;
ae_int_t n2;
if( (x.length()!=y.length()))
throw ap_error("Error while calling 'spline1dconvcubic': looks like one of arguments has wrong size");
n = x.length();
boundltype = 0;
boundl = 0;
boundrtype = 0;
boundr = 0;
n2 = x2.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dconvcubic(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, boundltype, boundl, boundrtype, boundr, const_cast<alglib_impl::ae_vector*>(x2.c_ptr()), n2, const_cast<alglib_impl::ae_vector*>(y2.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function solves following problem: given table y[] of function values
at old nodes x[] and new nodes x2[], it calculates and returns table of
function values y2[] and derivatives d2[] (calculated at x2[]).
This function yields same result as Spline1DBuildCubic() call followed by
sequence of Spline1DDiff() calls, but it can be several times faster when
called for ordered X[] and X2[].
INPUT PARAMETERS:
X - old spline nodes
Y - function values
X2 - new spline nodes
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points from X/Y are used
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundLType - boundary condition type for the left boundary
BoundL - left boundary condition (first or second derivative,
depending on the BoundLType)
BoundRType - boundary condition type for the right boundary
BoundR - right boundary condition (first or second derivative,
depending on the BoundRType)
N2 - new points count:
* N2>=2
* if given, only first N2 points from X2 are used
* if not given, automatically detected from X2 size
OUTPUT PARAMETERS:
F2 - function values at X2[]
D2 - first derivatives at X2[]
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
Function values are correctly reordered on return, so F2[I] is always
equal to S(X2[I]) independently of points order.
SETTING BOUNDARY VALUES:
The BoundLType/BoundRType parameters can have the following values:
* -1, which corresonds to the periodic (cyclic) boundary conditions.
In this case:
* both BoundLType and BoundRType must be equal to -1.
* BoundL/BoundR are ignored
* Y[last] is ignored (it is assumed to be equal to Y[first]).
* 0, which corresponds to the parabolically terminated spline
(BoundL and/or BoundR are ignored).
* 1, which corresponds to the first derivative boundary condition
* 2, which corresponds to the second derivative boundary condition
* by default, BoundType=0 is used
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dconvdiffcubic(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t boundltype, const double boundl, const ae_int_t boundrtype, const double boundr, const real_1d_array &x2, const ae_int_t n2, real_1d_array &y2, real_1d_array &d2)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dconvdiffcubic(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, boundltype, boundl, boundrtype, boundr, const_cast<alglib_impl::ae_vector*>(x2.c_ptr()), n2, const_cast<alglib_impl::ae_vector*>(y2.c_ptr()), const_cast<alglib_impl::ae_vector*>(d2.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function solves following problem: given table y[] of function values
at old nodes x[] and new nodes x2[], it calculates and returns table of
function values y2[] and derivatives d2[] (calculated at x2[]).
This function yields same result as Spline1DBuildCubic() call followed by
sequence of Spline1DDiff() calls, but it can be several times faster when
called for ordered X[] and X2[].
INPUT PARAMETERS:
X - old spline nodes
Y - function values
X2 - new spline nodes
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points from X/Y are used
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundLType - boundary condition type for the left boundary
BoundL - left boundary condition (first or second derivative,
depending on the BoundLType)
BoundRType - boundary condition type for the right boundary
BoundR - right boundary condition (first or second derivative,
depending on the BoundRType)
N2 - new points count:
* N2>=2
* if given, only first N2 points from X2 are used
* if not given, automatically detected from X2 size
OUTPUT PARAMETERS:
F2 - function values at X2[]
D2 - first derivatives at X2[]
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
Function values are correctly reordered on return, so F2[I] is always
equal to S(X2[I]) independently of points order.
SETTING BOUNDARY VALUES:
The BoundLType/BoundRType parameters can have the following values:
* -1, which corresonds to the periodic (cyclic) boundary conditions.
In this case:
* both BoundLType and BoundRType must be equal to -1.
* BoundL/BoundR are ignored
* Y[last] is ignored (it is assumed to be equal to Y[first]).
* 0, which corresponds to the parabolically terminated spline
(BoundL and/or BoundR are ignored).
* 1, which corresponds to the first derivative boundary condition
* 2, which corresponds to the second derivative boundary condition
* by default, BoundType=0 is used
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dconvdiffcubic(const real_1d_array &x, const real_1d_array &y, const real_1d_array &x2, real_1d_array &y2, real_1d_array &d2)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t boundltype;
double boundl;
ae_int_t boundrtype;
double boundr;
ae_int_t n2;
if( (x.length()!=y.length()))
throw ap_error("Error while calling 'spline1dconvdiffcubic': looks like one of arguments has wrong size");
n = x.length();
boundltype = 0;
boundl = 0;
boundrtype = 0;
boundr = 0;
n2 = x2.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dconvdiffcubic(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, boundltype, boundl, boundrtype, boundr, const_cast<alglib_impl::ae_vector*>(x2.c_ptr()), n2, const_cast<alglib_impl::ae_vector*>(y2.c_ptr()), const_cast<alglib_impl::ae_vector*>(d2.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function solves following problem: given table y[] of function values
at old nodes x[] and new nodes x2[], it calculates and returns table of
function values y2[], first and second derivatives d2[] and dd2[]
(calculated at x2[]).
This function yields same result as Spline1DBuildCubic() call followed by
sequence of Spline1DDiff() calls, but it can be several times faster when
called for ordered X[] and X2[].
INPUT PARAMETERS:
X - old spline nodes
Y - function values
X2 - new spline nodes
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points from X/Y are used
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundLType - boundary condition type for the left boundary
BoundL - left boundary condition (first or second derivative,
depending on the BoundLType)
BoundRType - boundary condition type for the right boundary
BoundR - right boundary condition (first or second derivative,
depending on the BoundRType)
N2 - new points count:
* N2>=2
* if given, only first N2 points from X2 are used
* if not given, automatically detected from X2 size
OUTPUT PARAMETERS:
F2 - function values at X2[]
D2 - first derivatives at X2[]
DD2 - second derivatives at X2[]
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
Function values are correctly reordered on return, so F2[I] is always
equal to S(X2[I]) independently of points order.
SETTING BOUNDARY VALUES:
The BoundLType/BoundRType parameters can have the following values:
* -1, which corresonds to the periodic (cyclic) boundary conditions.
In this case:
* both BoundLType and BoundRType must be equal to -1.
* BoundL/BoundR are ignored
* Y[last] is ignored (it is assumed to be equal to Y[first]).
* 0, which corresponds to the parabolically terminated spline
(BoundL and/or BoundR are ignored).
* 1, which corresponds to the first derivative boundary condition
* 2, which corresponds to the second derivative boundary condition
* by default, BoundType=0 is used
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dconvdiff2cubic(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t boundltype, const double boundl, const ae_int_t boundrtype, const double boundr, const real_1d_array &x2, const ae_int_t n2, real_1d_array &y2, real_1d_array &d2, real_1d_array &dd2)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dconvdiff2cubic(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, boundltype, boundl, boundrtype, boundr, const_cast<alglib_impl::ae_vector*>(x2.c_ptr()), n2, const_cast<alglib_impl::ae_vector*>(y2.c_ptr()), const_cast<alglib_impl::ae_vector*>(d2.c_ptr()), const_cast<alglib_impl::ae_vector*>(dd2.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function solves following problem: given table y[] of function values
at old nodes x[] and new nodes x2[], it calculates and returns table of
function values y2[], first and second derivatives d2[] and dd2[]
(calculated at x2[]).
This function yields same result as Spline1DBuildCubic() call followed by
sequence of Spline1DDiff() calls, but it can be several times faster when
called for ordered X[] and X2[].
INPUT PARAMETERS:
X - old spline nodes
Y - function values
X2 - new spline nodes
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points from X/Y are used
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundLType - boundary condition type for the left boundary
BoundL - left boundary condition (first or second derivative,
depending on the BoundLType)
BoundRType - boundary condition type for the right boundary
BoundR - right boundary condition (first or second derivative,
depending on the BoundRType)
N2 - new points count:
* N2>=2
* if given, only first N2 points from X2 are used
* if not given, automatically detected from X2 size
OUTPUT PARAMETERS:
F2 - function values at X2[]
D2 - first derivatives at X2[]
DD2 - second derivatives at X2[]
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
Function values are correctly reordered on return, so F2[I] is always
equal to S(X2[I]) independently of points order.
SETTING BOUNDARY VALUES:
The BoundLType/BoundRType parameters can have the following values:
* -1, which corresonds to the periodic (cyclic) boundary conditions.
In this case:
* both BoundLType and BoundRType must be equal to -1.
* BoundL/BoundR are ignored
* Y[last] is ignored (it is assumed to be equal to Y[first]).
* 0, which corresponds to the parabolically terminated spline
(BoundL and/or BoundR are ignored).
* 1, which corresponds to the first derivative boundary condition
* 2, which corresponds to the second derivative boundary condition
* by default, BoundType=0 is used
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dconvdiff2cubic(const real_1d_array &x, const real_1d_array &y, const real_1d_array &x2, real_1d_array &y2, real_1d_array &d2, real_1d_array &dd2)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t boundltype;
double boundl;
ae_int_t boundrtype;
double boundr;
ae_int_t n2;
if( (x.length()!=y.length()))
throw ap_error("Error while calling 'spline1dconvdiff2cubic': looks like one of arguments has wrong size");
n = x.length();
boundltype = 0;
boundl = 0;
boundrtype = 0;
boundr = 0;
n2 = x2.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dconvdiff2cubic(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, boundltype, boundl, boundrtype, boundr, const_cast<alglib_impl::ae_vector*>(x2.c_ptr()), n2, const_cast<alglib_impl::ae_vector*>(y2.c_ptr()), const_cast<alglib_impl::ae_vector*>(d2.c_ptr()), const_cast<alglib_impl::ae_vector*>(dd2.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine builds Catmull-Rom spline interpolant.
INPUT PARAMETERS:
X - spline nodes, array[0..N-1].
Y - function values, array[0..N-1].
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points are used to build spline
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundType - boundary condition type:
* -1 for periodic boundary condition
* 0 for parabolically terminated spline (default)
Tension - tension parameter:
* tension=0 corresponds to classic Catmull-Rom spline (default)
* 0<tension<1 corresponds to more general form - cardinal spline
OUTPUT PARAMETERS:
C - spline interpolant
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildcatmullrom(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t boundtype, const double tension, spline1dinterpolant &c)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dbuildcatmullrom(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, boundtype, tension, const_cast<alglib_impl::spline1dinterpolant*>(c.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine builds Catmull-Rom spline interpolant.
INPUT PARAMETERS:
X - spline nodes, array[0..N-1].
Y - function values, array[0..N-1].
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points are used to build spline
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundType - boundary condition type:
* -1 for periodic boundary condition
* 0 for parabolically terminated spline (default)
Tension - tension parameter:
* tension=0 corresponds to classic Catmull-Rom spline (default)
* 0<tension<1 corresponds to more general form - cardinal spline
OUTPUT PARAMETERS:
C - spline interpolant
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildcatmullrom(const real_1d_array &x, const real_1d_array &y, spline1dinterpolant &c)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t boundtype;
double tension;
if( (x.length()!=y.length()))
throw ap_error("Error while calling 'spline1dbuildcatmullrom': looks like one of arguments has wrong size");
n = x.length();
boundtype = 0;
tension = 0;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dbuildcatmullrom(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, boundtype, tension, const_cast<alglib_impl::spline1dinterpolant*>(c.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine builds Hermite spline interpolant.
INPUT PARAMETERS:
X - spline nodes, array[0..N-1]
Y - function values, array[0..N-1]
D - derivatives, array[0..N-1]
N - points count (optional):
* N>=2
* if given, only first N points are used to build spline
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
OUTPUT PARAMETERS:
C - spline interpolant.
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
-- ALGLIB PROJECT --
Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildhermite(const real_1d_array &x, const real_1d_array &y, const real_1d_array &d, const ae_int_t n, spline1dinterpolant &c)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dbuildhermite(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(d.c_ptr()), n, const_cast<alglib_impl::spline1dinterpolant*>(c.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine builds Hermite spline interpolant.
INPUT PARAMETERS:
X - spline nodes, array[0..N-1]
Y - function values, array[0..N-1]
D - derivatives, array[0..N-1]
N - points count (optional):
* N>=2
* if given, only first N points are used to build spline
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
OUTPUT PARAMETERS:
C - spline interpolant.
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
-- ALGLIB PROJECT --
Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildhermite(const real_1d_array &x, const real_1d_array &y, const real_1d_array &d, spline1dinterpolant &c)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
if( (x.length()!=y.length()) || (x.length()!=d.length()))
throw ap_error("Error while calling 'spline1dbuildhermite': looks like one of arguments has wrong size");
n = x.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dbuildhermite(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(d.c_ptr()), n, const_cast<alglib_impl::spline1dinterpolant*>(c.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine builds Akima spline interpolant
INPUT PARAMETERS:
X - spline nodes, array[0..N-1]
Y - function values, array[0..N-1]
N - points count (optional):
* N>=2
* if given, only first N points are used to build spline
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
OUTPUT PARAMETERS:
C - spline interpolant
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
-- ALGLIB PROJECT --
Copyright 24.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildakima(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, spline1dinterpolant &c)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dbuildakima(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, const_cast<alglib_impl::spline1dinterpolant*>(c.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine builds Akima spline interpolant
INPUT PARAMETERS:
X - spline nodes, array[0..N-1]
Y - function values, array[0..N-1]
N - points count (optional):
* N>=2
* if given, only first N points are used to build spline
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
OUTPUT PARAMETERS:
C - spline interpolant
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
-- ALGLIB PROJECT --
Copyright 24.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildakima(const real_1d_array &x, const real_1d_array &y, spline1dinterpolant &c)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
if( (x.length()!=y.length()))
throw ap_error("Error while calling 'spline1dbuildakima': looks like one of arguments has wrong size");
n = x.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dbuildakima(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, const_cast<alglib_impl::spline1dinterpolant*>(c.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine calculates the value of the spline at the given point X.
INPUT PARAMETERS:
C - spline interpolant
X - point
Result:
S(x)
-- ALGLIB PROJECT --
Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
double spline1dcalc(const spline1dinterpolant &c, const double x)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
double result = alglib_impl::spline1dcalc(const_cast<alglib_impl::spline1dinterpolant*>(c.c_ptr()), x, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return *(reinterpret_cast<double*>(&result));
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine differentiates the spline.
INPUT PARAMETERS:
C - spline interpolant.
X - point
Result:
S - S(x)
DS - S'(x)
D2S - S''(x)
-- ALGLIB PROJECT --
Copyright 24.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1ddiff(const spline1dinterpolant &c, const double x, double &s, double &ds, double &d2s)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1ddiff(const_cast<alglib_impl::spline1dinterpolant*>(c.c_ptr()), x, &s, &ds, &d2s, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine unpacks the spline into the coefficients table.
INPUT PARAMETERS:
C - spline interpolant.
X - point
OUTPUT PARAMETERS:
Tbl - coefficients table, unpacked format, array[0..N-2, 0..5].
For I = 0...N-2:
Tbl[I,0] = X[i]
Tbl[I,1] = X[i+1]
Tbl[I,2] = C0
Tbl[I,3] = C1
Tbl[I,4] = C2
Tbl[I,5] = C3
On [x[i], x[i+1]] spline is equals to:
S(x) = C0 + C1*t + C2*t^2 + C3*t^3
t = x-x[i]
NOTE:
You can rebuild spline with Spline1DBuildHermite() function, which
accepts as inputs function values and derivatives at nodes, which are
easy to calculate when you have coefficients.
-- ALGLIB PROJECT --
Copyright 29.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dunpack(const spline1dinterpolant &c, ae_int_t &n, real_2d_array &tbl)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dunpack(const_cast<alglib_impl::spline1dinterpolant*>(c.c_ptr()), &n, const_cast<alglib_impl::ae_matrix*>(tbl.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine performs linear transformation of the spline argument.
INPUT PARAMETERS:
C - spline interpolant.
A, B- transformation coefficients: x = A*t + B
Result:
C - transformed spline
-- ALGLIB PROJECT --
Copyright 30.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dlintransx(const spline1dinterpolant &c, const double a, const double b)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dlintransx(const_cast<alglib_impl::spline1dinterpolant*>(c.c_ptr()), a, b, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine performs linear transformation of the spline.
INPUT PARAMETERS:
C - spline interpolant.
A, B- transformation coefficients: S2(x) = A*S(x) + B
Result:
C - transformed spline
-- ALGLIB PROJECT --
Copyright 30.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dlintransy(const spline1dinterpolant &c, const double a, const double b)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dlintransy(const_cast<alglib_impl::spline1dinterpolant*>(c.c_ptr()), a, b, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine integrates the spline.
INPUT PARAMETERS:
C - spline interpolant.
X - right bound of the integration interval [a, x],
here 'a' denotes min(x[])
Result:
integral(S(t)dt,a,x)
-- ALGLIB PROJECT --
Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
double spline1dintegrate(const spline1dinterpolant &c, const double x)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
double result = alglib_impl::spline1dintegrate(const_cast<alglib_impl::spline1dinterpolant*>(c.c_ptr()), x, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return *(reinterpret_cast<double*>(&result));
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function builds monotone cubic Hermite interpolant. This interpolant
is monotonic in [x(0),x(n-1)] and is constant outside of this interval.
In case y[] form non-monotonic sequence, interpolant is piecewise
monotonic. Say, for x=(0,1,2,3,4) and y=(0,1,2,1,0) interpolant will
monotonically grow at [0..2] and monotonically decrease at [2..4].
INPUT PARAMETERS:
X - spline nodes, array[0..N-1]. Subroutine automatically
sorts points, so caller may pass unsorted array.
Y - function values, array[0..N-1]
N - the number of points(N>=2).
OUTPUT PARAMETERS:
C - spline interpolant.
-- ALGLIB PROJECT --
Copyright 21.06.2012 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildmonotone(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, spline1dinterpolant &c)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dbuildmonotone(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, const_cast<alglib_impl::spline1dinterpolant*>(c.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function builds monotone cubic Hermite interpolant. This interpolant
is monotonic in [x(0),x(n-1)] and is constant outside of this interval.
In case y[] form non-monotonic sequence, interpolant is piecewise
monotonic. Say, for x=(0,1,2,3,4) and y=(0,1,2,1,0) interpolant will
monotonically grow at [0..2] and monotonically decrease at [2..4].
INPUT PARAMETERS:
X - spline nodes, array[0..N-1]. Subroutine automatically
sorts points, so caller may pass unsorted array.
Y - function values, array[0..N-1]
N - the number of points(N>=2).
OUTPUT PARAMETERS:
C - spline interpolant.
-- ALGLIB PROJECT --
Copyright 21.06.2012 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildmonotone(const real_1d_array &x, const real_1d_array &y, spline1dinterpolant &c)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
if( (x.length()!=y.length()))
throw ap_error("Error while calling 'spline1dbuildmonotone': looks like one of arguments has wrong size");
n = x.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dbuildmonotone(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, const_cast<alglib_impl::spline1dinterpolant*>(c.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Polynomial fitting report:
TaskRCond reciprocal of task's condition number
RMSError RMS error
AvgError average error
AvgRelError average relative error (for non-zero Y[I])
MaxError maximum error
*************************************************************************/
_polynomialfitreport_owner::_polynomialfitreport_owner()
{
p_struct = (alglib_impl::polynomialfitreport*)alglib_impl::ae_malloc(sizeof(alglib_impl::polynomialfitreport), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_polynomialfitreport_init(p_struct, NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_polynomialfitreport_owner::_polynomialfitreport_owner(const _polynomialfitreport_owner &rhs)
{
p_struct = (alglib_impl::polynomialfitreport*)alglib_impl::ae_malloc(sizeof(alglib_impl::polynomialfitreport), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_polynomialfitreport_init_copy(p_struct, const_cast<alglib_impl::polynomialfitreport*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_polynomialfitreport_owner& _polynomialfitreport_owner::operator=(const _polynomialfitreport_owner &rhs)
{
if( this==&rhs )
return *this;
alglib_impl::_polynomialfitreport_clear(p_struct);
if( !alglib_impl::_polynomialfitreport_init_copy(p_struct, const_cast<alglib_impl::polynomialfitreport*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
return *this;
}
_polynomialfitreport_owner::~_polynomialfitreport_owner()
{
alglib_impl::_polynomialfitreport_clear(p_struct);
ae_free(p_struct);
}
alglib_impl::polynomialfitreport* _polynomialfitreport_owner::c_ptr()
{
return p_struct;
}
alglib_impl::polynomialfitreport* _polynomialfitreport_owner::c_ptr() const
{
return const_cast<alglib_impl::polynomialfitreport*>(p_struct);
}
polynomialfitreport::polynomialfitreport() : _polynomialfitreport_owner() ,taskrcond(p_struct->taskrcond),rmserror(p_struct->rmserror),avgerror(p_struct->avgerror),avgrelerror(p_struct->avgrelerror),maxerror(p_struct->maxerror)
{
}
polynomialfitreport::polynomialfitreport(const polynomialfitreport &rhs):_polynomialfitreport_owner(rhs) ,taskrcond(p_struct->taskrcond),rmserror(p_struct->rmserror),avgerror(p_struct->avgerror),avgrelerror(p_struct->avgrelerror),maxerror(p_struct->maxerror)
{
}
polynomialfitreport& polynomialfitreport::operator=(const polynomialfitreport &rhs)
{
if( this==&rhs )
return *this;
_polynomialfitreport_owner::operator=(rhs);
return *this;
}
polynomialfitreport::~polynomialfitreport()
{
}
/*************************************************************************
Barycentric fitting report:
RMSError RMS error
AvgError average error
AvgRelError average relative error (for non-zero Y[I])
MaxError maximum error
TaskRCond reciprocal of task's condition number
*************************************************************************/
_barycentricfitreport_owner::_barycentricfitreport_owner()
{
p_struct = (alglib_impl::barycentricfitreport*)alglib_impl::ae_malloc(sizeof(alglib_impl::barycentricfitreport), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_barycentricfitreport_init(p_struct, NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_barycentricfitreport_owner::_barycentricfitreport_owner(const _barycentricfitreport_owner &rhs)
{
p_struct = (alglib_impl::barycentricfitreport*)alglib_impl::ae_malloc(sizeof(alglib_impl::barycentricfitreport), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_barycentricfitreport_init_copy(p_struct, const_cast<alglib_impl::barycentricfitreport*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_barycentricfitreport_owner& _barycentricfitreport_owner::operator=(const _barycentricfitreport_owner &rhs)
{
if( this==&rhs )
return *this;
alglib_impl::_barycentricfitreport_clear(p_struct);
if( !alglib_impl::_barycentricfitreport_init_copy(p_struct, const_cast<alglib_impl::barycentricfitreport*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
return *this;
}
_barycentricfitreport_owner::~_barycentricfitreport_owner()
{
alglib_impl::_barycentricfitreport_clear(p_struct);
ae_free(p_struct);
}
alglib_impl::barycentricfitreport* _barycentricfitreport_owner::c_ptr()
{
return p_struct;
}
alglib_impl::barycentricfitreport* _barycentricfitreport_owner::c_ptr() const
{
return const_cast<alglib_impl::barycentricfitreport*>(p_struct);
}
barycentricfitreport::barycentricfitreport() : _barycentricfitreport_owner() ,taskrcond(p_struct->taskrcond),dbest(p_struct->dbest),rmserror(p_struct->rmserror),avgerror(p_struct->avgerror),avgrelerror(p_struct->avgrelerror),maxerror(p_struct->maxerror)
{
}
barycentricfitreport::barycentricfitreport(const barycentricfitreport &rhs):_barycentricfitreport_owner(rhs) ,taskrcond(p_struct->taskrcond),dbest(p_struct->dbest),rmserror(p_struct->rmserror),avgerror(p_struct->avgerror),avgrelerror(p_struct->avgrelerror),maxerror(p_struct->maxerror)
{
}
barycentricfitreport& barycentricfitreport::operator=(const barycentricfitreport &rhs)
{
if( this==&rhs )
return *this;
_barycentricfitreport_owner::operator=(rhs);
return *this;
}
barycentricfitreport::~barycentricfitreport()
{
}
/*************************************************************************
Spline fitting report:
RMSError RMS error
AvgError average error
AvgRelError average relative error (for non-zero Y[I])
MaxError maximum error
Fields below are filled by obsolete functions (Spline1DFitCubic,
Spline1DFitHermite). Modern fitting functions do NOT fill these fields:
TaskRCond reciprocal of task's condition number
*************************************************************************/
_spline1dfitreport_owner::_spline1dfitreport_owner()
{
p_struct = (alglib_impl::spline1dfitreport*)alglib_impl::ae_malloc(sizeof(alglib_impl::spline1dfitreport), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_spline1dfitreport_init(p_struct, NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_spline1dfitreport_owner::_spline1dfitreport_owner(const _spline1dfitreport_owner &rhs)
{
p_struct = (alglib_impl::spline1dfitreport*)alglib_impl::ae_malloc(sizeof(alglib_impl::spline1dfitreport), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_spline1dfitreport_init_copy(p_struct, const_cast<alglib_impl::spline1dfitreport*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_spline1dfitreport_owner& _spline1dfitreport_owner::operator=(const _spline1dfitreport_owner &rhs)
{
if( this==&rhs )
return *this;
alglib_impl::_spline1dfitreport_clear(p_struct);
if( !alglib_impl::_spline1dfitreport_init_copy(p_struct, const_cast<alglib_impl::spline1dfitreport*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
return *this;
}
_spline1dfitreport_owner::~_spline1dfitreport_owner()
{
alglib_impl::_spline1dfitreport_clear(p_struct);
ae_free(p_struct);
}
alglib_impl::spline1dfitreport* _spline1dfitreport_owner::c_ptr()
{
return p_struct;
}
alglib_impl::spline1dfitreport* _spline1dfitreport_owner::c_ptr() const
{
return const_cast<alglib_impl::spline1dfitreport*>(p_struct);
}
spline1dfitreport::spline1dfitreport() : _spline1dfitreport_owner() ,taskrcond(p_struct->taskrcond),rmserror(p_struct->rmserror),avgerror(p_struct->avgerror),avgrelerror(p_struct->avgrelerror),maxerror(p_struct->maxerror)
{
}
spline1dfitreport::spline1dfitreport(const spline1dfitreport &rhs):_spline1dfitreport_owner(rhs) ,taskrcond(p_struct->taskrcond),rmserror(p_struct->rmserror),avgerror(p_struct->avgerror),avgrelerror(p_struct->avgrelerror),maxerror(p_struct->maxerror)
{
}
spline1dfitreport& spline1dfitreport::operator=(const spline1dfitreport &rhs)
{
if( this==&rhs )
return *this;
_spline1dfitreport_owner::operator=(rhs);
return *this;
}
spline1dfitreport::~spline1dfitreport()
{
}
/*************************************************************************
Least squares fitting report. This structure contains informational fields
which are set by fitting functions provided by this unit.
Different functions initialize different sets of fields, so you should
read documentation on specific function you used in order to know which
fields are initialized.
TaskRCond reciprocal of task's condition number
IterationsCount number of internal iterations
VarIdx if user-supplied gradient contains errors which were
detected by nonlinear fitter, this field is set to
index of the first component of gradient which is
suspected to be spoiled by bugs.
RMSError RMS error
AvgError average error
AvgRelError average relative error (for non-zero Y[I])
MaxError maximum error
WRMSError weighted RMS error
CovPar covariance matrix for parameters, filled by some solvers
ErrPar vector of errors in parameters, filled by some solvers
ErrCurve vector of fit errors - variability of the best-fit
curve, filled by some solvers.
Noise vector of per-point noise estimates, filled by
some solvers.
R2 coefficient of determination (non-weighted, non-adjusted),
filled by some solvers.
*************************************************************************/
_lsfitreport_owner::_lsfitreport_owner()
{
p_struct = (alglib_impl::lsfitreport*)alglib_impl::ae_malloc(sizeof(alglib_impl::lsfitreport), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_lsfitreport_init(p_struct, NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_lsfitreport_owner::_lsfitreport_owner(const _lsfitreport_owner &rhs)
{
p_struct = (alglib_impl::lsfitreport*)alglib_impl::ae_malloc(sizeof(alglib_impl::lsfitreport), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_lsfitreport_init_copy(p_struct, const_cast<alglib_impl::lsfitreport*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_lsfitreport_owner& _lsfitreport_owner::operator=(const _lsfitreport_owner &rhs)
{
if( this==&rhs )
return *this;
alglib_impl::_lsfitreport_clear(p_struct);
if( !alglib_impl::_lsfitreport_init_copy(p_struct, const_cast<alglib_impl::lsfitreport*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
return *this;
}
_lsfitreport_owner::~_lsfitreport_owner()
{
alglib_impl::_lsfitreport_clear(p_struct);
ae_free(p_struct);
}
alglib_impl::lsfitreport* _lsfitreport_owner::c_ptr()
{
return p_struct;
}
alglib_impl::lsfitreport* _lsfitreport_owner::c_ptr() const
{
return const_cast<alglib_impl::lsfitreport*>(p_struct);
}
lsfitreport::lsfitreport() : _lsfitreport_owner() ,taskrcond(p_struct->taskrcond),iterationscount(p_struct->iterationscount),varidx(p_struct->varidx),rmserror(p_struct->rmserror),avgerror(p_struct->avgerror),avgrelerror(p_struct->avgrelerror),maxerror(p_struct->maxerror),wrmserror(p_struct->wrmserror),covpar(&p_struct->covpar),errpar(&p_struct->errpar),errcurve(&p_struct->errcurve),noise(&p_struct->noise),r2(p_struct->r2)
{
}
lsfitreport::lsfitreport(const lsfitreport &rhs):_lsfitreport_owner(rhs) ,taskrcond(p_struct->taskrcond),iterationscount(p_struct->iterationscount),varidx(p_struct->varidx),rmserror(p_struct->rmserror),avgerror(p_struct->avgerror),avgrelerror(p_struct->avgrelerror),maxerror(p_struct->maxerror),wrmserror(p_struct->wrmserror),covpar(&p_struct->covpar),errpar(&p_struct->errpar),errcurve(&p_struct->errcurve),noise(&p_struct->noise),r2(p_struct->r2)
{
}
lsfitreport& lsfitreport::operator=(const lsfitreport &rhs)
{
if( this==&rhs )
return *this;
_lsfitreport_owner::operator=(rhs);
return *this;
}
lsfitreport::~lsfitreport()
{
}
/*************************************************************************
Nonlinear fitter.
You should use ALGLIB functions to work with fitter.
Never try to access its fields directly!
*************************************************************************/
_lsfitstate_owner::_lsfitstate_owner()
{
p_struct = (alglib_impl::lsfitstate*)alglib_impl::ae_malloc(sizeof(alglib_impl::lsfitstate), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_lsfitstate_init(p_struct, NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_lsfitstate_owner::_lsfitstate_owner(const _lsfitstate_owner &rhs)
{
p_struct = (alglib_impl::lsfitstate*)alglib_impl::ae_malloc(sizeof(alglib_impl::lsfitstate), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_lsfitstate_init_copy(p_struct, const_cast<alglib_impl::lsfitstate*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_lsfitstate_owner& _lsfitstate_owner::operator=(const _lsfitstate_owner &rhs)
{
if( this==&rhs )
return *this;
alglib_impl::_lsfitstate_clear(p_struct);
if( !alglib_impl::_lsfitstate_init_copy(p_struct, const_cast<alglib_impl::lsfitstate*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
return *this;
}
_lsfitstate_owner::~_lsfitstate_owner()
{
alglib_impl::_lsfitstate_clear(p_struct);
ae_free(p_struct);
}
alglib_impl::lsfitstate* _lsfitstate_owner::c_ptr()
{
return p_struct;
}
alglib_impl::lsfitstate* _lsfitstate_owner::c_ptr() const
{
return const_cast<alglib_impl::lsfitstate*>(p_struct);
}
lsfitstate::lsfitstate() : _lsfitstate_owner() ,needf(p_struct->needf),needfg(p_struct->needfg),needfgh(p_struct->needfgh),xupdated(p_struct->xupdated),c(&p_struct->c),f(p_struct->f),g(&p_struct->g),h(&p_struct->h),x(&p_struct->x)
{
}
lsfitstate::lsfitstate(const lsfitstate &rhs):_lsfitstate_owner(rhs) ,needf(p_struct->needf),needfg(p_struct->needfg),needfgh(p_struct->needfgh),xupdated(p_struct->xupdated),c(&p_struct->c),f(p_struct->f),g(&p_struct->g),h(&p_struct->h),x(&p_struct->x)
{
}
lsfitstate& lsfitstate::operator=(const lsfitstate &rhs)
{
if( this==&rhs )
return *this;
_lsfitstate_owner::operator=(rhs);
return *this;
}
lsfitstate::~lsfitstate()
{
}
/*************************************************************************
Fitting by polynomials in barycentric form. This function provides simple
unterface for unconstrained unweighted fitting. See PolynomialFitWC() if
you need constrained fitting.
Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver
SEE ALSO:
PolynomialFitWC()
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
N - number of points, N>0
* if given, only leading N elements of X/Y are used
* if not given, automatically determined from sizes of X/Y
M - number of basis functions (= polynomial_degree + 1), M>=1
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearW() subroutine:
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
P - interpolant in barycentric form.
Rep - report, same format as in LSFitLinearW() subroutine.
Following fields are set:
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
NOTES:
you can convert P from barycentric form to the power or Chebyshev
basis with PolynomialBar2Pow() or PolynomialBar2Cheb() functions from
POLINT subpackage.
-- ALGLIB PROJECT --
Copyright 10.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialfit(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t m, ae_int_t &info, barycentricinterpolant &p, polynomialfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::polynomialfit(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, m, &info, const_cast<alglib_impl::barycentricinterpolant*>(p.c_ptr()), const_cast<alglib_impl::polynomialfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Fitting by polynomials in barycentric form. This function provides simple
unterface for unconstrained unweighted fitting. See PolynomialFitWC() if
you need constrained fitting.
Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver
SEE ALSO:
PolynomialFitWC()
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
N - number of points, N>0
* if given, only leading N elements of X/Y are used
* if not given, automatically determined from sizes of X/Y
M - number of basis functions (= polynomial_degree + 1), M>=1
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearW() subroutine:
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
P - interpolant in barycentric form.
Rep - report, same format as in LSFitLinearW() subroutine.
Following fields are set:
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
NOTES:
you can convert P from barycentric form to the power or Chebyshev
basis with PolynomialBar2Pow() or PolynomialBar2Cheb() functions from
POLINT subpackage.
-- ALGLIB PROJECT --
Copyright 10.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialfit(const real_1d_array &x, const real_1d_array &y, const ae_int_t m, ae_int_t &info, barycentricinterpolant &p, polynomialfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
if( (x.length()!=y.length()))
throw ap_error("Error while calling 'polynomialfit': looks like one of arguments has wrong size");
n = x.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::polynomialfit(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, m, &info, const_cast<alglib_impl::barycentricinterpolant*>(p.c_ptr()), const_cast<alglib_impl::polynomialfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Weighted fitting by polynomials in barycentric form, with constraints on
function values or first derivatives.
Small regularizing term is used when solving constrained tasks (to improve
stability).
Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver
SEE ALSO:
PolynomialFit()
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
W - weights, array[0..N-1]
Each summand in square sum of approximation deviations from
given values is multiplied by the square of corresponding
weight. Fill it by 1's if you don't want to solve weighted
task.
N - number of points, N>0.
* if given, only leading N elements of X/Y/W are used
* if not given, automatically determined from sizes of X/Y/W
XC - points where polynomial values/derivatives are constrained,
array[0..K-1].
YC - values of constraints, array[0..K-1]
DC - array[0..K-1], types of constraints:
* DC[i]=0 means that P(XC[i])=YC[i]
* DC[i]=1 means that P'(XC[i])=YC[i]
SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
K - number of constraints, 0<=K<M.
K=0 means no constraints (XC/YC/DC are not used in such cases)
M - number of basis functions (= polynomial_degree + 1), M>=1
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearW() subroutine:
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
-3 means inconsistent constraints
P - interpolant in barycentric form.
Rep - report, same format as in LSFitLinearW() subroutine.
Following fields are set:
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
NOTES:
you can convert P from barycentric form to the power or Chebyshev
basis with PolynomialBar2Pow() or PolynomialBar2Cheb() functions from
POLINT subpackage.
SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:
Setting constraints can lead to undesired results, like ill-conditioned
behavior, or inconsistency being detected. From the other side, it allows
us to improve quality of the fit. Here we summarize our experience with
constrained regression splines:
* even simple constraints can be inconsistent, see Wikipedia article on
this subject: http://en.wikipedia.org/wiki/Birkhoff_interpolation
* the greater is M (given fixed constraints), the more chances that
constraints will be consistent
* in the general case, consistency of constraints is NOT GUARANTEED.
* in the one special cases, however, we can guarantee consistency. This
case is: M>1 and constraints on the function values (NOT DERIVATIVES)
Our final recommendation is to use constraints WHEN AND ONLY when you
can't solve your task without them. Anything beyond special cases given
above is not guaranteed and may result in inconsistency.
-- ALGLIB PROJECT --
Copyright 10.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialfitwc(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const ae_int_t n, const real_1d_array &xc, const real_1d_array &yc, const integer_1d_array &dc, const ae_int_t k, const ae_int_t m, ae_int_t &info, barycentricinterpolant &p, polynomialfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::polynomialfitwc(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), n, const_cast<alglib_impl::ae_vector*>(xc.c_ptr()), const_cast<alglib_impl::ae_vector*>(yc.c_ptr()), const_cast<alglib_impl::ae_vector*>(dc.c_ptr()), k, m, &info, const_cast<alglib_impl::barycentricinterpolant*>(p.c_ptr()), const_cast<alglib_impl::polynomialfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Weighted fitting by polynomials in barycentric form, with constraints on
function values or first derivatives.
Small regularizing term is used when solving constrained tasks (to improve
stability).
Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver
SEE ALSO:
PolynomialFit()
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
W - weights, array[0..N-1]
Each summand in square sum of approximation deviations from
given values is multiplied by the square of corresponding
weight. Fill it by 1's if you don't want to solve weighted
task.
N - number of points, N>0.
* if given, only leading N elements of X/Y/W are used
* if not given, automatically determined from sizes of X/Y/W
XC - points where polynomial values/derivatives are constrained,
array[0..K-1].
YC - values of constraints, array[0..K-1]
DC - array[0..K-1], types of constraints:
* DC[i]=0 means that P(XC[i])=YC[i]
* DC[i]=1 means that P'(XC[i])=YC[i]
SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
K - number of constraints, 0<=K<M.
K=0 means no constraints (XC/YC/DC are not used in such cases)
M - number of basis functions (= polynomial_degree + 1), M>=1
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearW() subroutine:
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
-3 means inconsistent constraints
P - interpolant in barycentric form.
Rep - report, same format as in LSFitLinearW() subroutine.
Following fields are set:
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
NOTES:
you can convert P from barycentric form to the power or Chebyshev
basis with PolynomialBar2Pow() or PolynomialBar2Cheb() functions from
POLINT subpackage.
SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:
Setting constraints can lead to undesired results, like ill-conditioned
behavior, or inconsistency being detected. From the other side, it allows
us to improve quality of the fit. Here we summarize our experience with
constrained regression splines:
* even simple constraints can be inconsistent, see Wikipedia article on
this subject: http://en.wikipedia.org/wiki/Birkhoff_interpolation
* the greater is M (given fixed constraints), the more chances that
constraints will be consistent
* in the general case, consistency of constraints is NOT GUARANTEED.
* in the one special cases, however, we can guarantee consistency. This
case is: M>1 and constraints on the function values (NOT DERIVATIVES)
Our final recommendation is to use constraints WHEN AND ONLY when you
can't solve your task without them. Anything beyond special cases given
above is not guaranteed and may result in inconsistency.
-- ALGLIB PROJECT --
Copyright 10.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialfitwc(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const real_1d_array &xc, const real_1d_array &yc, const integer_1d_array &dc, const ae_int_t m, ae_int_t &info, barycentricinterpolant &p, polynomialfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t k;
if( (x.length()!=y.length()) || (x.length()!=w.length()))
throw ap_error("Error while calling 'polynomialfitwc': looks like one of arguments has wrong size");
if( (xc.length()!=yc.length()) || (xc.length()!=dc.length()))
throw ap_error("Error while calling 'polynomialfitwc': looks like one of arguments has wrong size");
n = x.length();
k = xc.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::polynomialfitwc(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), n, const_cast<alglib_impl::ae_vector*>(xc.c_ptr()), const_cast<alglib_impl::ae_vector*>(yc.c_ptr()), const_cast<alglib_impl::ae_vector*>(dc.c_ptr()), k, m, &info, const_cast<alglib_impl::barycentricinterpolant*>(p.c_ptr()), const_cast<alglib_impl::polynomialfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Weghted rational least squares fitting using Floater-Hormann rational
functions with optimal D chosen from [0,9], with constraints and
individual weights.
Equidistant grid with M node on [min(x),max(x)] is used to build basis
functions. Different values of D are tried, optimal D (least WEIGHTED root
mean square error) is chosen. Task is linear, so linear least squares
solver is used. Complexity of this computational scheme is O(N*M^2)
(mostly dominated by the least squares solver).
SEE ALSO
* BarycentricFitFloaterHormann(), "lightweight" fitting without invididual
weights and constraints.
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
W - weights, array[0..N-1]
Each summand in square sum of approximation deviations from
given values is multiplied by the square of corresponding
weight. Fill it by 1's if you don't want to solve weighted
task.
N - number of points, N>0.
XC - points where function values/derivatives are constrained,
array[0..K-1].
YC - values of constraints, array[0..K-1]
DC - array[0..K-1], types of constraints:
* DC[i]=0 means that S(XC[i])=YC[i]
* DC[i]=1 means that S'(XC[i])=YC[i]
SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
K - number of constraints, 0<=K<M.
K=0 means no constraints (XC/YC/DC are not used in such cases)
M - number of basis functions ( = number_of_nodes), M>=2.
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearWC() subroutine.
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
-3 means inconsistent constraints
-1 means another errors in parameters passed
(N<=0, for example)
B - barycentric interpolant.
Rep - report, same format as in LSFitLinearWC() subroutine.
Following fields are set:
* DBest best value of the D parameter
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroutine doesn't calculate task's condition number for K<>0.
SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:
Setting constraints can lead to undesired results, like ill-conditioned
behavior, or inconsistency being detected. From the other side, it allows
us to improve quality of the fit. Here we summarize our experience with
constrained barycentric interpolants:
* excessive constraints can be inconsistent. Floater-Hormann basis
functions aren't as flexible as splines (although they are very smooth).
* the more evenly constraints are spread across [min(x),max(x)], the more
chances that they will be consistent
* the greater is M (given fixed constraints), the more chances that
constraints will be consistent
* in the general case, consistency of constraints IS NOT GUARANTEED.
* in the several special cases, however, we CAN guarantee consistency.
* one of this cases is constraints on the function VALUES at the interval
boundaries. Note that consustency of the constraints on the function
DERIVATIVES is NOT guaranteed (you can use in such cases cubic splines
which are more flexible).
* another special case is ONE constraint on the function value (OR, but
not AND, derivative) anywhere in the interval
Our final recommendation is to use constraints WHEN AND ONLY WHEN you
can't solve your task without them. Anything beyond special cases given
above is not guaranteed and may result in inconsistency.
-- ALGLIB PROJECT --
Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentricfitfloaterhormannwc(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const ae_int_t n, const real_1d_array &xc, const real_1d_array &yc, const integer_1d_array &dc, const ae_int_t k, const ae_int_t m, ae_int_t &info, barycentricinterpolant &b, barycentricfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::barycentricfitfloaterhormannwc(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), n, const_cast<alglib_impl::ae_vector*>(xc.c_ptr()), const_cast<alglib_impl::ae_vector*>(yc.c_ptr()), const_cast<alglib_impl::ae_vector*>(dc.c_ptr()), k, m, &info, const_cast<alglib_impl::barycentricinterpolant*>(b.c_ptr()), const_cast<alglib_impl::barycentricfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Rational least squares fitting using Floater-Hormann rational functions
with optimal D chosen from [0,9].
Equidistant grid with M node on [min(x),max(x)] is used to build basis
functions. Different values of D are tried, optimal D (least root mean
square error) is chosen. Task is linear, so linear least squares solver
is used. Complexity of this computational scheme is O(N*M^2) (mostly
dominated by the least squares solver).
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
N - number of points, N>0.
M - number of basis functions ( = number_of_nodes), M>=2.
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearWC() subroutine.
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
-3 means inconsistent constraints
B - barycentric interpolant.
Rep - report, same format as in LSFitLinearWC() subroutine.
Following fields are set:
* DBest best value of the D parameter
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
-- ALGLIB PROJECT --
Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentricfitfloaterhormann(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t m, ae_int_t &info, barycentricinterpolant &b, barycentricfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::barycentricfitfloaterhormann(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, m, &info, const_cast<alglib_impl::barycentricinterpolant*>(b.c_ptr()), const_cast<alglib_impl::barycentricfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Rational least squares fitting using Floater-Hormann rational functions
with optimal D chosen from [0,9].
Equidistant grid with M node on [min(x),max(x)] is used to build basis
functions. Different values of D are tried, optimal D (least root mean
square error) is chosen. Task is linear, so linear least squares solver
is used. Complexity of this computational scheme is O(N*M^2) (mostly
dominated by the least squares solver).
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
N - number of points, N>0.
M - number of basis functions ( = number_of_nodes), M>=2.
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearWC() subroutine.
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
-3 means inconsistent constraints
B - barycentric interpolant.
Rep - report, same format as in LSFitLinearWC() subroutine.
Following fields are set:
* DBest best value of the D parameter
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
-- ALGLIB PROJECT --
Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfitpenalized(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t m, const double rho, ae_int_t &info, spline1dinterpolant &s, spline1dfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dfitpenalized(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, m, rho, &info, const_cast<alglib_impl::spline1dinterpolant*>(s.c_ptr()), const_cast<alglib_impl::spline1dfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Rational least squares fitting using Floater-Hormann rational functions
with optimal D chosen from [0,9].
Equidistant grid with M node on [min(x),max(x)] is used to build basis
functions. Different values of D are tried, optimal D (least root mean
square error) is chosen. Task is linear, so linear least squares solver
is used. Complexity of this computational scheme is O(N*M^2) (mostly
dominated by the least squares solver).
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
N - number of points, N>0.
M - number of basis functions ( = number_of_nodes), M>=2.
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearWC() subroutine.
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
-3 means inconsistent constraints
B - barycentric interpolant.
Rep - report, same format as in LSFitLinearWC() subroutine.
Following fields are set:
* DBest best value of the D parameter
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
-- ALGLIB PROJECT --
Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfitpenalized(const real_1d_array &x, const real_1d_array &y, const ae_int_t m, const double rho, ae_int_t &info, spline1dinterpolant &s, spline1dfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
if( (x.length()!=y.length()))
throw ap_error("Error while calling 'spline1dfitpenalized': looks like one of arguments has wrong size");
n = x.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dfitpenalized(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, m, rho, &info, const_cast<alglib_impl::spline1dinterpolant*>(s.c_ptr()), const_cast<alglib_impl::spline1dfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Weighted fitting by penalized cubic spline.
Equidistant grid with M nodes on [min(x,xc),max(x,xc)] is used to build
basis functions. Basis functions are cubic splines with natural boundary
conditions. Problem is regularized by adding non-linearity penalty to the
usual least squares penalty function:
S(x) = arg min { LS + P }, where
LS = SUM { w[i]^2*(y[i] - S(x[i]))^2 } - least squares penalty
P = C*10^rho*integral{ S''(x)^2*dx } - non-linearity penalty
rho - tunable constant given by user
C - automatically determined scale parameter,
makes penalty invariant with respect to scaling of X, Y, W.
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
W - weights, array[0..N-1]
Each summand in square sum of approximation deviations from
given values is multiplied by the square of corresponding
weight. Fill it by 1's if you don't want to solve weighted
problem.
N - number of points (optional):
* N>0
* if given, only first N elements of X/Y/W are processed
* if not given, automatically determined from X/Y/W sizes
M - number of basis functions ( = number_of_nodes), M>=4.
Rho - regularization constant passed by user. It penalizes
nonlinearity in the regression spline. It is logarithmically
scaled, i.e. actual value of regularization constant is
calculated as 10^Rho. It is automatically scaled so that:
* Rho=2.0 corresponds to moderate amount of nonlinearity
* generally, it should be somewhere in the [-8.0,+8.0]
If you do not want to penalize nonlineary,
pass small Rho. Values as low as -15 should work.
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearWC() subroutine.
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD or
Cholesky decomposition; problem may be
too ill-conditioned (very rare)
S - spline interpolant.
Rep - Following fields are set:
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
NOTE 1: additional nodes are added to the spline outside of the fitting
interval to force linearity when x<min(x,xc) or x>max(x,xc). It is done
for consistency - we penalize non-linearity at [min(x,xc),max(x,xc)], so
it is natural to force linearity outside of this interval.
NOTE 2: function automatically sorts points, so caller may pass unsorted
array.
-- ALGLIB PROJECT --
Copyright 19.10.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dfitpenalizedw(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const ae_int_t n, const ae_int_t m, const double rho, ae_int_t &info, spline1dinterpolant &s, spline1dfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dfitpenalizedw(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), n, m, rho, &info, const_cast<alglib_impl::spline1dinterpolant*>(s.c_ptr()), const_cast<alglib_impl::spline1dfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Weighted fitting by penalized cubic spline.
Equidistant grid with M nodes on [min(x,xc),max(x,xc)] is used to build
basis functions. Basis functions are cubic splines with natural boundary
conditions. Problem is regularized by adding non-linearity penalty to the
usual least squares penalty function:
S(x) = arg min { LS + P }, where
LS = SUM { w[i]^2*(y[i] - S(x[i]))^2 } - least squares penalty
P = C*10^rho*integral{ S''(x)^2*dx } - non-linearity penalty
rho - tunable constant given by user
C - automatically determined scale parameter,
makes penalty invariant with respect to scaling of X, Y, W.
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
W - weights, array[0..N-1]
Each summand in square sum of approximation deviations from
given values is multiplied by the square of corresponding
weight. Fill it by 1's if you don't want to solve weighted
problem.
N - number of points (optional):
* N>0
* if given, only first N elements of X/Y/W are processed
* if not given, automatically determined from X/Y/W sizes
M - number of basis functions ( = number_of_nodes), M>=4.
Rho - regularization constant passed by user. It penalizes
nonlinearity in the regression spline. It is logarithmically
scaled, i.e. actual value of regularization constant is
calculated as 10^Rho. It is automatically scaled so that:
* Rho=2.0 corresponds to moderate amount of nonlinearity
* generally, it should be somewhere in the [-8.0,+8.0]
If you do not want to penalize nonlineary,
pass small Rho. Values as low as -15 should work.
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearWC() subroutine.
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD or
Cholesky decomposition; problem may be
too ill-conditioned (very rare)
S - spline interpolant.
Rep - Following fields are set:
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
NOTE 1: additional nodes are added to the spline outside of the fitting
interval to force linearity when x<min(x,xc) or x>max(x,xc). It is done
for consistency - we penalize non-linearity at [min(x,xc),max(x,xc)], so
it is natural to force linearity outside of this interval.
NOTE 2: function automatically sorts points, so caller may pass unsorted
array.
-- ALGLIB PROJECT --
Copyright 19.10.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dfitpenalizedw(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const ae_int_t m, const double rho, ae_int_t &info, spline1dinterpolant &s, spline1dfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
if( (x.length()!=y.length()) || (x.length()!=w.length()))
throw ap_error("Error while calling 'spline1dfitpenalizedw': looks like one of arguments has wrong size");
n = x.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dfitpenalizedw(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), n, m, rho, &info, const_cast<alglib_impl::spline1dinterpolant*>(s.c_ptr()), const_cast<alglib_impl::spline1dfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Weighted fitting by cubic spline, with constraints on function values or
derivatives.
Equidistant grid with M-2 nodes on [min(x,xc),max(x,xc)] is used to build
basis functions. Basis functions are cubic splines with continuous second
derivatives and non-fixed first derivatives at interval ends. Small
regularizing term is used when solving constrained tasks (to improve
stability).
Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver
SEE ALSO
Spline1DFitHermiteWC() - fitting by Hermite splines (more flexible,
less smooth)
Spline1DFitCubic() - "lightweight" fitting by cubic splines,
without invididual weights and constraints
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
W - weights, array[0..N-1]
Each summand in square sum of approximation deviations from
given values is multiplied by the square of corresponding
weight. Fill it by 1's if you don't want to solve weighted
task.
N - number of points (optional):
* N>0
* if given, only first N elements of X/Y/W are processed
* if not given, automatically determined from X/Y/W sizes
XC - points where spline values/derivatives are constrained,
array[0..K-1].
YC - values of constraints, array[0..K-1]
DC - array[0..K-1], types of constraints:
* DC[i]=0 means that S(XC[i])=YC[i]
* DC[i]=1 means that S'(XC[i])=YC[i]
SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
K - number of constraints (optional):
* 0<=K<M.
* K=0 means no constraints (XC/YC/DC are not used)
* if given, only first K elements of XC/YC/DC are used
* if not given, automatically determined from XC/YC/DC
M - number of basis functions ( = number_of_nodes+2), M>=4.
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearWC() subroutine.
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
-3 means inconsistent constraints
S - spline interpolant.
Rep - report, same format as in LSFitLinearWC() subroutine.
Following fields are set:
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:
Setting constraints can lead to undesired results, like ill-conditioned
behavior, or inconsistency being detected. From the other side, it allows
us to improve quality of the fit. Here we summarize our experience with
constrained regression splines:
* excessive constraints can be inconsistent. Splines are piecewise cubic
functions, and it is easy to create an example, where large number of
constraints concentrated in small area will result in inconsistency.
Just because spline is not flexible enough to satisfy all of them. And
same constraints spread across the [min(x),max(x)] will be perfectly
consistent.
* the more evenly constraints are spread across [min(x),max(x)], the more
chances that they will be consistent
* the greater is M (given fixed constraints), the more chances that
constraints will be consistent
* in the general case, consistency of constraints IS NOT GUARANTEED.
* in the several special cases, however, we CAN guarantee consistency.
* one of this cases is constraints on the function values AND/OR its
derivatives at the interval boundaries.
* another special case is ONE constraint on the function value (OR, but
not AND, derivative) anywhere in the interval
Our final recommendation is to use constraints WHEN AND ONLY WHEN you
can't solve your task without them. Anything beyond special cases given
above is not guaranteed and may result in inconsistency.
-- ALGLIB PROJECT --
Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfitcubicwc(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const ae_int_t n, const real_1d_array &xc, const real_1d_array &yc, const integer_1d_array &dc, const ae_int_t k, const ae_int_t m, ae_int_t &info, spline1dinterpolant &s, spline1dfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dfitcubicwc(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), n, const_cast<alglib_impl::ae_vector*>(xc.c_ptr()), const_cast<alglib_impl::ae_vector*>(yc.c_ptr()), const_cast<alglib_impl::ae_vector*>(dc.c_ptr()), k, m, &info, const_cast<alglib_impl::spline1dinterpolant*>(s.c_ptr()), const_cast<alglib_impl::spline1dfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Weighted fitting by cubic spline, with constraints on function values or
derivatives.
Equidistant grid with M-2 nodes on [min(x,xc),max(x,xc)] is used to build
basis functions. Basis functions are cubic splines with continuous second
derivatives and non-fixed first derivatives at interval ends. Small
regularizing term is used when solving constrained tasks (to improve
stability).
Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver
SEE ALSO
Spline1DFitHermiteWC() - fitting by Hermite splines (more flexible,
less smooth)
Spline1DFitCubic() - "lightweight" fitting by cubic splines,
without invididual weights and constraints
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
W - weights, array[0..N-1]
Each summand in square sum of approximation deviations from
given values is multiplied by the square of corresponding
weight. Fill it by 1's if you don't want to solve weighted
task.
N - number of points (optional):
* N>0
* if given, only first N elements of X/Y/W are processed
* if not given, automatically determined from X/Y/W sizes
XC - points where spline values/derivatives are constrained,
array[0..K-1].
YC - values of constraints, array[0..K-1]
DC - array[0..K-1], types of constraints:
* DC[i]=0 means that S(XC[i])=YC[i]
* DC[i]=1 means that S'(XC[i])=YC[i]
SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
K - number of constraints (optional):
* 0<=K<M.
* K=0 means no constraints (XC/YC/DC are not used)
* if given, only first K elements of XC/YC/DC are used
* if not given, automatically determined from XC/YC/DC
M - number of basis functions ( = number_of_nodes+2), M>=4.
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearWC() subroutine.
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
-3 means inconsistent constraints
S - spline interpolant.
Rep - report, same format as in LSFitLinearWC() subroutine.
Following fields are set:
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:
Setting constraints can lead to undesired results, like ill-conditioned
behavior, or inconsistency being detected. From the other side, it allows
us to improve quality of the fit. Here we summarize our experience with
constrained regression splines:
* excessive constraints can be inconsistent. Splines are piecewise cubic
functions, and it is easy to create an example, where large number of
constraints concentrated in small area will result in inconsistency.
Just because spline is not flexible enough to satisfy all of them. And
same constraints spread across the [min(x),max(x)] will be perfectly
consistent.
* the more evenly constraints are spread across [min(x),max(x)], the more
chances that they will be consistent
* the greater is M (given fixed constraints), the more chances that
constraints will be consistent
* in the general case, consistency of constraints IS NOT GUARANTEED.
* in the several special cases, however, we CAN guarantee consistency.
* one of this cases is constraints on the function values AND/OR its
derivatives at the interval boundaries.
* another special case is ONE constraint on the function value (OR, but
not AND, derivative) anywhere in the interval
Our final recommendation is to use constraints WHEN AND ONLY WHEN you
can't solve your task without them. Anything beyond special cases given
above is not guaranteed and may result in inconsistency.
-- ALGLIB PROJECT --
Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfitcubicwc(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const real_1d_array &xc, const real_1d_array &yc, const integer_1d_array &dc, const ae_int_t m, ae_int_t &info, spline1dinterpolant &s, spline1dfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t k;
if( (x.length()!=y.length()) || (x.length()!=w.length()))
throw ap_error("Error while calling 'spline1dfitcubicwc': looks like one of arguments has wrong size");
if( (xc.length()!=yc.length()) || (xc.length()!=dc.length()))
throw ap_error("Error while calling 'spline1dfitcubicwc': looks like one of arguments has wrong size");
n = x.length();
k = xc.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dfitcubicwc(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), n, const_cast<alglib_impl::ae_vector*>(xc.c_ptr()), const_cast<alglib_impl::ae_vector*>(yc.c_ptr()), const_cast<alglib_impl::ae_vector*>(dc.c_ptr()), k, m, &info, const_cast<alglib_impl::spline1dinterpolant*>(s.c_ptr()), const_cast<alglib_impl::spline1dfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Weighted fitting by Hermite spline, with constraints on function values
or first derivatives.
Equidistant grid with M nodes on [min(x,xc),max(x,xc)] is used to build
basis functions. Basis functions are Hermite splines. Small regularizing
term is used when solving constrained tasks (to improve stability).
Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver
SEE ALSO
Spline1DFitCubicWC() - fitting by Cubic splines (less flexible,
more smooth)
Spline1DFitHermite() - "lightweight" Hermite fitting, without
invididual weights and constraints
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
W - weights, array[0..N-1]
Each summand in square sum of approximation deviations from
given values is multiplied by the square of corresponding
weight. Fill it by 1's if you don't want to solve weighted
task.
N - number of points (optional):
* N>0
* if given, only first N elements of X/Y/W are processed
* if not given, automatically determined from X/Y/W sizes
XC - points where spline values/derivatives are constrained,
array[0..K-1].
YC - values of constraints, array[0..K-1]
DC - array[0..K-1], types of constraints:
* DC[i]=0 means that S(XC[i])=YC[i]
* DC[i]=1 means that S'(XC[i])=YC[i]
SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
K - number of constraints (optional):
* 0<=K<M.
* K=0 means no constraints (XC/YC/DC are not used)
* if given, only first K elements of XC/YC/DC are used
* if not given, automatically determined from XC/YC/DC
M - number of basis functions (= 2 * number of nodes),
M>=4,
M IS EVEN!
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearW() subroutine:
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
-3 means inconsistent constraints
-2 means odd M was passed (which is not supported)
-1 means another errors in parameters passed
(N<=0, for example)
S - spline interpolant.
Rep - report, same format as in LSFitLinearW() subroutine.
Following fields are set:
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
IMPORTANT:
this subroitine supports only even M's
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:
Setting constraints can lead to undesired results, like ill-conditioned
behavior, or inconsistency being detected. From the other side, it allows
us to improve quality of the fit. Here we summarize our experience with
constrained regression splines:
* excessive constraints can be inconsistent. Splines are piecewise cubic
functions, and it is easy to create an example, where large number of
constraints concentrated in small area will result in inconsistency.
Just because spline is not flexible enough to satisfy all of them. And
same constraints spread across the [min(x),max(x)] will be perfectly
consistent.
* the more evenly constraints are spread across [min(x),max(x)], the more
chances that they will be consistent
* the greater is M (given fixed constraints), the more chances that
constraints will be consistent
* in the general case, consistency of constraints is NOT GUARANTEED.
* in the several special cases, however, we can guarantee consistency.
* one of this cases is M>=4 and constraints on the function value
(AND/OR its derivative) at the interval boundaries.
* another special case is M>=4 and ONE constraint on the function value
(OR, BUT NOT AND, derivative) anywhere in [min(x),max(x)]
Our final recommendation is to use constraints WHEN AND ONLY when you
can't solve your task without them. Anything beyond special cases given
above is not guaranteed and may result in inconsistency.
-- ALGLIB PROJECT --
Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfithermitewc(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const ae_int_t n, const real_1d_array &xc, const real_1d_array &yc, const integer_1d_array &dc, const ae_int_t k, const ae_int_t m, ae_int_t &info, spline1dinterpolant &s, spline1dfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dfithermitewc(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), n, const_cast<alglib_impl::ae_vector*>(xc.c_ptr()), const_cast<alglib_impl::ae_vector*>(yc.c_ptr()), const_cast<alglib_impl::ae_vector*>(dc.c_ptr()), k, m, &info, const_cast<alglib_impl::spline1dinterpolant*>(s.c_ptr()), const_cast<alglib_impl::spline1dfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Weighted fitting by Hermite spline, with constraints on function values
or first derivatives.
Equidistant grid with M nodes on [min(x,xc),max(x,xc)] is used to build
basis functions. Basis functions are Hermite splines. Small regularizing
term is used when solving constrained tasks (to improve stability).
Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver
SEE ALSO
Spline1DFitCubicWC() - fitting by Cubic splines (less flexible,
more smooth)
Spline1DFitHermite() - "lightweight" Hermite fitting, without
invididual weights and constraints
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
W - weights, array[0..N-1]
Each summand in square sum of approximation deviations from
given values is multiplied by the square of corresponding
weight. Fill it by 1's if you don't want to solve weighted
task.
N - number of points (optional):
* N>0
* if given, only first N elements of X/Y/W are processed
* if not given, automatically determined from X/Y/W sizes
XC - points where spline values/derivatives are constrained,
array[0..K-1].
YC - values of constraints, array[0..K-1]
DC - array[0..K-1], types of constraints:
* DC[i]=0 means that S(XC[i])=YC[i]
* DC[i]=1 means that S'(XC[i])=YC[i]
SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
K - number of constraints (optional):
* 0<=K<M.
* K=0 means no constraints (XC/YC/DC are not used)
* if given, only first K elements of XC/YC/DC are used
* if not given, automatically determined from XC/YC/DC
M - number of basis functions (= 2 * number of nodes),
M>=4,
M IS EVEN!
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearW() subroutine:
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
-3 means inconsistent constraints
-2 means odd M was passed (which is not supported)
-1 means another errors in parameters passed
(N<=0, for example)
S - spline interpolant.
Rep - report, same format as in LSFitLinearW() subroutine.
Following fields are set:
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
IMPORTANT:
this subroitine supports only even M's
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:
Setting constraints can lead to undesired results, like ill-conditioned
behavior, or inconsistency being detected. From the other side, it allows
us to improve quality of the fit. Here we summarize our experience with
constrained regression splines:
* excessive constraints can be inconsistent. Splines are piecewise cubic
functions, and it is easy to create an example, where large number of
constraints concentrated in small area will result in inconsistency.
Just because spline is not flexible enough to satisfy all of them. And
same constraints spread across the [min(x),max(x)] will be perfectly
consistent.
* the more evenly constraints are spread across [min(x),max(x)], the more
chances that they will be consistent
* the greater is M (given fixed constraints), the more chances that
constraints will be consistent
* in the general case, consistency of constraints is NOT GUARANTEED.
* in the several special cases, however, we can guarantee consistency.
* one of this cases is M>=4 and constraints on the function value
(AND/OR its derivative) at the interval boundaries.
* another special case is M>=4 and ONE constraint on the function value
(OR, BUT NOT AND, derivative) anywhere in [min(x),max(x)]
Our final recommendation is to use constraints WHEN AND ONLY when you
can't solve your task without them. Anything beyond special cases given
above is not guaranteed and may result in inconsistency.
-- ALGLIB PROJECT --
Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfithermitewc(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const real_1d_array &xc, const real_1d_array &yc, const integer_1d_array &dc, const ae_int_t m, ae_int_t &info, spline1dinterpolant &s, spline1dfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t k;
if( (x.length()!=y.length()) || (x.length()!=w.length()))
throw ap_error("Error while calling 'spline1dfithermitewc': looks like one of arguments has wrong size");
if( (xc.length()!=yc.length()) || (xc.length()!=dc.length()))
throw ap_error("Error while calling 'spline1dfithermitewc': looks like one of arguments has wrong size");
n = x.length();
k = xc.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dfithermitewc(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), n, const_cast<alglib_impl::ae_vector*>(xc.c_ptr()), const_cast<alglib_impl::ae_vector*>(yc.c_ptr()), const_cast<alglib_impl::ae_vector*>(dc.c_ptr()), k, m, &info, const_cast<alglib_impl::spline1dinterpolant*>(s.c_ptr()), const_cast<alglib_impl::spline1dfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Least squares fitting by cubic spline.
This subroutine is "lightweight" alternative for more complex and feature-
rich Spline1DFitCubicWC(). See Spline1DFitCubicWC() for more information
about subroutine parameters (we don't duplicate it here because of length)
-- ALGLIB PROJECT --
Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfitcubic(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t m, ae_int_t &info, spline1dinterpolant &s, spline1dfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dfitcubic(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, m, &info, const_cast<alglib_impl::spline1dinterpolant*>(s.c_ptr()), const_cast<alglib_impl::spline1dfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Least squares fitting by cubic spline.
This subroutine is "lightweight" alternative for more complex and feature-
rich Spline1DFitCubicWC(). See Spline1DFitCubicWC() for more information
about subroutine parameters (we don't duplicate it here because of length)
-- ALGLIB PROJECT --
Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfitcubic(const real_1d_array &x, const real_1d_array &y, const ae_int_t m, ae_int_t &info, spline1dinterpolant &s, spline1dfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
if( (x.length()!=y.length()))
throw ap_error("Error while calling 'spline1dfitcubic': looks like one of arguments has wrong size");
n = x.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dfitcubic(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, m, &info, const_cast<alglib_impl::spline1dinterpolant*>(s.c_ptr()), const_cast<alglib_impl::spline1dfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Least squares fitting by Hermite spline.
This subroutine is "lightweight" alternative for more complex and feature-
rich Spline1DFitHermiteWC(). See Spline1DFitHermiteWC() description for
more information about subroutine parameters (we don't duplicate it here
because of length).
-- ALGLIB PROJECT --
Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfithermite(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t m, ae_int_t &info, spline1dinterpolant &s, spline1dfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dfithermite(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, m, &info, const_cast<alglib_impl::spline1dinterpolant*>(s.c_ptr()), const_cast<alglib_impl::spline1dfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Least squares fitting by Hermite spline.
This subroutine is "lightweight" alternative for more complex and feature-
rich Spline1DFitHermiteWC(). See Spline1DFitHermiteWC() description for
more information about subroutine parameters (we don't duplicate it here
because of length).
-- ALGLIB PROJECT --
Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfithermite(const real_1d_array &x, const real_1d_array &y, const ae_int_t m, ae_int_t &info, spline1dinterpolant &s, spline1dfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
if( (x.length()!=y.length()))
throw ap_error("Error while calling 'spline1dfithermite': looks like one of arguments has wrong size");
n = x.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline1dfithermite(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), n, m, &info, const_cast<alglib_impl::spline1dinterpolant*>(s.c_ptr()), const_cast<alglib_impl::spline1dfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Weighted linear least squares fitting.
QR decomposition is used to reduce task to MxM, then triangular solver or
SVD-based solver is used depending on condition number of the system. It
allows to maximize speed and retain decent accuracy.
IMPORTANT: if you want to perform polynomial fitting, it may be more
convenient to use PolynomialFit() function. This function gives
best results on polynomial problems and solves numerical
stability issues which arise when you fit high-degree
polynomials to your data.
INPUT PARAMETERS:
Y - array[0..N-1] Function values in N points.
W - array[0..N-1] Weights corresponding to function values.
Each summand in square sum of approximation deviations
from given values is multiplied by the square of
corresponding weight.
FMatrix - a table of basis functions values, array[0..N-1, 0..M-1].
FMatrix[I, J] - value of J-th basis function in I-th point.
N - number of points used. N>=1.
M - number of basis functions, M>=1.
OUTPUT PARAMETERS:
Info - error code:
* -4 internal SVD decomposition subroutine failed (very
rare and for degenerate systems only)
* -1 incorrect N/M were specified
* 1 task is solved
C - decomposition coefficients, array[0..M-1]
Rep - fitting report. Following fields are set:
* Rep.TaskRCond reciprocal of condition number
* R2 non-adjusted coefficient of determination
(non-weighted)
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
ERRORS IN PARAMETERS
This solver also calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar covariance matrix for parameters, array[K,K].
* Rep.ErrPar errors in parameters, array[K],
errpar = sqrt(diag(CovPar))
* Rep.ErrCurve vector of fit errors - standard deviations of empirical
best-fit curve from "ideal" best-fit curve built with
infinite number of samples, array[N].
errcurve = sqrt(diag(F*CovPar*F')),
where F is functions matrix.
* Rep.Noise vector of per-point estimates of noise, array[N]
NOTE: noise in the data is estimated as follows:
* for fitting without user-supplied weights all points are
assumed to have same level of noise, which is estimated from
the data
* for fitting with user-supplied weights we assume that noise
level in I-th point is inversely proportional to Ith weight.
Coefficient of proportionality is estimated from the data.
NOTE: we apply small amount of regularization when we invert squared
Jacobian and calculate covariance matrix. It guarantees that
algorithm won't divide by zero during inversion, but skews
error estimates a bit (fractional error is about 10^-9).
However, we believe that this difference is insignificant for
all practical purposes except for the situation when you want
to compare ALGLIB results with "reference" implementation up
to the last significant digit.
NOTE: covariance matrix is estimated using correction for degrees
of freedom (covariances are divided by N-M instead of dividing
by N).
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitlinearw(const real_1d_array &y, const real_1d_array &w, const real_2d_array &fmatrix, const ae_int_t n, const ae_int_t m, ae_int_t &info, real_1d_array &c, lsfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitlinearw(const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), const_cast<alglib_impl::ae_matrix*>(fmatrix.c_ptr()), n, m, &info, const_cast<alglib_impl::ae_vector*>(c.c_ptr()), const_cast<alglib_impl::lsfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Weighted linear least squares fitting.
QR decomposition is used to reduce task to MxM, then triangular solver or
SVD-based solver is used depending on condition number of the system. It
allows to maximize speed and retain decent accuracy.
IMPORTANT: if you want to perform polynomial fitting, it may be more
convenient to use PolynomialFit() function. This function gives
best results on polynomial problems and solves numerical
stability issues which arise when you fit high-degree
polynomials to your data.
INPUT PARAMETERS:
Y - array[0..N-1] Function values in N points.
W - array[0..N-1] Weights corresponding to function values.
Each summand in square sum of approximation deviations
from given values is multiplied by the square of
corresponding weight.
FMatrix - a table of basis functions values, array[0..N-1, 0..M-1].
FMatrix[I, J] - value of J-th basis function in I-th point.
N - number of points used. N>=1.
M - number of basis functions, M>=1.
OUTPUT PARAMETERS:
Info - error code:
* -4 internal SVD decomposition subroutine failed (very
rare and for degenerate systems only)
* -1 incorrect N/M were specified
* 1 task is solved
C - decomposition coefficients, array[0..M-1]
Rep - fitting report. Following fields are set:
* Rep.TaskRCond reciprocal of condition number
* R2 non-adjusted coefficient of determination
(non-weighted)
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
ERRORS IN PARAMETERS
This solver also calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar covariance matrix for parameters, array[K,K].
* Rep.ErrPar errors in parameters, array[K],
errpar = sqrt(diag(CovPar))
* Rep.ErrCurve vector of fit errors - standard deviations of empirical
best-fit curve from "ideal" best-fit curve built with
infinite number of samples, array[N].
errcurve = sqrt(diag(F*CovPar*F')),
where F is functions matrix.
* Rep.Noise vector of per-point estimates of noise, array[N]
NOTE: noise in the data is estimated as follows:
* for fitting without user-supplied weights all points are
assumed to have same level of noise, which is estimated from
the data
* for fitting with user-supplied weights we assume that noise
level in I-th point is inversely proportional to Ith weight.
Coefficient of proportionality is estimated from the data.
NOTE: we apply small amount of regularization when we invert squared
Jacobian and calculate covariance matrix. It guarantees that
algorithm won't divide by zero during inversion, but skews
error estimates a bit (fractional error is about 10^-9).
However, we believe that this difference is insignificant for
all practical purposes except for the situation when you want
to compare ALGLIB results with "reference" implementation up
to the last significant digit.
NOTE: covariance matrix is estimated using correction for degrees
of freedom (covariances are divided by N-M instead of dividing
by N).
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitlinearw(const real_1d_array &y, const real_1d_array &w, const real_2d_array &fmatrix, ae_int_t &info, real_1d_array &c, lsfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t m;
if( (y.length()!=w.length()) || (y.length()!=fmatrix.rows()))
throw ap_error("Error while calling 'lsfitlinearw': looks like one of arguments has wrong size");
n = y.length();
m = fmatrix.cols();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitlinearw(const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), const_cast<alglib_impl::ae_matrix*>(fmatrix.c_ptr()), n, m, &info, const_cast<alglib_impl::ae_vector*>(c.c_ptr()), const_cast<alglib_impl::lsfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Weighted constained linear least squares fitting.
This is variation of LSFitLinearW(), which searchs for min|A*x=b| given
that K additional constaints C*x=bc are satisfied. It reduces original
task to modified one: min|B*y-d| WITHOUT constraints, then LSFitLinearW()
is called.
IMPORTANT: if you want to perform polynomial fitting, it may be more
convenient to use PolynomialFit() function. This function gives
best results on polynomial problems and solves numerical
stability issues which arise when you fit high-degree
polynomials to your data.
INPUT PARAMETERS:
Y - array[0..N-1] Function values in N points.
W - array[0..N-1] Weights corresponding to function values.
Each summand in square sum of approximation deviations
from given values is multiplied by the square of
corresponding weight.
FMatrix - a table of basis functions values, array[0..N-1, 0..M-1].
FMatrix[I,J] - value of J-th basis function in I-th point.
CMatrix - a table of constaints, array[0..K-1,0..M].
I-th row of CMatrix corresponds to I-th linear constraint:
CMatrix[I,0]*C[0] + ... + CMatrix[I,M-1]*C[M-1] = CMatrix[I,M]
N - number of points used. N>=1.
M - number of basis functions, M>=1.
K - number of constraints, 0 <= K < M
K=0 corresponds to absence of constraints.
OUTPUT PARAMETERS:
Info - error code:
* -4 internal SVD decomposition subroutine failed (very
rare and for degenerate systems only)
* -3 either too many constraints (M or more),
degenerate constraints (some constraints are
repetead twice) or inconsistent constraints were
specified.
* 1 task is solved
C - decomposition coefficients, array[0..M-1]
Rep - fitting report. Following fields are set:
* R2 non-adjusted coefficient of determination
(non-weighted)
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
ERRORS IN PARAMETERS
This solver also calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar covariance matrix for parameters, array[K,K].
* Rep.ErrPar errors in parameters, array[K],
errpar = sqrt(diag(CovPar))
* Rep.ErrCurve vector of fit errors - standard deviations of empirical
best-fit curve from "ideal" best-fit curve built with
infinite number of samples, array[N].
errcurve = sqrt(diag(F*CovPar*F')),
where F is functions matrix.
* Rep.Noise vector of per-point estimates of noise, array[N]
IMPORTANT: errors in parameters are calculated without taking into
account boundary/linear constraints! Presence of constraints
changes distribution of errors, but there is no easy way to
account for constraints when you calculate covariance matrix.
NOTE: noise in the data is estimated as follows:
* for fitting without user-supplied weights all points are
assumed to have same level of noise, which is estimated from
the data
* for fitting with user-supplied weights we assume that noise
level in I-th point is inversely proportional to Ith weight.
Coefficient of proportionality is estimated from the data.
NOTE: we apply small amount of regularization when we invert squared
Jacobian and calculate covariance matrix. It guarantees that
algorithm won't divide by zero during inversion, but skews
error estimates a bit (fractional error is about 10^-9).
However, we believe that this difference is insignificant for
all practical purposes except for the situation when you want
to compare ALGLIB results with "reference" implementation up
to the last significant digit.
NOTE: covariance matrix is estimated using correction for degrees
of freedom (covariances are divided by N-M instead of dividing
by N).
-- ALGLIB --
Copyright 07.09.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitlinearwc(const real_1d_array &y, const real_1d_array &w, const real_2d_array &fmatrix, const real_2d_array &cmatrix, const ae_int_t n, const ae_int_t m, const ae_int_t k, ae_int_t &info, real_1d_array &c, lsfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitlinearwc(const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), const_cast<alglib_impl::ae_matrix*>(fmatrix.c_ptr()), const_cast<alglib_impl::ae_matrix*>(cmatrix.c_ptr()), n, m, k, &info, const_cast<alglib_impl::ae_vector*>(c.c_ptr()), const_cast<alglib_impl::lsfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Weighted constained linear least squares fitting.
This is variation of LSFitLinearW(), which searchs for min|A*x=b| given
that K additional constaints C*x=bc are satisfied. It reduces original
task to modified one: min|B*y-d| WITHOUT constraints, then LSFitLinearW()
is called.
IMPORTANT: if you want to perform polynomial fitting, it may be more
convenient to use PolynomialFit() function. This function gives
best results on polynomial problems and solves numerical
stability issues which arise when you fit high-degree
polynomials to your data.
INPUT PARAMETERS:
Y - array[0..N-1] Function values in N points.
W - array[0..N-1] Weights corresponding to function values.
Each summand in square sum of approximation deviations
from given values is multiplied by the square of
corresponding weight.
FMatrix - a table of basis functions values, array[0..N-1, 0..M-1].
FMatrix[I,J] - value of J-th basis function in I-th point.
CMatrix - a table of constaints, array[0..K-1,0..M].
I-th row of CMatrix corresponds to I-th linear constraint:
CMatrix[I,0]*C[0] + ... + CMatrix[I,M-1]*C[M-1] = CMatrix[I,M]
N - number of points used. N>=1.
M - number of basis functions, M>=1.
K - number of constraints, 0 <= K < M
K=0 corresponds to absence of constraints.
OUTPUT PARAMETERS:
Info - error code:
* -4 internal SVD decomposition subroutine failed (very
rare and for degenerate systems only)
* -3 either too many constraints (M or more),
degenerate constraints (some constraints are
repetead twice) or inconsistent constraints were
specified.
* 1 task is solved
C - decomposition coefficients, array[0..M-1]
Rep - fitting report. Following fields are set:
* R2 non-adjusted coefficient of determination
(non-weighted)
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
ERRORS IN PARAMETERS
This solver also calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar covariance matrix for parameters, array[K,K].
* Rep.ErrPar errors in parameters, array[K],
errpar = sqrt(diag(CovPar))
* Rep.ErrCurve vector of fit errors - standard deviations of empirical
best-fit curve from "ideal" best-fit curve built with
infinite number of samples, array[N].
errcurve = sqrt(diag(F*CovPar*F')),
where F is functions matrix.
* Rep.Noise vector of per-point estimates of noise, array[N]
IMPORTANT: errors in parameters are calculated without taking into
account boundary/linear constraints! Presence of constraints
changes distribution of errors, but there is no easy way to
account for constraints when you calculate covariance matrix.
NOTE: noise in the data is estimated as follows:
* for fitting without user-supplied weights all points are
assumed to have same level of noise, which is estimated from
the data
* for fitting with user-supplied weights we assume that noise
level in I-th point is inversely proportional to Ith weight.
Coefficient of proportionality is estimated from the data.
NOTE: we apply small amount of regularization when we invert squared
Jacobian and calculate covariance matrix. It guarantees that
algorithm won't divide by zero during inversion, but skews
error estimates a bit (fractional error is about 10^-9).
However, we believe that this difference is insignificant for
all practical purposes except for the situation when you want
to compare ALGLIB results with "reference" implementation up
to the last significant digit.
NOTE: covariance matrix is estimated using correction for degrees
of freedom (covariances are divided by N-M instead of dividing
by N).
-- ALGLIB --
Copyright 07.09.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitlinearwc(const real_1d_array &y, const real_1d_array &w, const real_2d_array &fmatrix, const real_2d_array &cmatrix, ae_int_t &info, real_1d_array &c, lsfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t m;
ae_int_t k;
if( (y.length()!=w.length()) || (y.length()!=fmatrix.rows()))
throw ap_error("Error while calling 'lsfitlinearwc': looks like one of arguments has wrong size");
if( (fmatrix.cols()!=cmatrix.cols()-1))
throw ap_error("Error while calling 'lsfitlinearwc': looks like one of arguments has wrong size");
n = y.length();
m = fmatrix.cols();
k = cmatrix.rows();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitlinearwc(const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), const_cast<alglib_impl::ae_matrix*>(fmatrix.c_ptr()), const_cast<alglib_impl::ae_matrix*>(cmatrix.c_ptr()), n, m, k, &info, const_cast<alglib_impl::ae_vector*>(c.c_ptr()), const_cast<alglib_impl::lsfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Linear least squares fitting.
QR decomposition is used to reduce task to MxM, then triangular solver or
SVD-based solver is used depending on condition number of the system. It
allows to maximize speed and retain decent accuracy.
IMPORTANT: if you want to perform polynomial fitting, it may be more
convenient to use PolynomialFit() function. This function gives
best results on polynomial problems and solves numerical
stability issues which arise when you fit high-degree
polynomials to your data.
INPUT PARAMETERS:
Y - array[0..N-1] Function values in N points.
FMatrix - a table of basis functions values, array[0..N-1, 0..M-1].
FMatrix[I, J] - value of J-th basis function in I-th point.
N - number of points used. N>=1.
M - number of basis functions, M>=1.
OUTPUT PARAMETERS:
Info - error code:
* -4 internal SVD decomposition subroutine failed (very
rare and for degenerate systems only)
* 1 task is solved
C - decomposition coefficients, array[0..M-1]
Rep - fitting report. Following fields are set:
* Rep.TaskRCond reciprocal of condition number
* R2 non-adjusted coefficient of determination
(non-weighted)
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
ERRORS IN PARAMETERS
This solver also calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar covariance matrix for parameters, array[K,K].
* Rep.ErrPar errors in parameters, array[K],
errpar = sqrt(diag(CovPar))
* Rep.ErrCurve vector of fit errors - standard deviations of empirical
best-fit curve from "ideal" best-fit curve built with
infinite number of samples, array[N].
errcurve = sqrt(diag(F*CovPar*F')),
where F is functions matrix.
* Rep.Noise vector of per-point estimates of noise, array[N]
NOTE: noise in the data is estimated as follows:
* for fitting without user-supplied weights all points are
assumed to have same level of noise, which is estimated from
the data
* for fitting with user-supplied weights we assume that noise
level in I-th point is inversely proportional to Ith weight.
Coefficient of proportionality is estimated from the data.
NOTE: we apply small amount of regularization when we invert squared
Jacobian and calculate covariance matrix. It guarantees that
algorithm won't divide by zero during inversion, but skews
error estimates a bit (fractional error is about 10^-9).
However, we believe that this difference is insignificant for
all practical purposes except for the situation when you want
to compare ALGLIB results with "reference" implementation up
to the last significant digit.
NOTE: covariance matrix is estimated using correction for degrees
of freedom (covariances are divided by N-M instead of dividing
by N).
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitlinear(const real_1d_array &y, const real_2d_array &fmatrix, const ae_int_t n, const ae_int_t m, ae_int_t &info, real_1d_array &c, lsfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitlinear(const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_matrix*>(fmatrix.c_ptr()), n, m, &info, const_cast<alglib_impl::ae_vector*>(c.c_ptr()), const_cast<alglib_impl::lsfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Linear least squares fitting.
QR decomposition is used to reduce task to MxM, then triangular solver or
SVD-based solver is used depending on condition number of the system. It
allows to maximize speed and retain decent accuracy.
IMPORTANT: if you want to perform polynomial fitting, it may be more
convenient to use PolynomialFit() function. This function gives
best results on polynomial problems and solves numerical
stability issues which arise when you fit high-degree
polynomials to your data.
INPUT PARAMETERS:
Y - array[0..N-1] Function values in N points.
FMatrix - a table of basis functions values, array[0..N-1, 0..M-1].
FMatrix[I, J] - value of J-th basis function in I-th point.
N - number of points used. N>=1.
M - number of basis functions, M>=1.
OUTPUT PARAMETERS:
Info - error code:
* -4 internal SVD decomposition subroutine failed (very
rare and for degenerate systems only)
* 1 task is solved
C - decomposition coefficients, array[0..M-1]
Rep - fitting report. Following fields are set:
* Rep.TaskRCond reciprocal of condition number
* R2 non-adjusted coefficient of determination
(non-weighted)
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
ERRORS IN PARAMETERS
This solver also calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar covariance matrix for parameters, array[K,K].
* Rep.ErrPar errors in parameters, array[K],
errpar = sqrt(diag(CovPar))
* Rep.ErrCurve vector of fit errors - standard deviations of empirical
best-fit curve from "ideal" best-fit curve built with
infinite number of samples, array[N].
errcurve = sqrt(diag(F*CovPar*F')),
where F is functions matrix.
* Rep.Noise vector of per-point estimates of noise, array[N]
NOTE: noise in the data is estimated as follows:
* for fitting without user-supplied weights all points are
assumed to have same level of noise, which is estimated from
the data
* for fitting with user-supplied weights we assume that noise
level in I-th point is inversely proportional to Ith weight.
Coefficient of proportionality is estimated from the data.
NOTE: we apply small amount of regularization when we invert squared
Jacobian and calculate covariance matrix. It guarantees that
algorithm won't divide by zero during inversion, but skews
error estimates a bit (fractional error is about 10^-9).
However, we believe that this difference is insignificant for
all practical purposes except for the situation when you want
to compare ALGLIB results with "reference" implementation up
to the last significant digit.
NOTE: covariance matrix is estimated using correction for degrees
of freedom (covariances are divided by N-M instead of dividing
by N).
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitlinear(const real_1d_array &y, const real_2d_array &fmatrix, ae_int_t &info, real_1d_array &c, lsfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t m;
if( (y.length()!=fmatrix.rows()))
throw ap_error("Error while calling 'lsfitlinear': looks like one of arguments has wrong size");
n = y.length();
m = fmatrix.cols();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitlinear(const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_matrix*>(fmatrix.c_ptr()), n, m, &info, const_cast<alglib_impl::ae_vector*>(c.c_ptr()), const_cast<alglib_impl::lsfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Constained linear least squares fitting.
This is variation of LSFitLinear(), which searchs for min|A*x=b| given
that K additional constaints C*x=bc are satisfied. It reduces original
task to modified one: min|B*y-d| WITHOUT constraints, then LSFitLinear()
is called.
IMPORTANT: if you want to perform polynomial fitting, it may be more
convenient to use PolynomialFit() function. This function gives
best results on polynomial problems and solves numerical
stability issues which arise when you fit high-degree
polynomials to your data.
INPUT PARAMETERS:
Y - array[0..N-1] Function values in N points.
FMatrix - a table of basis functions values, array[0..N-1, 0..M-1].
FMatrix[I,J] - value of J-th basis function in I-th point.
CMatrix - a table of constaints, array[0..K-1,0..M].
I-th row of CMatrix corresponds to I-th linear constraint:
CMatrix[I,0]*C[0] + ... + CMatrix[I,M-1]*C[M-1] = CMatrix[I,M]
N - number of points used. N>=1.
M - number of basis functions, M>=1.
K - number of constraints, 0 <= K < M
K=0 corresponds to absence of constraints.
OUTPUT PARAMETERS:
Info - error code:
* -4 internal SVD decomposition subroutine failed (very
rare and for degenerate systems only)
* -3 either too many constraints (M or more),
degenerate constraints (some constraints are
repetead twice) or inconsistent constraints were
specified.
* 1 task is solved
C - decomposition coefficients, array[0..M-1]
Rep - fitting report. Following fields are set:
* R2 non-adjusted coefficient of determination
(non-weighted)
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
ERRORS IN PARAMETERS
This solver also calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar covariance matrix for parameters, array[K,K].
* Rep.ErrPar errors in parameters, array[K],
errpar = sqrt(diag(CovPar))
* Rep.ErrCurve vector of fit errors - standard deviations of empirical
best-fit curve from "ideal" best-fit curve built with
infinite number of samples, array[N].
errcurve = sqrt(diag(F*CovPar*F')),
where F is functions matrix.
* Rep.Noise vector of per-point estimates of noise, array[N]
IMPORTANT: errors in parameters are calculated without taking into
account boundary/linear constraints! Presence of constraints
changes distribution of errors, but there is no easy way to
account for constraints when you calculate covariance matrix.
NOTE: noise in the data is estimated as follows:
* for fitting without user-supplied weights all points are
assumed to have same level of noise, which is estimated from
the data
* for fitting with user-supplied weights we assume that noise
level in I-th point is inversely proportional to Ith weight.
Coefficient of proportionality is estimated from the data.
NOTE: we apply small amount of regularization when we invert squared
Jacobian and calculate covariance matrix. It guarantees that
algorithm won't divide by zero during inversion, but skews
error estimates a bit (fractional error is about 10^-9).
However, we believe that this difference is insignificant for
all practical purposes except for the situation when you want
to compare ALGLIB results with "reference" implementation up
to the last significant digit.
NOTE: covariance matrix is estimated using correction for degrees
of freedom (covariances are divided by N-M instead of dividing
by N).
-- ALGLIB --
Copyright 07.09.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitlinearc(const real_1d_array &y, const real_2d_array &fmatrix, const real_2d_array &cmatrix, const ae_int_t n, const ae_int_t m, const ae_int_t k, ae_int_t &info, real_1d_array &c, lsfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitlinearc(const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_matrix*>(fmatrix.c_ptr()), const_cast<alglib_impl::ae_matrix*>(cmatrix.c_ptr()), n, m, k, &info, const_cast<alglib_impl::ae_vector*>(c.c_ptr()), const_cast<alglib_impl::lsfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Constained linear least squares fitting.
This is variation of LSFitLinear(), which searchs for min|A*x=b| given
that K additional constaints C*x=bc are satisfied. It reduces original
task to modified one: min|B*y-d| WITHOUT constraints, then LSFitLinear()
is called.
IMPORTANT: if you want to perform polynomial fitting, it may be more
convenient to use PolynomialFit() function. This function gives
best results on polynomial problems and solves numerical
stability issues which arise when you fit high-degree
polynomials to your data.
INPUT PARAMETERS:
Y - array[0..N-1] Function values in N points.
FMatrix - a table of basis functions values, array[0..N-1, 0..M-1].
FMatrix[I,J] - value of J-th basis function in I-th point.
CMatrix - a table of constaints, array[0..K-1,0..M].
I-th row of CMatrix corresponds to I-th linear constraint:
CMatrix[I,0]*C[0] + ... + CMatrix[I,M-1]*C[M-1] = CMatrix[I,M]
N - number of points used. N>=1.
M - number of basis functions, M>=1.
K - number of constraints, 0 <= K < M
K=0 corresponds to absence of constraints.
OUTPUT PARAMETERS:
Info - error code:
* -4 internal SVD decomposition subroutine failed (very
rare and for degenerate systems only)
* -3 either too many constraints (M or more),
degenerate constraints (some constraints are
repetead twice) or inconsistent constraints were
specified.
* 1 task is solved
C - decomposition coefficients, array[0..M-1]
Rep - fitting report. Following fields are set:
* R2 non-adjusted coefficient of determination
(non-weighted)
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
ERRORS IN PARAMETERS
This solver also calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar covariance matrix for parameters, array[K,K].
* Rep.ErrPar errors in parameters, array[K],
errpar = sqrt(diag(CovPar))
* Rep.ErrCurve vector of fit errors - standard deviations of empirical
best-fit curve from "ideal" best-fit curve built with
infinite number of samples, array[N].
errcurve = sqrt(diag(F*CovPar*F')),
where F is functions matrix.
* Rep.Noise vector of per-point estimates of noise, array[N]
IMPORTANT: errors in parameters are calculated without taking into
account boundary/linear constraints! Presence of constraints
changes distribution of errors, but there is no easy way to
account for constraints when you calculate covariance matrix.
NOTE: noise in the data is estimated as follows:
* for fitting without user-supplied weights all points are
assumed to have same level of noise, which is estimated from
the data
* for fitting with user-supplied weights we assume that noise
level in I-th point is inversely proportional to Ith weight.
Coefficient of proportionality is estimated from the data.
NOTE: we apply small amount of regularization when we invert squared
Jacobian and calculate covariance matrix. It guarantees that
algorithm won't divide by zero during inversion, but skews
error estimates a bit (fractional error is about 10^-9).
However, we believe that this difference is insignificant for
all practical purposes except for the situation when you want
to compare ALGLIB results with "reference" implementation up
to the last significant digit.
NOTE: covariance matrix is estimated using correction for degrees
of freedom (covariances are divided by N-M instead of dividing
by N).
-- ALGLIB --
Copyright 07.09.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitlinearc(const real_1d_array &y, const real_2d_array &fmatrix, const real_2d_array &cmatrix, ae_int_t &info, real_1d_array &c, lsfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t m;
ae_int_t k;
if( (y.length()!=fmatrix.rows()))
throw ap_error("Error while calling 'lsfitlinearc': looks like one of arguments has wrong size");
if( (fmatrix.cols()!=cmatrix.cols()-1))
throw ap_error("Error while calling 'lsfitlinearc': looks like one of arguments has wrong size");
n = y.length();
m = fmatrix.cols();
k = cmatrix.rows();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitlinearc(const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_matrix*>(fmatrix.c_ptr()), const_cast<alglib_impl::ae_matrix*>(cmatrix.c_ptr()), n, m, k, &info, const_cast<alglib_impl::ae_vector*>(c.c_ptr()), const_cast<alglib_impl::lsfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Weighted nonlinear least squares fitting using function values only.
Combination of numerical differentiation and secant updates is used to
obtain function Jacobian.
Nonlinear task min(F(c)) is solved, where
F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2,
* N is a number of points,
* M is a dimension of a space points belong to,
* K is a dimension of a space of parameters being fitted,
* w is an N-dimensional vector of weight coefficients,
* x is a set of N points, each of them is an M-dimensional vector,
* c is a K-dimensional vector of parameters being fitted
This subroutine uses only f(c,x[i]).
INPUT PARAMETERS:
X - array[0..N-1,0..M-1], points (one row = one point)
Y - array[0..N-1], function values.
W - weights, array[0..N-1]
C - array[0..K-1], initial approximation to the solution,
N - number of points, N>1
M - dimension of space
K - number of parameters being fitted
DiffStep- numerical differentiation step;
should not be very small or large;
large = loss of accuracy
small = growth of round-off errors
OUTPUT PARAMETERS:
State - structure which stores algorithm state
-- ALGLIB --
Copyright 18.10.2008 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatewf(const real_2d_array &x, const real_1d_array &y, const real_1d_array &w, const real_1d_array &c, const ae_int_t n, const ae_int_t m, const ae_int_t k, const double diffstep, lsfitstate &state)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitcreatewf(const_cast<alglib_impl::ae_matrix*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), const_cast<alglib_impl::ae_vector*>(c.c_ptr()), n, m, k, diffstep, const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Weighted nonlinear least squares fitting using function values only.
Combination of numerical differentiation and secant updates is used to
obtain function Jacobian.
Nonlinear task min(F(c)) is solved, where
F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2,
* N is a number of points,
* M is a dimension of a space points belong to,
* K is a dimension of a space of parameters being fitted,
* w is an N-dimensional vector of weight coefficients,
* x is a set of N points, each of them is an M-dimensional vector,
* c is a K-dimensional vector of parameters being fitted
This subroutine uses only f(c,x[i]).
INPUT PARAMETERS:
X - array[0..N-1,0..M-1], points (one row = one point)
Y - array[0..N-1], function values.
W - weights, array[0..N-1]
C - array[0..K-1], initial approximation to the solution,
N - number of points, N>1
M - dimension of space
K - number of parameters being fitted
DiffStep- numerical differentiation step;
should not be very small or large;
large = loss of accuracy
small = growth of round-off errors
OUTPUT PARAMETERS:
State - structure which stores algorithm state
-- ALGLIB --
Copyright 18.10.2008 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatewf(const real_2d_array &x, const real_1d_array &y, const real_1d_array &w, const real_1d_array &c, const double diffstep, lsfitstate &state)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t m;
ae_int_t k;
if( (x.rows()!=y.length()) || (x.rows()!=w.length()))
throw ap_error("Error while calling 'lsfitcreatewf': looks like one of arguments has wrong size");
n = x.rows();
m = x.cols();
k = c.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitcreatewf(const_cast<alglib_impl::ae_matrix*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), const_cast<alglib_impl::ae_vector*>(c.c_ptr()), n, m, k, diffstep, const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Nonlinear least squares fitting using function values only.
Combination of numerical differentiation and secant updates is used to
obtain function Jacobian.
Nonlinear task min(F(c)) is solved, where
F(c) = (f(c,x[0])-y[0])^2 + ... + (f(c,x[n-1])-y[n-1])^2,
* N is a number of points,
* M is a dimension of a space points belong to,
* K is a dimension of a space of parameters being fitted,
* w is an N-dimensional vector of weight coefficients,
* x is a set of N points, each of them is an M-dimensional vector,
* c is a K-dimensional vector of parameters being fitted
This subroutine uses only f(c,x[i]).
INPUT PARAMETERS:
X - array[0..N-1,0..M-1], points (one row = one point)
Y - array[0..N-1], function values.
C - array[0..K-1], initial approximation to the solution,
N - number of points, N>1
M - dimension of space
K - number of parameters being fitted
DiffStep- numerical differentiation step;
should not be very small or large;
large = loss of accuracy
small = growth of round-off errors
OUTPUT PARAMETERS:
State - structure which stores algorithm state
-- ALGLIB --
Copyright 18.10.2008 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatef(const real_2d_array &x, const real_1d_array &y, const real_1d_array &c, const ae_int_t n, const ae_int_t m, const ae_int_t k, const double diffstep, lsfitstate &state)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitcreatef(const_cast<alglib_impl::ae_matrix*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(c.c_ptr()), n, m, k, diffstep, const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Nonlinear least squares fitting using function values only.
Combination of numerical differentiation and secant updates is used to
obtain function Jacobian.
Nonlinear task min(F(c)) is solved, where
F(c) = (f(c,x[0])-y[0])^2 + ... + (f(c,x[n-1])-y[n-1])^2,
* N is a number of points,
* M is a dimension of a space points belong to,
* K is a dimension of a space of parameters being fitted,
* w is an N-dimensional vector of weight coefficients,
* x is a set of N points, each of them is an M-dimensional vector,
* c is a K-dimensional vector of parameters being fitted
This subroutine uses only f(c,x[i]).
INPUT PARAMETERS:
X - array[0..N-1,0..M-1], points (one row = one point)
Y - array[0..N-1], function values.
C - array[0..K-1], initial approximation to the solution,
N - number of points, N>1
M - dimension of space
K - number of parameters being fitted
DiffStep- numerical differentiation step;
should not be very small or large;
large = loss of accuracy
small = growth of round-off errors
OUTPUT PARAMETERS:
State - structure which stores algorithm state
-- ALGLIB --
Copyright 18.10.2008 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatef(const real_2d_array &x, const real_1d_array &y, const real_1d_array &c, const double diffstep, lsfitstate &state)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t m;
ae_int_t k;
if( (x.rows()!=y.length()))
throw ap_error("Error while calling 'lsfitcreatef': looks like one of arguments has wrong size");
n = x.rows();
m = x.cols();
k = c.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitcreatef(const_cast<alglib_impl::ae_matrix*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(c.c_ptr()), n, m, k, diffstep, const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Weighted nonlinear least squares fitting using gradient only.
Nonlinear task min(F(c)) is solved, where
F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2,
* N is a number of points,
* M is a dimension of a space points belong to,
* K is a dimension of a space of parameters being fitted,
* w is an N-dimensional vector of weight coefficients,
* x is a set of N points, each of them is an M-dimensional vector,
* c is a K-dimensional vector of parameters being fitted
This subroutine uses only f(c,x[i]) and its gradient.
INPUT PARAMETERS:
X - array[0..N-1,0..M-1], points (one row = one point)
Y - array[0..N-1], function values.
W - weights, array[0..N-1]
C - array[0..K-1], initial approximation to the solution,
N - number of points, N>1
M - dimension of space
K - number of parameters being fitted
CheapFG - boolean flag, which is:
* True if both function and gradient calculation complexity
are less than O(M^2). An improved algorithm can
be used which corresponds to FGJ scheme from
MINLM unit.
* False otherwise.
Standard Jacibian-bases Levenberg-Marquardt algo
will be used (FJ scheme).
OUTPUT PARAMETERS:
State - structure which stores algorithm state
See also:
LSFitResults
LSFitCreateFG (fitting without weights)
LSFitCreateWFGH (fitting using Hessian)
LSFitCreateFGH (fitting using Hessian, without weights)
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatewfg(const real_2d_array &x, const real_1d_array &y, const real_1d_array &w, const real_1d_array &c, const ae_int_t n, const ae_int_t m, const ae_int_t k, const bool cheapfg, lsfitstate &state)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitcreatewfg(const_cast<alglib_impl::ae_matrix*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), const_cast<alglib_impl::ae_vector*>(c.c_ptr()), n, m, k, cheapfg, const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Weighted nonlinear least squares fitting using gradient only.
Nonlinear task min(F(c)) is solved, where
F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2,
* N is a number of points,
* M is a dimension of a space points belong to,
* K is a dimension of a space of parameters being fitted,
* w is an N-dimensional vector of weight coefficients,
* x is a set of N points, each of them is an M-dimensional vector,
* c is a K-dimensional vector of parameters being fitted
This subroutine uses only f(c,x[i]) and its gradient.
INPUT PARAMETERS:
X - array[0..N-1,0..M-1], points (one row = one point)
Y - array[0..N-1], function values.
W - weights, array[0..N-1]
C - array[0..K-1], initial approximation to the solution,
N - number of points, N>1
M - dimension of space
K - number of parameters being fitted
CheapFG - boolean flag, which is:
* True if both function and gradient calculation complexity
are less than O(M^2). An improved algorithm can
be used which corresponds to FGJ scheme from
MINLM unit.
* False otherwise.
Standard Jacibian-bases Levenberg-Marquardt algo
will be used (FJ scheme).
OUTPUT PARAMETERS:
State - structure which stores algorithm state
See also:
LSFitResults
LSFitCreateFG (fitting without weights)
LSFitCreateWFGH (fitting using Hessian)
LSFitCreateFGH (fitting using Hessian, without weights)
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatewfg(const real_2d_array &x, const real_1d_array &y, const real_1d_array &w, const real_1d_array &c, const bool cheapfg, lsfitstate &state)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t m;
ae_int_t k;
if( (x.rows()!=y.length()) || (x.rows()!=w.length()))
throw ap_error("Error while calling 'lsfitcreatewfg': looks like one of arguments has wrong size");
n = x.rows();
m = x.cols();
k = c.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitcreatewfg(const_cast<alglib_impl::ae_matrix*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), const_cast<alglib_impl::ae_vector*>(c.c_ptr()), n, m, k, cheapfg, const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Nonlinear least squares fitting using gradient only, without individual
weights.
Nonlinear task min(F(c)) is solved, where
F(c) = ((f(c,x[0])-y[0]))^2 + ... + ((f(c,x[n-1])-y[n-1]))^2,
* N is a number of points,
* M is a dimension of a space points belong to,
* K is a dimension of a space of parameters being fitted,
* x is a set of N points, each of them is an M-dimensional vector,
* c is a K-dimensional vector of parameters being fitted
This subroutine uses only f(c,x[i]) and its gradient.
INPUT PARAMETERS:
X - array[0..N-1,0..M-1], points (one row = one point)
Y - array[0..N-1], function values.
C - array[0..K-1], initial approximation to the solution,
N - number of points, N>1
M - dimension of space
K - number of parameters being fitted
CheapFG - boolean flag, which is:
* True if both function and gradient calculation complexity
are less than O(M^2). An improved algorithm can
be used which corresponds to FGJ scheme from
MINLM unit.
* False otherwise.
Standard Jacibian-bases Levenberg-Marquardt algo
will be used (FJ scheme).
OUTPUT PARAMETERS:
State - structure which stores algorithm state
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatefg(const real_2d_array &x, const real_1d_array &y, const real_1d_array &c, const ae_int_t n, const ae_int_t m, const ae_int_t k, const bool cheapfg, lsfitstate &state)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitcreatefg(const_cast<alglib_impl::ae_matrix*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(c.c_ptr()), n, m, k, cheapfg, const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Nonlinear least squares fitting using gradient only, without individual
weights.
Nonlinear task min(F(c)) is solved, where
F(c) = ((f(c,x[0])-y[0]))^2 + ... + ((f(c,x[n-1])-y[n-1]))^2,
* N is a number of points,
* M is a dimension of a space points belong to,
* K is a dimension of a space of parameters being fitted,
* x is a set of N points, each of them is an M-dimensional vector,
* c is a K-dimensional vector of parameters being fitted
This subroutine uses only f(c,x[i]) and its gradient.
INPUT PARAMETERS:
X - array[0..N-1,0..M-1], points (one row = one point)
Y - array[0..N-1], function values.
C - array[0..K-1], initial approximation to the solution,
N - number of points, N>1
M - dimension of space
K - number of parameters being fitted
CheapFG - boolean flag, which is:
* True if both function and gradient calculation complexity
are less than O(M^2). An improved algorithm can
be used which corresponds to FGJ scheme from
MINLM unit.
* False otherwise.
Standard Jacibian-bases Levenberg-Marquardt algo
will be used (FJ scheme).
OUTPUT PARAMETERS:
State - structure which stores algorithm state
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatefg(const real_2d_array &x, const real_1d_array &y, const real_1d_array &c, const bool cheapfg, lsfitstate &state)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t m;
ae_int_t k;
if( (x.rows()!=y.length()))
throw ap_error("Error while calling 'lsfitcreatefg': looks like one of arguments has wrong size");
n = x.rows();
m = x.cols();
k = c.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitcreatefg(const_cast<alglib_impl::ae_matrix*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(c.c_ptr()), n, m, k, cheapfg, const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Weighted nonlinear least squares fitting using gradient/Hessian.
Nonlinear task min(F(c)) is solved, where
F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2,
* N is a number of points,
* M is a dimension of a space points belong to,
* K is a dimension of a space of parameters being fitted,
* w is an N-dimensional vector of weight coefficients,
* x is a set of N points, each of them is an M-dimensional vector,
* c is a K-dimensional vector of parameters being fitted
This subroutine uses f(c,x[i]), its gradient and its Hessian.
INPUT PARAMETERS:
X - array[0..N-1,0..M-1], points (one row = one point)
Y - array[0..N-1], function values.
W - weights, array[0..N-1]
C - array[0..K-1], initial approximation to the solution,
N - number of points, N>1
M - dimension of space
K - number of parameters being fitted
OUTPUT PARAMETERS:
State - structure which stores algorithm state
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatewfgh(const real_2d_array &x, const real_1d_array &y, const real_1d_array &w, const real_1d_array &c, const ae_int_t n, const ae_int_t m, const ae_int_t k, lsfitstate &state)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitcreatewfgh(const_cast<alglib_impl::ae_matrix*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), const_cast<alglib_impl::ae_vector*>(c.c_ptr()), n, m, k, const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Weighted nonlinear least squares fitting using gradient/Hessian.
Nonlinear task min(F(c)) is solved, where
F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2,
* N is a number of points,
* M is a dimension of a space points belong to,
* K is a dimension of a space of parameters being fitted,
* w is an N-dimensional vector of weight coefficients,
* x is a set of N points, each of them is an M-dimensional vector,
* c is a K-dimensional vector of parameters being fitted
This subroutine uses f(c,x[i]), its gradient and its Hessian.
INPUT PARAMETERS:
X - array[0..N-1,0..M-1], points (one row = one point)
Y - array[0..N-1], function values.
W - weights, array[0..N-1]
C - array[0..K-1], initial approximation to the solution,
N - number of points, N>1
M - dimension of space
K - number of parameters being fitted
OUTPUT PARAMETERS:
State - structure which stores algorithm state
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatewfgh(const real_2d_array &x, const real_1d_array &y, const real_1d_array &w, const real_1d_array &c, lsfitstate &state)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t m;
ae_int_t k;
if( (x.rows()!=y.length()) || (x.rows()!=w.length()))
throw ap_error("Error while calling 'lsfitcreatewfgh': looks like one of arguments has wrong size");
n = x.rows();
m = x.cols();
k = c.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitcreatewfgh(const_cast<alglib_impl::ae_matrix*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(w.c_ptr()), const_cast<alglib_impl::ae_vector*>(c.c_ptr()), n, m, k, const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Nonlinear least squares fitting using gradient/Hessian, without individial
weights.
Nonlinear task min(F(c)) is solved, where
F(c) = ((f(c,x[0])-y[0]))^2 + ... + ((f(c,x[n-1])-y[n-1]))^2,
* N is a number of points,
* M is a dimension of a space points belong to,
* K is a dimension of a space of parameters being fitted,
* x is a set of N points, each of them is an M-dimensional vector,
* c is a K-dimensional vector of parameters being fitted
This subroutine uses f(c,x[i]), its gradient and its Hessian.
INPUT PARAMETERS:
X - array[0..N-1,0..M-1], points (one row = one point)
Y - array[0..N-1], function values.
C - array[0..K-1], initial approximation to the solution,
N - number of points, N>1
M - dimension of space
K - number of parameters being fitted
OUTPUT PARAMETERS:
State - structure which stores algorithm state
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatefgh(const real_2d_array &x, const real_1d_array &y, const real_1d_array &c, const ae_int_t n, const ae_int_t m, const ae_int_t k, lsfitstate &state)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitcreatefgh(const_cast<alglib_impl::ae_matrix*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(c.c_ptr()), n, m, k, const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Nonlinear least squares fitting using gradient/Hessian, without individial
weights.
Nonlinear task min(F(c)) is solved, where
F(c) = ((f(c,x[0])-y[0]))^2 + ... + ((f(c,x[n-1])-y[n-1]))^2,
* N is a number of points,
* M is a dimension of a space points belong to,
* K is a dimension of a space of parameters being fitted,
* x is a set of N points, each of them is an M-dimensional vector,
* c is a K-dimensional vector of parameters being fitted
This subroutine uses f(c,x[i]), its gradient and its Hessian.
INPUT PARAMETERS:
X - array[0..N-1,0..M-1], points (one row = one point)
Y - array[0..N-1], function values.
C - array[0..K-1], initial approximation to the solution,
N - number of points, N>1
M - dimension of space
K - number of parameters being fitted
OUTPUT PARAMETERS:
State - structure which stores algorithm state
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatefgh(const real_2d_array &x, const real_1d_array &y, const real_1d_array &c, lsfitstate &state)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
ae_int_t m;
ae_int_t k;
if( (x.rows()!=y.length()))
throw ap_error("Error while calling 'lsfitcreatefgh': looks like one of arguments has wrong size");
n = x.rows();
m = x.cols();
k = c.length();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitcreatefgh(const_cast<alglib_impl::ae_matrix*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_vector*>(c.c_ptr()), n, m, k, const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Stopping conditions for nonlinear least squares fitting.
INPUT PARAMETERS:
State - structure which stores algorithm state
EpsF - stopping criterion. Algorithm stops if
|F(k+1)-F(k)| <= EpsF*max{|F(k)|, |F(k+1)|, 1}
EpsX - >=0
The subroutine finishes its work if on k+1-th iteration
the condition |v|<=EpsX is fulfilled, where:
* |.| means Euclidian norm
* v - scaled step vector, v[i]=dx[i]/s[i]
* dx - ste pvector, dx=X(k+1)-X(k)
* s - scaling coefficients set by LSFitSetScale()
MaxIts - maximum number of iterations. If MaxIts=0, the number of
iterations is unlimited. Only Levenberg-Marquardt
iterations are counted (L-BFGS/CG iterations are NOT
counted because their cost is very low compared to that of
LM).
NOTE
Passing EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic
stopping criterion selection (according to the scheme used by MINLM unit).
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitsetcond(const lsfitstate &state, const double epsf, const double epsx, const ae_int_t maxits)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitsetcond(const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), epsf, epsx, maxits, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function sets maximum step length
INPUT PARAMETERS:
State - structure which stores algorithm state
StpMax - maximum step length, >=0. Set StpMax to 0.0, if you don't
want to limit step length.
Use this subroutine when you optimize target function which contains exp()
or other fast growing functions, and optimization algorithm makes too
large steps which leads to overflow. This function allows us to reject
steps that are too large (and therefore expose us to the possible
overflow) without actually calculating function value at the x+stp*d.
NOTE: non-zero StpMax leads to moderate performance degradation because
intermediate step of preconditioned L-BFGS optimization is incompatible
with limits on step size.
-- ALGLIB --
Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void lsfitsetstpmax(const lsfitstate &state, const double stpmax)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitsetstpmax(const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), stpmax, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function turns on/off reporting.
INPUT PARAMETERS:
State - structure which stores algorithm state
NeedXRep- whether iteration reports are needed or not
When reports are needed, State.C (current parameters) and State.F (current
value of fitting function) are reported.
-- ALGLIB --
Copyright 15.08.2010 by Bochkanov Sergey
*************************************************************************/
void lsfitsetxrep(const lsfitstate &state, const bool needxrep)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitsetxrep(const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), needxrep, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function sets scaling coefficients for underlying optimizer.
ALGLIB optimizers use scaling matrices to test stopping conditions (step
size and gradient are scaled before comparison with tolerances). Scale of
the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function
Generally, scale is NOT considered to be a form of preconditioner. But LM
optimizer is unique in that it uses scaling matrix both in the stopping
condition tests and as Marquardt damping factor.
Proper scaling is very important for the algorithm performance. It is less
important for the quality of results, but still has some influence (it is
easier to converge when variables are properly scaled, so premature
stopping is possible when very badly scalled variables are combined with
relaxed stopping conditions).
INPUT PARAMETERS:
State - structure stores algorithm state
S - array[N], non-zero scaling coefficients
S[i] may be negative, sign doesn't matter.
-- ALGLIB --
Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void lsfitsetscale(const lsfitstate &state, const real_1d_array &s)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitsetscale(const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), const_cast<alglib_impl::ae_vector*>(s.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function sets boundary constraints for underlying optimizer
Boundary constraints are inactive by default (after initial creation).
They are preserved until explicitly turned off with another SetBC() call.
INPUT PARAMETERS:
State - structure stores algorithm state
BndL - lower bounds, array[K].
If some (all) variables are unbounded, you may specify
very small number or -INF (latter is recommended because
it will allow solver to use better algorithm).
BndU - upper bounds, array[K].
If some (all) variables are unbounded, you may specify
very large number or +INF (latter is recommended because
it will allow solver to use better algorithm).
NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th
variable will be "frozen" at X[i]=BndL[i]=BndU[i].
NOTE 2: unlike other constrained optimization algorithms, this solver has
following useful properties:
* bound constraints are always satisfied exactly
* function is evaluated only INSIDE area specified by bound constraints
-- ALGLIB --
Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void lsfitsetbc(const lsfitstate &state, const real_1d_array &bndl, const real_1d_array &bndu)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitsetbc(const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), const_cast<alglib_impl::ae_vector*>(bndl.c_ptr()), const_cast<alglib_impl::ae_vector*>(bndu.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function provides reverse communication interface
Reverse communication interface is not documented or recommended to use.
See below for functions which provide better documented API
*************************************************************************/
bool lsfititeration(const lsfitstate &state)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
ae_bool result = alglib_impl::lsfititeration(const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return *(reinterpret_cast<bool*>(&result));
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
void lsfitfit(lsfitstate &state,
void (*func)(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr),
void (*rep)(const real_1d_array &c, double func, void *ptr),
void *ptr)
{
alglib_impl::ae_state _alglib_env_state;
if( func==NULL )
throw ap_error("ALGLIB: error in 'lsfitfit()' (func is NULL)");
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
while( alglib_impl::lsfititeration(state.c_ptr(), &_alglib_env_state) )
{
if( state.needf )
{
func(state.c, state.x, state.f, ptr);
continue;
}
if( state.xupdated )
{
if( rep!=NULL )
rep(state.c, state.f, ptr);
continue;
}
throw ap_error("ALGLIB: error in 'lsfitfit' (some derivatives were not provided?)");
}
alglib_impl::ae_state_clear(&_alglib_env_state);
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
void lsfitfit(lsfitstate &state,
void (*func)(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr),
void (*grad)(const real_1d_array &c, const real_1d_array &x, double &func, real_1d_array &grad, void *ptr),
void (*rep)(const real_1d_array &c, double func, void *ptr),
void *ptr)
{
alglib_impl::ae_state _alglib_env_state;
if( func==NULL )
throw ap_error("ALGLIB: error in 'lsfitfit()' (func is NULL)");
if( grad==NULL )
throw ap_error("ALGLIB: error in 'lsfitfit()' (grad is NULL)");
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
while( alglib_impl::lsfititeration(state.c_ptr(), &_alglib_env_state) )
{
if( state.needf )
{
func(state.c, state.x, state.f, ptr);
continue;
}
if( state.needfg )
{
grad(state.c, state.x, state.f, state.g, ptr);
continue;
}
if( state.xupdated )
{
if( rep!=NULL )
rep(state.c, state.f, ptr);
continue;
}
throw ap_error("ALGLIB: error in 'lsfitfit' (some derivatives were not provided?)");
}
alglib_impl::ae_state_clear(&_alglib_env_state);
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
void lsfitfit(lsfitstate &state,
void (*func)(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr),
void (*grad)(const real_1d_array &c, const real_1d_array &x, double &func, real_1d_array &grad, void *ptr),
void (*hess)(const real_1d_array &c, const real_1d_array &x, double &func, real_1d_array &grad, real_2d_array &hess, void *ptr),
void (*rep)(const real_1d_array &c, double func, void *ptr),
void *ptr)
{
alglib_impl::ae_state _alglib_env_state;
if( func==NULL )
throw ap_error("ALGLIB: error in 'lsfitfit()' (func is NULL)");
if( grad==NULL )
throw ap_error("ALGLIB: error in 'lsfitfit()' (grad is NULL)");
if( hess==NULL )
throw ap_error("ALGLIB: error in 'lsfitfit()' (hess is NULL)");
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
while( alglib_impl::lsfititeration(state.c_ptr(), &_alglib_env_state) )
{
if( state.needf )
{
func(state.c, state.x, state.f, ptr);
continue;
}
if( state.needfg )
{
grad(state.c, state.x, state.f, state.g, ptr);
continue;
}
if( state.needfgh )
{
hess(state.c, state.x, state.f, state.g, state.h, ptr);
continue;
}
if( state.xupdated )
{
if( rep!=NULL )
rep(state.c, state.f, ptr);
continue;
}
throw ap_error("ALGLIB: error in 'lsfitfit' (some derivatives were not provided?)");
}
alglib_impl::ae_state_clear(&_alglib_env_state);
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Nonlinear least squares fitting results.
Called after return from LSFitFit().
INPUT PARAMETERS:
State - algorithm state
OUTPUT PARAMETERS:
Info - completion code:
* -7 gradient verification failed.
See LSFitSetGradientCheck() for more information.
* 1 relative function improvement is no more than
EpsF.
* 2 relative step is no more than EpsX.
* 4 gradient norm is no more than EpsG
* 5 MaxIts steps was taken
* 7 stopping conditions are too stringent,
further improvement is impossible
C - array[0..K-1], solution
Rep - optimization report. On success following fields are set:
* R2 non-adjusted coefficient of determination
(non-weighted)
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
* WRMSError weighted rms error on the (X,Y).
ERRORS IN PARAMETERS
This solver also calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar covariance matrix for parameters, array[K,K].
* Rep.ErrPar errors in parameters, array[K],
errpar = sqrt(diag(CovPar))
* Rep.ErrCurve vector of fit errors - standard deviations of empirical
best-fit curve from "ideal" best-fit curve built with
infinite number of samples, array[N].
errcurve = sqrt(diag(J*CovPar*J')),
where J is Jacobian matrix.
* Rep.Noise vector of per-point estimates of noise, array[N]
IMPORTANT: errors in parameters are calculated without taking into
account boundary/linear constraints! Presence of constraints
changes distribution of errors, but there is no easy way to
account for constraints when you calculate covariance matrix.
NOTE: noise in the data is estimated as follows:
* for fitting without user-supplied weights all points are
assumed to have same level of noise, which is estimated from
the data
* for fitting with user-supplied weights we assume that noise
level in I-th point is inversely proportional to Ith weight.
Coefficient of proportionality is estimated from the data.
NOTE: we apply small amount of regularization when we invert squared
Jacobian and calculate covariance matrix. It guarantees that
algorithm won't divide by zero during inversion, but skews
error estimates a bit (fractional error is about 10^-9).
However, we believe that this difference is insignificant for
all practical purposes except for the situation when you want
to compare ALGLIB results with "reference" implementation up
to the last significant digit.
NOTE: covariance matrix is estimated using correction for degrees
of freedom (covariances are divided by N-M instead of dividing
by N).
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitresults(const lsfitstate &state, ae_int_t &info, real_1d_array &c, lsfitreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitresults(const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), &info, const_cast<alglib_impl::ae_vector*>(c.c_ptr()), const_cast<alglib_impl::lsfitreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine turns on verification of the user-supplied analytic
gradient:
* user calls this subroutine before fitting begins
* LSFitFit() is called
* prior to actual fitting, for each point in data set X_i and each
component of parameters being fited C_j algorithm performs following
steps:
* two trial steps are made to C_j-TestStep*S[j] and C_j+TestStep*S[j],
where C_j is j-th parameter and S[j] is a scale of j-th parameter
* if needed, steps are bounded with respect to constraints on C[]
* F(X_i|C) is evaluated at these trial points
* we perform one more evaluation in the middle point of the interval
* we build cubic model using function values and derivatives at trial
points and we compare its prediction with actual value in the middle
point
* in case difference between prediction and actual value is higher than
some predetermined threshold, algorithm stops with completion code -7;
Rep.VarIdx is set to index of the parameter with incorrect derivative.
* after verification is over, algorithm proceeds to the actual optimization.
NOTE 1: verification needs N*K (points count * parameters count) gradient
evaluations. It is very costly and you should use it only for low
dimensional problems, when you want to be sure that you've
correctly calculated analytic derivatives. You should not use it
in the production code (unless you want to check derivatives
provided by some third party).
NOTE 2: you should carefully choose TestStep. Value which is too large
(so large that function behaviour is significantly non-cubic) will
lead to false alarms. You may use different step for different
parameters by means of setting scale with LSFitSetScale().
NOTE 3: this function may lead to false positives. In case it reports that
I-th derivative was calculated incorrectly, you may decrease test
step and try one more time - maybe your function changes too
sharply and your step is too large for such rapidly chanding
function.
NOTE 4: this function works only for optimizers created with LSFitCreateWFG()
or LSFitCreateFG() constructors.
INPUT PARAMETERS:
State - structure used to store algorithm state
TestStep - verification step:
* TestStep=0 turns verification off
* TestStep>0 activates verification
-- ALGLIB --
Copyright 15.06.2012 by Bochkanov Sergey
*************************************************************************/
void lsfitsetgradientcheck(const lsfitstate &state, const double teststep)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::lsfitsetgradientcheck(const_cast<alglib_impl::lsfitstate*>(state.c_ptr()), teststep, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Parametric spline inteprolant: 2-dimensional curve.
You should not try to access its members directly - use PSpline2XXXXXXXX()
functions instead.
*************************************************************************/
_pspline2interpolant_owner::_pspline2interpolant_owner()
{
p_struct = (alglib_impl::pspline2interpolant*)alglib_impl::ae_malloc(sizeof(alglib_impl::pspline2interpolant), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_pspline2interpolant_init(p_struct, NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_pspline2interpolant_owner::_pspline2interpolant_owner(const _pspline2interpolant_owner &rhs)
{
p_struct = (alglib_impl::pspline2interpolant*)alglib_impl::ae_malloc(sizeof(alglib_impl::pspline2interpolant), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_pspline2interpolant_init_copy(p_struct, const_cast<alglib_impl::pspline2interpolant*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_pspline2interpolant_owner& _pspline2interpolant_owner::operator=(const _pspline2interpolant_owner &rhs)
{
if( this==&rhs )
return *this;
alglib_impl::_pspline2interpolant_clear(p_struct);
if( !alglib_impl::_pspline2interpolant_init_copy(p_struct, const_cast<alglib_impl::pspline2interpolant*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
return *this;
}
_pspline2interpolant_owner::~_pspline2interpolant_owner()
{
alglib_impl::_pspline2interpolant_clear(p_struct);
ae_free(p_struct);
}
alglib_impl::pspline2interpolant* _pspline2interpolant_owner::c_ptr()
{
return p_struct;
}
alglib_impl::pspline2interpolant* _pspline2interpolant_owner::c_ptr() const
{
return const_cast<alglib_impl::pspline2interpolant*>(p_struct);
}
pspline2interpolant::pspline2interpolant() : _pspline2interpolant_owner()
{
}
pspline2interpolant::pspline2interpolant(const pspline2interpolant &rhs):_pspline2interpolant_owner(rhs)
{
}
pspline2interpolant& pspline2interpolant::operator=(const pspline2interpolant &rhs)
{
if( this==&rhs )
return *this;
_pspline2interpolant_owner::operator=(rhs);
return *this;
}
pspline2interpolant::~pspline2interpolant()
{
}
/*************************************************************************
Parametric spline inteprolant: 3-dimensional curve.
You should not try to access its members directly - use PSpline3XXXXXXXX()
functions instead.
*************************************************************************/
_pspline3interpolant_owner::_pspline3interpolant_owner()
{
p_struct = (alglib_impl::pspline3interpolant*)alglib_impl::ae_malloc(sizeof(alglib_impl::pspline3interpolant), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_pspline3interpolant_init(p_struct, NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_pspline3interpolant_owner::_pspline3interpolant_owner(const _pspline3interpolant_owner &rhs)
{
p_struct = (alglib_impl::pspline3interpolant*)alglib_impl::ae_malloc(sizeof(alglib_impl::pspline3interpolant), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_pspline3interpolant_init_copy(p_struct, const_cast<alglib_impl::pspline3interpolant*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_pspline3interpolant_owner& _pspline3interpolant_owner::operator=(const _pspline3interpolant_owner &rhs)
{
if( this==&rhs )
return *this;
alglib_impl::_pspline3interpolant_clear(p_struct);
if( !alglib_impl::_pspline3interpolant_init_copy(p_struct, const_cast<alglib_impl::pspline3interpolant*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
return *this;
}
_pspline3interpolant_owner::~_pspline3interpolant_owner()
{
alglib_impl::_pspline3interpolant_clear(p_struct);
ae_free(p_struct);
}
alglib_impl::pspline3interpolant* _pspline3interpolant_owner::c_ptr()
{
return p_struct;
}
alglib_impl::pspline3interpolant* _pspline3interpolant_owner::c_ptr() const
{
return const_cast<alglib_impl::pspline3interpolant*>(p_struct);
}
pspline3interpolant::pspline3interpolant() : _pspline3interpolant_owner()
{
}
pspline3interpolant::pspline3interpolant(const pspline3interpolant &rhs):_pspline3interpolant_owner(rhs)
{
}
pspline3interpolant& pspline3interpolant::operator=(const pspline3interpolant &rhs)
{
if( this==&rhs )
return *this;
_pspline3interpolant_owner::operator=(rhs);
return *this;
}
pspline3interpolant::~pspline3interpolant()
{
}
/*************************************************************************
This function builds non-periodic 2-dimensional parametric spline which
starts at (X[0],Y[0]) and ends at (X[N-1],Y[N-1]).
INPUT PARAMETERS:
XY - points, array[0..N-1,0..1].
XY[I,0:1] corresponds to the Ith point.
Order of points is important!
N - points count, N>=5 for Akima splines, N>=2 for other types of
splines.
ST - spline type:
* 0 Akima spline
* 1 parabolically terminated Catmull-Rom spline (Tension=0)
* 2 parabolically terminated cubic spline
PT - parameterization type:
* 0 uniform
* 1 chord length
* 2 centripetal
OUTPUT PARAMETERS:
P - parametric spline interpolant
NOTES:
* this function assumes that there all consequent points are distinct.
I.e. (x0,y0)<>(x1,y1), (x1,y1)<>(x2,y2), (x2,y2)<>(x3,y3) and so on.
However, non-consequent points may coincide, i.e. we can have (x0,y0)=
=(x2,y2).
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2build(const real_2d_array &xy, const ae_int_t n, const ae_int_t st, const ae_int_t pt, pspline2interpolant &p)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::pspline2build(const_cast<alglib_impl::ae_matrix*>(xy.c_ptr()), n, st, pt, const_cast<alglib_impl::pspline2interpolant*>(p.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function builds non-periodic 3-dimensional parametric spline which
starts at (X[0],Y[0],Z[0]) and ends at (X[N-1],Y[N-1],Z[N-1]).
Same as PSpline2Build() function, but for 3D, so we won't duplicate its
description here.
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3build(const real_2d_array &xy, const ae_int_t n, const ae_int_t st, const ae_int_t pt, pspline3interpolant &p)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::pspline3build(const_cast<alglib_impl::ae_matrix*>(xy.c_ptr()), n, st, pt, const_cast<alglib_impl::pspline3interpolant*>(p.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function builds periodic 2-dimensional parametric spline which
starts at (X[0],Y[0]), goes through all points to (X[N-1],Y[N-1]) and then
back to (X[0],Y[0]).
INPUT PARAMETERS:
XY - points, array[0..N-1,0..1].
XY[I,0:1] corresponds to the Ith point.
XY[N-1,0:1] must be different from XY[0,0:1].
Order of points is important!
N - points count, N>=3 for other types of splines.
ST - spline type:
* 1 Catmull-Rom spline (Tension=0) with cyclic boundary conditions
* 2 cubic spline with cyclic boundary conditions
PT - parameterization type:
* 0 uniform
* 1 chord length
* 2 centripetal
OUTPUT PARAMETERS:
P - parametric spline interpolant
NOTES:
* this function assumes that there all consequent points are distinct.
I.e. (x0,y0)<>(x1,y1), (x1,y1)<>(x2,y2), (x2,y2)<>(x3,y3) and so on.
However, non-consequent points may coincide, i.e. we can have (x0,y0)=
=(x2,y2).
* last point of sequence is NOT equal to the first point. You shouldn't
make curve "explicitly periodic" by making them equal.
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2buildperiodic(const real_2d_array &xy, const ae_int_t n, const ae_int_t st, const ae_int_t pt, pspline2interpolant &p)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::pspline2buildperiodic(const_cast<alglib_impl::ae_matrix*>(xy.c_ptr()), n, st, pt, const_cast<alglib_impl::pspline2interpolant*>(p.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function builds periodic 3-dimensional parametric spline which
starts at (X[0],Y[0],Z[0]), goes through all points to (X[N-1],Y[N-1],Z[N-1])
and then back to (X[0],Y[0],Z[0]).
Same as PSpline2Build() function, but for 3D, so we won't duplicate its
description here.
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3buildperiodic(const real_2d_array &xy, const ae_int_t n, const ae_int_t st, const ae_int_t pt, pspline3interpolant &p)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::pspline3buildperiodic(const_cast<alglib_impl::ae_matrix*>(xy.c_ptr()), n, st, pt, const_cast<alglib_impl::pspline3interpolant*>(p.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function returns vector of parameter values correspoding to points.
I.e. for P created from (X[0],Y[0])...(X[N-1],Y[N-1]) and U=TValues(P) we
have
(X[0],Y[0]) = PSpline2Calc(P,U[0]),
(X[1],Y[1]) = PSpline2Calc(P,U[1]),
(X[2],Y[2]) = PSpline2Calc(P,U[2]),
...
INPUT PARAMETERS:
P - parametric spline interpolant
OUTPUT PARAMETERS:
N - array size
T - array[0..N-1]
NOTES:
* for non-periodic splines U[0]=0, U[0]<U[1]<...<U[N-1], U[N-1]=1
* for periodic splines U[0]=0, U[0]<U[1]<...<U[N-1], U[N-1]<1
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2parametervalues(const pspline2interpolant &p, ae_int_t &n, real_1d_array &t)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::pspline2parametervalues(const_cast<alglib_impl::pspline2interpolant*>(p.c_ptr()), &n, const_cast<alglib_impl::ae_vector*>(t.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function returns vector of parameter values correspoding to points.
Same as PSpline2ParameterValues(), but for 3D.
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3parametervalues(const pspline3interpolant &p, ae_int_t &n, real_1d_array &t)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::pspline3parametervalues(const_cast<alglib_impl::pspline3interpolant*>(p.c_ptr()), &n, const_cast<alglib_impl::ae_vector*>(t.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function calculates the value of the parametric spline for a given
value of parameter T
INPUT PARAMETERS:
P - parametric spline interpolant
T - point:
* T in [0,1] corresponds to interval spanned by points
* for non-periodic splines T<0 (or T>1) correspond to parts of
the curve before the first (after the last) point
* for periodic splines T<0 (or T>1) are projected into [0,1]
by making T=T-floor(T).
OUTPUT PARAMETERS:
X - X-position
Y - Y-position
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2calc(const pspline2interpolant &p, const double t, double &x, double &y)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::pspline2calc(const_cast<alglib_impl::pspline2interpolant*>(p.c_ptr()), t, &x, &y, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function calculates the value of the parametric spline for a given
value of parameter T.
INPUT PARAMETERS:
P - parametric spline interpolant
T - point:
* T in [0,1] corresponds to interval spanned by points
* for non-periodic splines T<0 (or T>1) correspond to parts of
the curve before the first (after the last) point
* for periodic splines T<0 (or T>1) are projected into [0,1]
by making T=T-floor(T).
OUTPUT PARAMETERS:
X - X-position
Y - Y-position
Z - Z-position
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3calc(const pspline3interpolant &p, const double t, double &x, double &y, double &z)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::pspline3calc(const_cast<alglib_impl::pspline3interpolant*>(p.c_ptr()), t, &x, &y, &z, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function calculates tangent vector for a given value of parameter T
INPUT PARAMETERS:
P - parametric spline interpolant
T - point:
* T in [0,1] corresponds to interval spanned by points
* for non-periodic splines T<0 (or T>1) correspond to parts of
the curve before the first (after the last) point
* for periodic splines T<0 (or T>1) are projected into [0,1]
by making T=T-floor(T).
OUTPUT PARAMETERS:
X - X-component of tangent vector (normalized)
Y - Y-component of tangent vector (normalized)
NOTE:
X^2+Y^2 is either 1 (for non-zero tangent vector) or 0.
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2tangent(const pspline2interpolant &p, const double t, double &x, double &y)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::pspline2tangent(const_cast<alglib_impl::pspline2interpolant*>(p.c_ptr()), t, &x, &y, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function calculates tangent vector for a given value of parameter T
INPUT PARAMETERS:
P - parametric spline interpolant
T - point:
* T in [0,1] corresponds to interval spanned by points
* for non-periodic splines T<0 (or T>1) correspond to parts of
the curve before the first (after the last) point
* for periodic splines T<0 (or T>1) are projected into [0,1]
by making T=T-floor(T).
OUTPUT PARAMETERS:
X - X-component of tangent vector (normalized)
Y - Y-component of tangent vector (normalized)
Z - Z-component of tangent vector (normalized)
NOTE:
X^2+Y^2+Z^2 is either 1 (for non-zero tangent vector) or 0.
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3tangent(const pspline3interpolant &p, const double t, double &x, double &y, double &z)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::pspline3tangent(const_cast<alglib_impl::pspline3interpolant*>(p.c_ptr()), t, &x, &y, &z, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function calculates derivative, i.e. it returns (dX/dT,dY/dT).
INPUT PARAMETERS:
P - parametric spline interpolant
T - point:
* T in [0,1] corresponds to interval spanned by points
* for non-periodic splines T<0 (or T>1) correspond to parts of
the curve before the first (after the last) point
* for periodic splines T<0 (or T>1) are projected into [0,1]
by making T=T-floor(T).
OUTPUT PARAMETERS:
X - X-value
DX - X-derivative
Y - Y-value
DY - Y-derivative
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2diff(const pspline2interpolant &p, const double t, double &x, double &dx, double &y, double &dy)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::pspline2diff(const_cast<alglib_impl::pspline2interpolant*>(p.c_ptr()), t, &x, &dx, &y, &dy, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function calculates derivative, i.e. it returns (dX/dT,dY/dT,dZ/dT).
INPUT PARAMETERS:
P - parametric spline interpolant
T - point:
* T in [0,1] corresponds to interval spanned by points
* for non-periodic splines T<0 (or T>1) correspond to parts of
the curve before the first (after the last) point
* for periodic splines T<0 (or T>1) are projected into [0,1]
by making T=T-floor(T).
OUTPUT PARAMETERS:
X - X-value
DX - X-derivative
Y - Y-value
DY - Y-derivative
Z - Z-value
DZ - Z-derivative
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3diff(const pspline3interpolant &p, const double t, double &x, double &dx, double &y, double &dy, double &z, double &dz)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::pspline3diff(const_cast<alglib_impl::pspline3interpolant*>(p.c_ptr()), t, &x, &dx, &y, &dy, &z, &dz, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function calculates first and second derivative with respect to T.
INPUT PARAMETERS:
P - parametric spline interpolant
T - point:
* T in [0,1] corresponds to interval spanned by points
* for non-periodic splines T<0 (or T>1) correspond to parts of
the curve before the first (after the last) point
* for periodic splines T<0 (or T>1) are projected into [0,1]
by making T=T-floor(T).
OUTPUT PARAMETERS:
X - X-value
DX - derivative
D2X - second derivative
Y - Y-value
DY - derivative
D2Y - second derivative
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2diff2(const pspline2interpolant &p, const double t, double &x, double &dx, double &d2x, double &y, double &dy, double &d2y)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::pspline2diff2(const_cast<alglib_impl::pspline2interpolant*>(p.c_ptr()), t, &x, &dx, &d2x, &y, &dy, &d2y, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function calculates first and second derivative with respect to T.
INPUT PARAMETERS:
P - parametric spline interpolant
T - point:
* T in [0,1] corresponds to interval spanned by points
* for non-periodic splines T<0 (or T>1) correspond to parts of
the curve before the first (after the last) point
* for periodic splines T<0 (or T>1) are projected into [0,1]
by making T=T-floor(T).
OUTPUT PARAMETERS:
X - X-value
DX - derivative
D2X - second derivative
Y - Y-value
DY - derivative
D2Y - second derivative
Z - Z-value
DZ - derivative
D2Z - second derivative
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3diff2(const pspline3interpolant &p, const double t, double &x, double &dx, double &d2x, double &y, double &dy, double &d2y, double &z, double &dz, double &d2z)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::pspline3diff2(const_cast<alglib_impl::pspline3interpolant*>(p.c_ptr()), t, &x, &dx, &d2x, &y, &dy, &d2y, &z, &dz, &d2z, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function calculates arc length, i.e. length of curve between t=a
and t=b.
INPUT PARAMETERS:
P - parametric spline interpolant
A,B - parameter values corresponding to arc ends:
* B>A will result in positive length returned
* B<A will result in negative length returned
RESULT:
length of arc starting at T=A and ending at T=B.
-- ALGLIB PROJECT --
Copyright 30.05.2010 by Bochkanov Sergey
*************************************************************************/
double pspline2arclength(const pspline2interpolant &p, const double a, const double b)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
double result = alglib_impl::pspline2arclength(const_cast<alglib_impl::pspline2interpolant*>(p.c_ptr()), a, b, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return *(reinterpret_cast<double*>(&result));
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function calculates arc length, i.e. length of curve between t=a
and t=b.
INPUT PARAMETERS:
P - parametric spline interpolant
A,B - parameter values corresponding to arc ends:
* B>A will result in positive length returned
* B<A will result in negative length returned
RESULT:
length of arc starting at T=A and ending at T=B.
-- ALGLIB PROJECT --
Copyright 30.05.2010 by Bochkanov Sergey
*************************************************************************/
double pspline3arclength(const pspline3interpolant &p, const double a, const double b)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
double result = alglib_impl::pspline3arclength(const_cast<alglib_impl::pspline3interpolant*>(p.c_ptr()), a, b, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return *(reinterpret_cast<double*>(&result));
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
RBF model.
Never try to directly work with fields of this object - always use ALGLIB
functions to use this object.
*************************************************************************/
_rbfmodel_owner::_rbfmodel_owner()
{
p_struct = (alglib_impl::rbfmodel*)alglib_impl::ae_malloc(sizeof(alglib_impl::rbfmodel), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_rbfmodel_init(p_struct, NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_rbfmodel_owner::_rbfmodel_owner(const _rbfmodel_owner &rhs)
{
p_struct = (alglib_impl::rbfmodel*)alglib_impl::ae_malloc(sizeof(alglib_impl::rbfmodel), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_rbfmodel_init_copy(p_struct, const_cast<alglib_impl::rbfmodel*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_rbfmodel_owner& _rbfmodel_owner::operator=(const _rbfmodel_owner &rhs)
{
if( this==&rhs )
return *this;
alglib_impl::_rbfmodel_clear(p_struct);
if( !alglib_impl::_rbfmodel_init_copy(p_struct, const_cast<alglib_impl::rbfmodel*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
return *this;
}
_rbfmodel_owner::~_rbfmodel_owner()
{
alglib_impl::_rbfmodel_clear(p_struct);
ae_free(p_struct);
}
alglib_impl::rbfmodel* _rbfmodel_owner::c_ptr()
{
return p_struct;
}
alglib_impl::rbfmodel* _rbfmodel_owner::c_ptr() const
{
return const_cast<alglib_impl::rbfmodel*>(p_struct);
}
rbfmodel::rbfmodel() : _rbfmodel_owner()
{
}
rbfmodel::rbfmodel(const rbfmodel &rhs):_rbfmodel_owner(rhs)
{
}
rbfmodel& rbfmodel::operator=(const rbfmodel &rhs)
{
if( this==&rhs )
return *this;
_rbfmodel_owner::operator=(rhs);
return *this;
}
rbfmodel::~rbfmodel()
{
}
/*************************************************************************
RBF solution report:
* TerminationType - termination type, positive values - success,
non-positive - failure.
*************************************************************************/
_rbfreport_owner::_rbfreport_owner()
{
p_struct = (alglib_impl::rbfreport*)alglib_impl::ae_malloc(sizeof(alglib_impl::rbfreport), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_rbfreport_init(p_struct, NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_rbfreport_owner::_rbfreport_owner(const _rbfreport_owner &rhs)
{
p_struct = (alglib_impl::rbfreport*)alglib_impl::ae_malloc(sizeof(alglib_impl::rbfreport), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_rbfreport_init_copy(p_struct, const_cast<alglib_impl::rbfreport*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_rbfreport_owner& _rbfreport_owner::operator=(const _rbfreport_owner &rhs)
{
if( this==&rhs )
return *this;
alglib_impl::_rbfreport_clear(p_struct);
if( !alglib_impl::_rbfreport_init_copy(p_struct, const_cast<alglib_impl::rbfreport*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
return *this;
}
_rbfreport_owner::~_rbfreport_owner()
{
alglib_impl::_rbfreport_clear(p_struct);
ae_free(p_struct);
}
alglib_impl::rbfreport* _rbfreport_owner::c_ptr()
{
return p_struct;
}
alglib_impl::rbfreport* _rbfreport_owner::c_ptr() const
{
return const_cast<alglib_impl::rbfreport*>(p_struct);
}
rbfreport::rbfreport() : _rbfreport_owner() ,arows(p_struct->arows),acols(p_struct->acols),annz(p_struct->annz),iterationscount(p_struct->iterationscount),nmv(p_struct->nmv),terminationtype(p_struct->terminationtype)
{
}
rbfreport::rbfreport(const rbfreport &rhs):_rbfreport_owner(rhs) ,arows(p_struct->arows),acols(p_struct->acols),annz(p_struct->annz),iterationscount(p_struct->iterationscount),nmv(p_struct->nmv),terminationtype(p_struct->terminationtype)
{
}
rbfreport& rbfreport::operator=(const rbfreport &rhs)
{
if( this==&rhs )
return *this;
_rbfreport_owner::operator=(rhs);
return *this;
}
rbfreport::~rbfreport()
{
}
/*************************************************************************
This function serializes data structure to string.
Important properties of s_out:
* it contains alphanumeric characters, dots, underscores, minus signs
* these symbols are grouped into words, which are separated by spaces
and Windows-style (CR+LF) newlines
* although serializer uses spaces and CR+LF as separators, you can
replace any separator character by arbitrary combination of spaces,
tabs, Windows or Unix newlines. It allows flexible reformatting of
the string in case you want to include it into text or XML file.
But you should not insert separators into the middle of the "words"
nor you should change case of letters.
* s_out can be freely moved between 32-bit and 64-bit systems, little
and big endian machines, and so on. You can serialize structure on
32-bit machine and unserialize it on 64-bit one (or vice versa), or
serialize it on SPARC and unserialize on x86. You can also
serialize it in C++ version of ALGLIB and unserialize in C# one,
and vice versa.
*************************************************************************/
void rbfserialize(rbfmodel &obj, std::string &s_out)
{
alglib_impl::ae_state state;
alglib_impl::ae_serializer serializer;
alglib_impl::ae_int_t ssize;
alglib_impl::ae_state_init(&state);
try
{
alglib_impl::ae_serializer_init(&serializer);
alglib_impl::ae_serializer_alloc_start(&serializer);
alglib_impl::rbfalloc(&serializer, obj.c_ptr(), &state);
ssize = alglib_impl::ae_serializer_get_alloc_size(&serializer);
s_out.clear();
s_out.reserve((size_t)(ssize+1));
alglib_impl::ae_serializer_sstart_str(&serializer, &s_out);
alglib_impl::rbfserialize(&serializer, obj.c_ptr(), &state);
alglib_impl::ae_serializer_stop(&serializer);
if( s_out.length()>(size_t)ssize )
throw ap_error("ALGLIB: serialization integrity error");
alglib_impl::ae_serializer_clear(&serializer);
alglib_impl::ae_state_clear(&state);
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(state.error_msg);
}
}
/*************************************************************************
This function unserializes data structure from string.
*************************************************************************/
void rbfunserialize(std::string &s_in, rbfmodel &obj)
{
alglib_impl::ae_state state;
alglib_impl::ae_serializer serializer;
alglib_impl::ae_state_init(&state);
try
{
alglib_impl::ae_serializer_init(&serializer);
alglib_impl::ae_serializer_ustart_str(&serializer, &s_in);
alglib_impl::rbfunserialize(&serializer, obj.c_ptr(), &state);
alglib_impl::ae_serializer_stop(&serializer);
alglib_impl::ae_serializer_clear(&serializer);
alglib_impl::ae_state_clear(&state);
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(state.error_msg);
}
}
/*************************************************************************
This function creates RBF model for a scalar (NY=1) or vector (NY>1)
function in a NX-dimensional space (NX=2 or NX=3).
Newly created model is empty. It can be used for interpolation right after
creation, but it just returns zeros. You have to add points to the model,
tune interpolation settings, and then call model construction function
RBFBuildModel() which will update model according to your specification.
USAGE:
1. User creates model with RBFCreate()
2. User adds dataset with RBFSetPoints() (points do NOT have to be on a
regular grid)
3. (OPTIONAL) User chooses polynomial term by calling:
* RBFLinTerm() to set linear term
* RBFConstTerm() to set constant term
* RBFZeroTerm() to set zero term
By default, linear term is used.
4. User chooses specific RBF algorithm to use: either QNN (RBFSetAlgoQNN)
or ML (RBFSetAlgoMultiLayer).
5. User calls RBFBuildModel() function which rebuilds model according to
the specification
6. User may call RBFCalc() to calculate model value at the specified point,
RBFGridCalc() to calculate model values at the points of the regular
grid. User may extract model coefficients with RBFUnpack() call.
INPUT PARAMETERS:
NX - dimension of the space, NX=2 or NX=3
NY - function dimension, NY>=1
OUTPUT PARAMETERS:
S - RBF model (initially equals to zero)
NOTE 1: memory requirements. RBF models require amount of memory which is
proportional to the number of data points. Memory is allocated
during model construction, but most of this memory is freed after
model coefficients are calculated.
Some approximate estimates for N centers with default settings are
given below:
* about 250*N*(sizeof(double)+2*sizeof(int)) bytes of memory is
needed during model construction stage.
* about 15*N*sizeof(double) bytes is needed after model is built.
For example, for N=100000 we may need 0.6 GB of memory to build
model, but just about 0.012 GB to store it.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfcreate(const ae_int_t nx, const ae_int_t ny, rbfmodel &s)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::rbfcreate(nx, ny, const_cast<alglib_impl::rbfmodel*>(s.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function adds dataset.
This function overrides results of the previous calls, i.e. multiple calls
of this function will result in only the last set being added.
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call.
XY - points, array[N,NX+NY]. One row corresponds to one point
in the dataset. First NX elements are coordinates, next
NY elements are function values. Array may be larger than
specific, in this case only leading [N,NX+NY] elements
will be used.
N - number of points in the dataset
After you've added dataset and (optionally) tuned algorithm settings you
should call RBFBuildModel() in order to build a model for you.
NOTE: this function has some serialization-related subtleties. We
recommend you to study serialization examples from ALGLIB Reference
Manual if you want to perform serialization of your models.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfsetpoints(const rbfmodel &s, const real_2d_array &xy, const ae_int_t n)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::rbfsetpoints(const_cast<alglib_impl::rbfmodel*>(s.c_ptr()), const_cast<alglib_impl::ae_matrix*>(xy.c_ptr()), n, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function adds dataset.
This function overrides results of the previous calls, i.e. multiple calls
of this function will result in only the last set being added.
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call.
XY - points, array[N,NX+NY]. One row corresponds to one point
in the dataset. First NX elements are coordinates, next
NY elements are function values. Array may be larger than
specific, in this case only leading [N,NX+NY] elements
will be used.
N - number of points in the dataset
After you've added dataset and (optionally) tuned algorithm settings you
should call RBFBuildModel() in order to build a model for you.
NOTE: this function has some serialization-related subtleties. We
recommend you to study serialization examples from ALGLIB Reference
Manual if you want to perform serialization of your models.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfsetpoints(const rbfmodel &s, const real_2d_array &xy)
{
alglib_impl::ae_state _alglib_env_state;
ae_int_t n;
n = xy.rows();
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::rbfsetpoints(const_cast<alglib_impl::rbfmodel*>(s.c_ptr()), const_cast<alglib_impl::ae_matrix*>(xy.c_ptr()), n, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function sets RBF interpolation algorithm. ALGLIB supports several
RBF algorithms with different properties.
This algorithm is called RBF-QNN and it is good for point sets with
following properties:
a) all points are distinct
b) all points are well separated.
c) points distribution is approximately uniform. There is no "contour
lines", clusters of points, or other small-scale structures.
Algorithm description:
1) interpolation centers are allocated to data points
2) interpolation radii are calculated as distances to the nearest centers
times Q coefficient (where Q is a value from [0.75,1.50]).
3) after performing (2) radii are transformed in order to avoid situation
when single outlier has very large radius and influences many points
across all dataset. Transformation has following form:
new_r[i] = min(r[i],Z*median(r[]))
where r[i] is I-th radius, median() is a median radius across entire
dataset, Z is user-specified value which controls amount of deviation
from median radius.
When (a) is violated, we will be unable to build RBF model. When (b) or
(c) are violated, model will be built, but interpolation quality will be
low. See http://www.alglib.net/interpolation/ for more information on this
subject.
This algorithm is used by default.
Additional Q parameter controls smoothness properties of the RBF basis:
* Q<0.75 will give perfectly conditioned basis, but terrible smoothness
properties (RBF interpolant will have sharp peaks around function values)
* Q around 1.0 gives good balance between smoothness and condition number
* Q>1.5 will lead to badly conditioned systems and slow convergence of the
underlying linear solver (although smoothness will be very good)
* Q>2.0 will effectively make optimizer useless because it won't converge
within reasonable amount of iterations. It is possible to set such large
Q, but it is advised not to do so.
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call
Q - Q parameter, Q>0, recommended value - 1.0
Z - Z parameter, Z>0, recommended value - 5.0
NOTE: this function has some serialization-related subtleties. We
recommend you to study serialization examples from ALGLIB Reference
Manual if you want to perform serialization of your models.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfsetalgoqnn(const rbfmodel &s, const double q, const double z)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::rbfsetalgoqnn(const_cast<alglib_impl::rbfmodel*>(s.c_ptr()), q, z, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function sets RBF interpolation algorithm. ALGLIB supports several
RBF algorithms with different properties.
This algorithm is called RBF-QNN and it is good for point sets with
following properties:
a) all points are distinct
b) all points are well separated.
c) points distribution is approximately uniform. There is no "contour
lines", clusters of points, or other small-scale structures.
Algorithm description:
1) interpolation centers are allocated to data points
2) interpolation radii are calculated as distances to the nearest centers
times Q coefficient (where Q is a value from [0.75,1.50]).
3) after performing (2) radii are transformed in order to avoid situation
when single outlier has very large radius and influences many points
across all dataset. Transformation has following form:
new_r[i] = min(r[i],Z*median(r[]))
where r[i] is I-th radius, median() is a median radius across entire
dataset, Z is user-specified value which controls amount of deviation
from median radius.
When (a) is violated, we will be unable to build RBF model. When (b) or
(c) are violated, model will be built, but interpolation quality will be
low. See http://www.alglib.net/interpolation/ for more information on this
subject.
This algorithm is used by default.
Additional Q parameter controls smoothness properties of the RBF basis:
* Q<0.75 will give perfectly conditioned basis, but terrible smoothness
properties (RBF interpolant will have sharp peaks around function values)
* Q around 1.0 gives good balance between smoothness and condition number
* Q>1.5 will lead to badly conditioned systems and slow convergence of the
underlying linear solver (although smoothness will be very good)
* Q>2.0 will effectively make optimizer useless because it won't converge
within reasonable amount of iterations. It is possible to set such large
Q, but it is advised not to do so.
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call
Q - Q parameter, Q>0, recommended value - 1.0
Z - Z parameter, Z>0, recommended value - 5.0
NOTE: this function has some serialization-related subtleties. We
recommend you to study serialization examples from ALGLIB Reference
Manual if you want to perform serialization of your models.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfsetalgoqnn(const rbfmodel &s)
{
alglib_impl::ae_state _alglib_env_state;
double q;
double z;
q = 1.0;
z = 5.0;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::rbfsetalgoqnn(const_cast<alglib_impl::rbfmodel*>(s.c_ptr()), q, z, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function sets RBF interpolation algorithm. ALGLIB supports several
RBF algorithms with different properties.
This algorithm is called RBF-ML. It builds multilayer RBF model, i.e.
model with subsequently decreasing radii, which allows us to combine
smoothness (due to large radii of the first layers) with exactness (due
to small radii of the last layers) and fast convergence.
Internally RBF-ML uses many different means of acceleration, from sparse
matrices to KD-trees, which results in algorithm whose working time is
roughly proportional to N*log(N)*Density*RBase^2*NLayers, where N is a
number of points, Density is an average density if points per unit of the
interpolation space, RBase is an initial radius, NLayers is a number of
layers.
RBF-ML is good for following kinds of interpolation problems:
1. "exact" problems (perfect fit) with well separated points
2. least squares problems with arbitrary distribution of points (algorithm
gives perfect fit where it is possible, and resorts to least squares
fit in the hard areas).
3. noisy problems where we want to apply some controlled amount of
smoothing.
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call
RBase - RBase parameter, RBase>0
NLayers - NLayers parameter, NLayers>0, recommended value to start
with - about 5.
LambdaV - regularization value, can be useful when solving problem
in the least squares sense. Optimal lambda is problem-
dependent and require trial and error. In our experience,
good lambda can be as large as 0.1, and you can use 0.001
as initial guess.
Default value - 0.01, which is used when LambdaV is not
given. You can specify zero value, but it is not
recommended to do so.
TUNING ALGORITHM
In order to use this algorithm you have to choose three parameters:
* initial radius RBase
* number of layers in the model NLayers
* regularization coefficient LambdaV
Initial radius is easy to choose - you can pick any number several times
larger than the average distance between points. Algorithm won't break
down if you choose radius which is too large (model construction time will
increase, but model will be built correctly).
Choose such number of layers that RLast=RBase/2^(NLayers-1) (radius used
by the last layer) will be smaller than the typical distance between
points. In case model error is too large, you can increase number of
layers. Having more layers will make model construction and evaluation
proportionally slower, but it will allow you to have model which precisely
fits your data. From the other side, if you want to suppress noise, you
can DECREASE number of layers to make your model less flexible.
Regularization coefficient LambdaV controls smoothness of the individual
models built for each layer. We recommend you to use default value in case
you don't want to tune this parameter, because having non-zero LambdaV
accelerates and stabilizes internal iterative algorithm. In case you want
to suppress noise you can use LambdaV as additional parameter (larger
value = more smoothness) to tune.
TYPICAL ERRORS
1. Using initial radius which is too large. Memory requirements of the
RBF-ML are roughly proportional to N*Density*RBase^2 (where Density is
an average density of points per unit of the interpolation space). In
the extreme case of the very large RBase we will need O(N^2) units of
memory - and many layers in order to decrease radius to some reasonably
small value.
2. Using too small number of layers - RBF models with large radius are not
flexible enough to reproduce small variations in the target function.
You need many layers with different radii, from large to small, in
order to have good model.
3. Using initial radius which is too small. You will get model with
"holes" in the areas which are too far away from interpolation centers.
However, algorithm will work correctly (and quickly) in this case.
4. Using too many layers - you will get too large and too slow model. This
model will perfectly reproduce your function, but maybe you will be
able to achieve similar results with less layers (and less memory).
-- ALGLIB --
Copyright 02.03.2012 by Bochkanov Sergey
*************************************************************************/
void rbfsetalgomultilayer(const rbfmodel &s, const double rbase, const ae_int_t nlayers, const double lambdav)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::rbfsetalgomultilayer(const_cast<alglib_impl::rbfmodel*>(s.c_ptr()), rbase, nlayers, lambdav, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function sets RBF interpolation algorithm. ALGLIB supports several
RBF algorithms with different properties.
This algorithm is called RBF-ML. It builds multilayer RBF model, i.e.
model with subsequently decreasing radii, which allows us to combine
smoothness (due to large radii of the first layers) with exactness (due
to small radii of the last layers) and fast convergence.
Internally RBF-ML uses many different means of acceleration, from sparse
matrices to KD-trees, which results in algorithm whose working time is
roughly proportional to N*log(N)*Density*RBase^2*NLayers, where N is a
number of points, Density is an average density if points per unit of the
interpolation space, RBase is an initial radius, NLayers is a number of
layers.
RBF-ML is good for following kinds of interpolation problems:
1. "exact" problems (perfect fit) with well separated points
2. least squares problems with arbitrary distribution of points (algorithm
gives perfect fit where it is possible, and resorts to least squares
fit in the hard areas).
3. noisy problems where we want to apply some controlled amount of
smoothing.
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call
RBase - RBase parameter, RBase>0
NLayers - NLayers parameter, NLayers>0, recommended value to start
with - about 5.
LambdaV - regularization value, can be useful when solving problem
in the least squares sense. Optimal lambda is problem-
dependent and require trial and error. In our experience,
good lambda can be as large as 0.1, and you can use 0.001
as initial guess.
Default value - 0.01, which is used when LambdaV is not
given. You can specify zero value, but it is not
recommended to do so.
TUNING ALGORITHM
In order to use this algorithm you have to choose three parameters:
* initial radius RBase
* number of layers in the model NLayers
* regularization coefficient LambdaV
Initial radius is easy to choose - you can pick any number several times
larger than the average distance between points. Algorithm won't break
down if you choose radius which is too large (model construction time will
increase, but model will be built correctly).
Choose such number of layers that RLast=RBase/2^(NLayers-1) (radius used
by the last layer) will be smaller than the typical distance between
points. In case model error is too large, you can increase number of
layers. Having more layers will make model construction and evaluation
proportionally slower, but it will allow you to have model which precisely
fits your data. From the other side, if you want to suppress noise, you
can DECREASE number of layers to make your model less flexible.
Regularization coefficient LambdaV controls smoothness of the individual
models built for each layer. We recommend you to use default value in case
you don't want to tune this parameter, because having non-zero LambdaV
accelerates and stabilizes internal iterative algorithm. In case you want
to suppress noise you can use LambdaV as additional parameter (larger
value = more smoothness) to tune.
TYPICAL ERRORS
1. Using initial radius which is too large. Memory requirements of the
RBF-ML are roughly proportional to N*Density*RBase^2 (where Density is
an average density of points per unit of the interpolation space). In
the extreme case of the very large RBase we will need O(N^2) units of
memory - and many layers in order to decrease radius to some reasonably
small value.
2. Using too small number of layers - RBF models with large radius are not
flexible enough to reproduce small variations in the target function.
You need many layers with different radii, from large to small, in
order to have good model.
3. Using initial radius which is too small. You will get model with
"holes" in the areas which are too far away from interpolation centers.
However, algorithm will work correctly (and quickly) in this case.
4. Using too many layers - you will get too large and too slow model. This
model will perfectly reproduce your function, but maybe you will be
able to achieve similar results with less layers (and less memory).
-- ALGLIB --
Copyright 02.03.2012 by Bochkanov Sergey
*************************************************************************/
void rbfsetalgomultilayer(const rbfmodel &s, const double rbase, const ae_int_t nlayers)
{
alglib_impl::ae_state _alglib_env_state;
double lambdav;
lambdav = 0.01;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::rbfsetalgomultilayer(const_cast<alglib_impl::rbfmodel*>(s.c_ptr()), rbase, nlayers, lambdav, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function sets linear term (model is a sum of radial basis functions
plus linear polynomial). This function won't have effect until next call
to RBFBuildModel().
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call
NOTE: this function has some serialization-related subtleties. We
recommend you to study serialization examples from ALGLIB Reference
Manual if you want to perform serialization of your models.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfsetlinterm(const rbfmodel &s)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::rbfsetlinterm(const_cast<alglib_impl::rbfmodel*>(s.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function sets constant term (model is a sum of radial basis functions
plus constant). This function won't have effect until next call to
RBFBuildModel().
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call
NOTE: this function has some serialization-related subtleties. We
recommend you to study serialization examples from ALGLIB Reference
Manual if you want to perform serialization of your models.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfsetconstterm(const rbfmodel &s)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::rbfsetconstterm(const_cast<alglib_impl::rbfmodel*>(s.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function sets zero term (model is a sum of radial basis functions
without polynomial term). This function won't have effect until next call
to RBFBuildModel().
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call
NOTE: this function has some serialization-related subtleties. We
recommend you to study serialization examples from ALGLIB Reference
Manual if you want to perform serialization of your models.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfsetzeroterm(const rbfmodel &s)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::rbfsetzeroterm(const_cast<alglib_impl::rbfmodel*>(s.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function builds RBF model and returns report (contains some
information which can be used for evaluation of the algorithm properties).
Call to this function modifies RBF model by calculating its centers/radii/
weights and saving them into RBFModel structure. Initially RBFModel
contain zero coefficients, but after call to this function we will have
coefficients which were calculated in order to fit our dataset.
After you called this function you can call RBFCalc(), RBFGridCalc() and
other model calculation functions.
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call
Rep - report:
* Rep.TerminationType:
* -5 - non-distinct basis function centers were detected,
interpolation aborted
* -4 - nonconvergence of the internal SVD solver
* 1 - successful termination
Fields are used for debugging purposes:
* Rep.IterationsCount - iterations count of the LSQR solver
* Rep.NMV - number of matrix-vector products
* Rep.ARows - rows count for the system matrix
* Rep.ACols - columns count for the system matrix
* Rep.ANNZ - number of significantly non-zero elements
(elements above some algorithm-determined threshold)
NOTE: failure to build model will leave current state of the structure
unchanged.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfbuildmodel(const rbfmodel &s, rbfreport &rep)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::rbfbuildmodel(const_cast<alglib_impl::rbfmodel*>(s.c_ptr()), const_cast<alglib_impl::rbfreport*>(rep.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function calculates values of the RBF model in the given point.
This function should be used when we have NY=1 (scalar function) and NX=2
(2-dimensional space). If you have 3-dimensional space, use RBFCalc3(). If
you have general situation (NX-dimensional space, NY-dimensional function)
you should use general, less efficient implementation RBFCalc().
If you want to calculate function values many times, consider using
RBFGridCalc2(), which is far more efficient than many subsequent calls to
RBFCalc2().
This function returns 0.0 when:
* model is not initialized
* NX<>2
*NY<>1
INPUT PARAMETERS:
S - RBF model
X0 - first coordinate, finite number
X1 - second coordinate, finite number
RESULT:
value of the model or 0.0 (as defined above)
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
double rbfcalc2(const rbfmodel &s, const double x0, const double x1)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
double result = alglib_impl::rbfcalc2(const_cast<alglib_impl::rbfmodel*>(s.c_ptr()), x0, x1, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return *(reinterpret_cast<double*>(&result));
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function calculates values of the RBF model in the given point.
This function should be used when we have NY=1 (scalar function) and NX=3
(3-dimensional space). If you have 2-dimensional space, use RBFCalc2(). If
you have general situation (NX-dimensional space, NY-dimensional function)
you should use general, less efficient implementation RBFCalc().
This function returns 0.0 when:
* model is not initialized
* NX<>3
*NY<>1
INPUT PARAMETERS:
S - RBF model
X0 - first coordinate, finite number
X1 - second coordinate, finite number
X2 - third coordinate, finite number
RESULT:
value of the model or 0.0 (as defined above)
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
double rbfcalc3(const rbfmodel &s, const double x0, const double x1, const double x2)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
double result = alglib_impl::rbfcalc3(const_cast<alglib_impl::rbfmodel*>(s.c_ptr()), x0, x1, x2, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return *(reinterpret_cast<double*>(&result));
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function calculates values of the RBF model at the given point.
This is general function which can be used for arbitrary NX (dimension of
the space of arguments) and NY (dimension of the function itself). However
when you have NY=1 you may find more convenient to use RBFCalc2() or
RBFCalc3().
This function returns 0.0 when model is not initialized.
INPUT PARAMETERS:
S - RBF model
X - coordinates, array[NX].
X may have more than NX elements, in this case only
leading NX will be used.
OUTPUT PARAMETERS:
Y - function value, array[NY]. Y is out-parameter and
reallocated after call to this function. In case you want
to reuse previously allocated Y, you may use RBFCalcBuf(),
which reallocates Y only when it is too small.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfcalc(const rbfmodel &s, const real_1d_array &x, real_1d_array &y)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::rbfcalc(const_cast<alglib_impl::rbfmodel*>(s.c_ptr()), const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function calculates values of the RBF model at the given point.
Same as RBFCalc(), but does not reallocate Y when in is large enough to
store function values.
INPUT PARAMETERS:
S - RBF model
X - coordinates, array[NX].
X may have more than NX elements, in this case only
leading NX will be used.
Y - possibly preallocated array
OUTPUT PARAMETERS:
Y - function value, array[NY]. Y is not reallocated when it
is larger than NY.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfcalcbuf(const rbfmodel &s, const real_1d_array &x, real_1d_array &y)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::rbfcalcbuf(const_cast<alglib_impl::rbfmodel*>(s.c_ptr()), const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function calculates values of the RBF model at the regular grid.
Grid have N0*N1 points, with Point[I,J] = (X0[I], X1[J])
This function returns 0.0 when:
* model is not initialized
* NX<>2
*NY<>1
INPUT PARAMETERS:
S - RBF model
X0 - array of grid nodes, first coordinates, array[N0]
N0 - grid size (number of nodes) in the first dimension
X1 - array of grid nodes, second coordinates, array[N1]
N1 - grid size (number of nodes) in the second dimension
OUTPUT PARAMETERS:
Y - function values, array[N0,N1]. Y is out-variable and
is reallocated by this function.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfgridcalc2(const rbfmodel &s, const real_1d_array &x0, const ae_int_t n0, const real_1d_array &x1, const ae_int_t n1, real_2d_array &y)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::rbfgridcalc2(const_cast<alglib_impl::rbfmodel*>(s.c_ptr()), const_cast<alglib_impl::ae_vector*>(x0.c_ptr()), n0, const_cast<alglib_impl::ae_vector*>(x1.c_ptr()), n1, const_cast<alglib_impl::ae_matrix*>(y.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This function "unpacks" RBF model by extracting its coefficients.
INPUT PARAMETERS:
S - RBF model
OUTPUT PARAMETERS:
NX - dimensionality of argument
NY - dimensionality of the target function
XWR - model information, array[NC,NX+NY+1].
One row of the array corresponds to one basis function:
* first NX columns - coordinates of the center
* next NY columns - weights, one per dimension of the
function being modelled
* last column - radius, same for all dimensions of
the function being modelled
NC - number of the centers
V - polynomial term , array[NY,NX+1]. One row per one
dimension of the function being modelled. First NX
elements are linear coefficients, V[NX] is equal to the
constant part.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfunpack(const rbfmodel &s, ae_int_t &nx, ae_int_t &ny, real_2d_array &xwr, ae_int_t &nc, real_2d_array &v)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::rbfunpack(const_cast<alglib_impl::rbfmodel*>(s.c_ptr()), &nx, &ny, const_cast<alglib_impl::ae_matrix*>(xwr.c_ptr()), &nc, const_cast<alglib_impl::ae_matrix*>(v.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
2-dimensional spline inteprolant
*************************************************************************/
_spline2dinterpolant_owner::_spline2dinterpolant_owner()
{
p_struct = (alglib_impl::spline2dinterpolant*)alglib_impl::ae_malloc(sizeof(alglib_impl::spline2dinterpolant), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_spline2dinterpolant_init(p_struct, NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_spline2dinterpolant_owner::_spline2dinterpolant_owner(const _spline2dinterpolant_owner &rhs)
{
p_struct = (alglib_impl::spline2dinterpolant*)alglib_impl::ae_malloc(sizeof(alglib_impl::spline2dinterpolant), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_spline2dinterpolant_init_copy(p_struct, const_cast<alglib_impl::spline2dinterpolant*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_spline2dinterpolant_owner& _spline2dinterpolant_owner::operator=(const _spline2dinterpolant_owner &rhs)
{
if( this==&rhs )
return *this;
alglib_impl::_spline2dinterpolant_clear(p_struct);
if( !alglib_impl::_spline2dinterpolant_init_copy(p_struct, const_cast<alglib_impl::spline2dinterpolant*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
return *this;
}
_spline2dinterpolant_owner::~_spline2dinterpolant_owner()
{
alglib_impl::_spline2dinterpolant_clear(p_struct);
ae_free(p_struct);
}
alglib_impl::spline2dinterpolant* _spline2dinterpolant_owner::c_ptr()
{
return p_struct;
}
alglib_impl::spline2dinterpolant* _spline2dinterpolant_owner::c_ptr() const
{
return const_cast<alglib_impl::spline2dinterpolant*>(p_struct);
}
spline2dinterpolant::spline2dinterpolant() : _spline2dinterpolant_owner()
{
}
spline2dinterpolant::spline2dinterpolant(const spline2dinterpolant &rhs):_spline2dinterpolant_owner(rhs)
{
}
spline2dinterpolant& spline2dinterpolant::operator=(const spline2dinterpolant &rhs)
{
if( this==&rhs )
return *this;
_spline2dinterpolant_owner::operator=(rhs);
return *this;
}
spline2dinterpolant::~spline2dinterpolant()
{
}
/*************************************************************************
This subroutine calculates the value of the bilinear or bicubic spline at
the given point X.
Input parameters:
C - coefficients table.
Built by BuildBilinearSpline or BuildBicubicSpline.
X, Y- point
Result:
S(x,y)
-- ALGLIB PROJECT --
Copyright 05.07.2007 by Bochkanov Sergey
*************************************************************************/
double spline2dcalc(const spline2dinterpolant &c, const double x, const double y)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
double result = alglib_impl::spline2dcalc(const_cast<alglib_impl::spline2dinterpolant*>(c.c_ptr()), x, y, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return *(reinterpret_cast<double*>(&result));
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine calculates the value of the bilinear or bicubic spline at
the given point X and its derivatives.
Input parameters:
C - spline interpolant.
X, Y- point
Output parameters:
F - S(x,y)
FX - dS(x,y)/dX
FY - dS(x,y)/dY
FXY - d2S(x,y)/dXdY
-- ALGLIB PROJECT --
Copyright 05.07.2007 by Bochkanov Sergey
*************************************************************************/
void spline2ddiff(const spline2dinterpolant &c, const double x, const double y, double &f, double &fx, double &fy, double &fxy)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline2ddiff(const_cast<alglib_impl::spline2dinterpolant*>(c.c_ptr()), x, y, &f, &fx, &fy, &fxy, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine performs linear transformation of the spline argument.
Input parameters:
C - spline interpolant
AX, BX - transformation coefficients: x = A*t + B
AY, BY - transformation coefficients: y = A*u + B
Result:
C - transformed spline
-- ALGLIB PROJECT --
Copyright 30.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline2dlintransxy(const spline2dinterpolant &c, const double ax, const double bx, const double ay, const double by)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline2dlintransxy(const_cast<alglib_impl::spline2dinterpolant*>(c.c_ptr()), ax, bx, ay, by, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine performs linear transformation of the spline.
Input parameters:
C - spline interpolant.
A, B- transformation coefficients: S2(x,y) = A*S(x,y) + B
Output parameters:
C - transformed spline
-- ALGLIB PROJECT --
Copyright 30.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline2dlintransf(const spline2dinterpolant &c, const double a, const double b)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline2dlintransf(const_cast<alglib_impl::spline2dinterpolant*>(c.c_ptr()), a, b, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine makes the copy of the spline model.
Input parameters:
C - spline interpolant
Output parameters:
CC - spline copy
-- ALGLIB PROJECT --
Copyright 29.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline2dcopy(const spline2dinterpolant &c, spline2dinterpolant &cc)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline2dcopy(const_cast<alglib_impl::spline2dinterpolant*>(c.c_ptr()), const_cast<alglib_impl::spline2dinterpolant*>(cc.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Bicubic spline resampling
Input parameters:
A - function values at the old grid,
array[0..OldHeight-1, 0..OldWidth-1]
OldHeight - old grid height, OldHeight>1
OldWidth - old grid width, OldWidth>1
NewHeight - new grid height, NewHeight>1
NewWidth - new grid width, NewWidth>1
Output parameters:
B - function values at the new grid,
array[0..NewHeight-1, 0..NewWidth-1]
-- ALGLIB routine --
15 May, 2007
Copyright by Bochkanov Sergey
*************************************************************************/
void spline2dresamplebicubic(const real_2d_array &a, const ae_int_t oldheight, const ae_int_t oldwidth, real_2d_array &b, const ae_int_t newheight, const ae_int_t newwidth)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline2dresamplebicubic(const_cast<alglib_impl::ae_matrix*>(a.c_ptr()), oldheight, oldwidth, const_cast<alglib_impl::ae_matrix*>(b.c_ptr()), newheight, newwidth, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Bilinear spline resampling
Input parameters:
A - function values at the old grid,
array[0..OldHeight-1, 0..OldWidth-1]
OldHeight - old grid height, OldHeight>1
OldWidth - old grid width, OldWidth>1
NewHeight - new grid height, NewHeight>1
NewWidth - new grid width, NewWidth>1
Output parameters:
B - function values at the new grid,
array[0..NewHeight-1, 0..NewWidth-1]
-- ALGLIB routine --
09.07.2007
Copyright by Bochkanov Sergey
*************************************************************************/
void spline2dresamplebilinear(const real_2d_array &a, const ae_int_t oldheight, const ae_int_t oldwidth, real_2d_array &b, const ae_int_t newheight, const ae_int_t newwidth)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline2dresamplebilinear(const_cast<alglib_impl::ae_matrix*>(a.c_ptr()), oldheight, oldwidth, const_cast<alglib_impl::ae_matrix*>(b.c_ptr()), newheight, newwidth, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine builds bilinear vector-valued spline.
Input parameters:
X - spline abscissas, array[0..N-1]
Y - spline ordinates, array[0..M-1]
F - function values, array[0..M*N*D-1]:
* first D elements store D values at (X[0],Y[0])
* next D elements store D values at (X[1],Y[0])
* general form - D function values at (X[i],Y[j]) are stored
at F[D*(J*N+I)...D*(J*N+I)+D-1].
M,N - grid size, M>=2, N>=2
D - vector dimension, D>=1
Output parameters:
C - spline interpolant
-- ALGLIB PROJECT --
Copyright 16.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildbilinearv(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, const real_1d_array &f, const ae_int_t d, spline2dinterpolant &c)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline2dbuildbilinearv(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), n, const_cast<alglib_impl::ae_vector*>(y.c_ptr()), m, const_cast<alglib_impl::ae_vector*>(f.c_ptr()), d, const_cast<alglib_impl::spline2dinterpolant*>(c.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine builds bicubic vector-valued spline.
Input parameters:
X - spline abscissas, array[0..N-1]
Y - spline ordinates, array[0..M-1]
F - function values, array[0..M*N*D-1]:
* first D elements store D values at (X[0],Y[0])
* next D elements store D values at (X[1],Y[0])
* general form - D function values at (X[i],Y[j]) are stored
at F[D*(J*N+I)...D*(J*N+I)+D-1].
M,N - grid size, M>=2, N>=2
D - vector dimension, D>=1
Output parameters:
C - spline interpolant
-- ALGLIB PROJECT --
Copyright 16.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildbicubicv(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, const real_1d_array &f, const ae_int_t d, spline2dinterpolant &c)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline2dbuildbicubicv(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), n, const_cast<alglib_impl::ae_vector*>(y.c_ptr()), m, const_cast<alglib_impl::ae_vector*>(f.c_ptr()), d, const_cast<alglib_impl::spline2dinterpolant*>(c.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine calculates bilinear or bicubic vector-valued spline at the
given point (X,Y).
INPUT PARAMETERS:
C - spline interpolant.
X, Y- point
F - output buffer, possibly preallocated array. In case array size
is large enough to store result, it is not reallocated. Array
which is too short will be reallocated
OUTPUT PARAMETERS:
F - array[D] (or larger) which stores function values
-- ALGLIB PROJECT --
Copyright 16.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline2dcalcvbuf(const spline2dinterpolant &c, const double x, const double y, real_1d_array &f)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline2dcalcvbuf(const_cast<alglib_impl::spline2dinterpolant*>(c.c_ptr()), x, y, const_cast<alglib_impl::ae_vector*>(f.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine calculates bilinear or bicubic vector-valued spline at the
given point (X,Y).
INPUT PARAMETERS:
C - spline interpolant.
X, Y- point
OUTPUT PARAMETERS:
F - array[D] which stores function values. F is out-parameter and
it is reallocated after call to this function. In case you
want to reuse previously allocated F, you may use
Spline2DCalcVBuf(), which reallocates F only when it is too
small.
-- ALGLIB PROJECT --
Copyright 16.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline2dcalcv(const spline2dinterpolant &c, const double x, const double y, real_1d_array &f)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline2dcalcv(const_cast<alglib_impl::spline2dinterpolant*>(c.c_ptr()), x, y, const_cast<alglib_impl::ae_vector*>(f.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine unpacks two-dimensional spline into the coefficients table
Input parameters:
C - spline interpolant.
Result:
M, N- grid size (x-axis and y-axis)
D - number of components
Tbl - coefficients table, unpacked format,
D - components: [0..(N-1)*(M-1)*D-1, 0..19].
For T=0..D-1 (component index), I = 0...N-2 (x index),
J=0..M-2 (y index):
K := T + I*D + J*D*(N-1)
K-th row stores decomposition for T-th component of the
vector-valued function
Tbl[K,0] = X[i]
Tbl[K,1] = X[i+1]
Tbl[K,2] = Y[j]
Tbl[K,3] = Y[j+1]
Tbl[K,4] = C00
Tbl[K,5] = C01
Tbl[K,6] = C02
Tbl[K,7] = C03
Tbl[K,8] = C10
Tbl[K,9] = C11
...
Tbl[K,19] = C33
On each grid square spline is equals to:
S(x) = SUM(c[i,j]*(t^i)*(u^j), i=0..3, j=0..3)
t = x-x[j]
u = y-y[i]
-- ALGLIB PROJECT --
Copyright 16.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline2dunpackv(const spline2dinterpolant &c, ae_int_t &m, ae_int_t &n, ae_int_t &d, real_2d_array &tbl)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline2dunpackv(const_cast<alglib_impl::spline2dinterpolant*>(c.c_ptr()), &m, &n, &d, const_cast<alglib_impl::ae_matrix*>(tbl.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine was deprecated in ALGLIB 3.6.0
We recommend you to switch to Spline2DBuildBilinearV(), which is more
flexible and accepts its arguments in more convenient order.
-- ALGLIB PROJECT --
Copyright 05.07.2007 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildbilinear(const real_1d_array &x, const real_1d_array &y, const real_2d_array &f, const ae_int_t m, const ae_int_t n, spline2dinterpolant &c)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline2dbuildbilinear(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_matrix*>(f.c_ptr()), m, n, const_cast<alglib_impl::spline2dinterpolant*>(c.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine was deprecated in ALGLIB 3.6.0
We recommend you to switch to Spline2DBuildBicubicV(), which is more
flexible and accepts its arguments in more convenient order.
-- ALGLIB PROJECT --
Copyright 05.07.2007 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildbicubic(const real_1d_array &x, const real_1d_array &y, const real_2d_array &f, const ae_int_t m, const ae_int_t n, spline2dinterpolant &c)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline2dbuildbicubic(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), const_cast<alglib_impl::ae_vector*>(y.c_ptr()), const_cast<alglib_impl::ae_matrix*>(f.c_ptr()), m, n, const_cast<alglib_impl::spline2dinterpolant*>(c.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine was deprecated in ALGLIB 3.6.0
We recommend you to switch to Spline2DUnpackV(), which is more flexible
and accepts its arguments in more convenient order.
-- ALGLIB PROJECT --
Copyright 29.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline2dunpack(const spline2dinterpolant &c, ae_int_t &m, ae_int_t &n, real_2d_array &tbl)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline2dunpack(const_cast<alglib_impl::spline2dinterpolant*>(c.c_ptr()), &m, &n, const_cast<alglib_impl::ae_matrix*>(tbl.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
3-dimensional spline inteprolant
*************************************************************************/
_spline3dinterpolant_owner::_spline3dinterpolant_owner()
{
p_struct = (alglib_impl::spline3dinterpolant*)alglib_impl::ae_malloc(sizeof(alglib_impl::spline3dinterpolant), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_spline3dinterpolant_init(p_struct, NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_spline3dinterpolant_owner::_spline3dinterpolant_owner(const _spline3dinterpolant_owner &rhs)
{
p_struct = (alglib_impl::spline3dinterpolant*)alglib_impl::ae_malloc(sizeof(alglib_impl::spline3dinterpolant), NULL);
if( p_struct==NULL )
throw ap_error("ALGLIB: malloc error");
if( !alglib_impl::_spline3dinterpolant_init_copy(p_struct, const_cast<alglib_impl::spline3dinterpolant*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
}
_spline3dinterpolant_owner& _spline3dinterpolant_owner::operator=(const _spline3dinterpolant_owner &rhs)
{
if( this==&rhs )
return *this;
alglib_impl::_spline3dinterpolant_clear(p_struct);
if( !alglib_impl::_spline3dinterpolant_init_copy(p_struct, const_cast<alglib_impl::spline3dinterpolant*>(rhs.p_struct), NULL, ae_false) )
throw ap_error("ALGLIB: malloc error");
return *this;
}
_spline3dinterpolant_owner::~_spline3dinterpolant_owner()
{
alglib_impl::_spline3dinterpolant_clear(p_struct);
ae_free(p_struct);
}
alglib_impl::spline3dinterpolant* _spline3dinterpolant_owner::c_ptr()
{
return p_struct;
}
alglib_impl::spline3dinterpolant* _spline3dinterpolant_owner::c_ptr() const
{
return const_cast<alglib_impl::spline3dinterpolant*>(p_struct);
}
spline3dinterpolant::spline3dinterpolant() : _spline3dinterpolant_owner()
{
}
spline3dinterpolant::spline3dinterpolant(const spline3dinterpolant &rhs):_spline3dinterpolant_owner(rhs)
{
}
spline3dinterpolant& spline3dinterpolant::operator=(const spline3dinterpolant &rhs)
{
if( this==&rhs )
return *this;
_spline3dinterpolant_owner::operator=(rhs);
return *this;
}
spline3dinterpolant::~spline3dinterpolant()
{
}
/*************************************************************************
This subroutine calculates the value of the trilinear or tricubic spline at
the given point (X,Y,Z).
INPUT PARAMETERS:
C - coefficients table.
Built by BuildBilinearSpline or BuildBicubicSpline.
X, Y,
Z - point
Result:
S(x,y,z)
-- ALGLIB PROJECT --
Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
double spline3dcalc(const spline3dinterpolant &c, const double x, const double y, const double z)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
double result = alglib_impl::spline3dcalc(const_cast<alglib_impl::spline3dinterpolant*>(c.c_ptr()), x, y, z, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return *(reinterpret_cast<double*>(&result));
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine performs linear transformation of the spline argument.
INPUT PARAMETERS:
C - spline interpolant
AX, BX - transformation coefficients: x = A*u + B
AY, BY - transformation coefficients: y = A*v + B
AZ, BZ - transformation coefficients: z = A*w + B
OUTPUT PARAMETERS:
C - transformed spline
-- ALGLIB PROJECT --
Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dlintransxyz(const spline3dinterpolant &c, const double ax, const double bx, const double ay, const double by, const double az, const double bz)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline3dlintransxyz(const_cast<alglib_impl::spline3dinterpolant*>(c.c_ptr()), ax, bx, ay, by, az, bz, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine performs linear transformation of the spline.
INPUT PARAMETERS:
C - spline interpolant.
A, B- transformation coefficients: S2(x,y) = A*S(x,y,z) + B
OUTPUT PARAMETERS:
C - transformed spline
-- ALGLIB PROJECT --
Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dlintransf(const spline3dinterpolant &c, const double a, const double b)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline3dlintransf(const_cast<alglib_impl::spline3dinterpolant*>(c.c_ptr()), a, b, &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
Trilinear spline resampling
INPUT PARAMETERS:
A - array[0..OldXCount*OldYCount*OldZCount-1], function
values at the old grid, :
A[0] x=0,y=0,z=0
A[1] x=1,y=0,z=0
A[..] ...
A[..] x=oldxcount-1,y=0,z=0
A[..] x=0,y=1,z=0
A[..] ...
...
OldZCount - old Z-count, OldZCount>1
OldYCount - old Y-count, OldYCount>1
OldXCount - old X-count, OldXCount>1
NewZCount - new Z-count, NewZCount>1
NewYCount - new Y-count, NewYCount>1
NewXCount - new X-count, NewXCount>1
OUTPUT PARAMETERS:
B - array[0..NewXCount*NewYCount*NewZCount-1], function
values at the new grid:
B[0] x=0,y=0,z=0
B[1] x=1,y=0,z=0
B[..] ...
B[..] x=newxcount-1,y=0,z=0
B[..] x=0,y=1,z=0
B[..] ...
...
-- ALGLIB routine --
26.04.2012
Copyright by Bochkanov Sergey
*************************************************************************/
void spline3dresampletrilinear(const real_1d_array &a, const ae_int_t oldzcount, const ae_int_t oldycount, const ae_int_t oldxcount, const ae_int_t newzcount, const ae_int_t newycount, const ae_int_t newxcount, real_1d_array &b)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline3dresampletrilinear(const_cast<alglib_impl::ae_vector*>(a.c_ptr()), oldzcount, oldycount, oldxcount, newzcount, newycount, newxcount, const_cast<alglib_impl::ae_vector*>(b.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine builds trilinear vector-valued spline.
INPUT PARAMETERS:
X - spline abscissas, array[0..N-1]
Y - spline ordinates, array[0..M-1]
Z - spline applicates, array[0..L-1]
F - function values, array[0..M*N*L*D-1]:
* first D elements store D values at (X[0],Y[0],Z[0])
* next D elements store D values at (X[1],Y[0],Z[0])
* next D elements store D values at (X[2],Y[0],Z[0])
* ...
* next D elements store D values at (X[0],Y[1],Z[0])
* next D elements store D values at (X[1],Y[1],Z[0])
* next D elements store D values at (X[2],Y[1],Z[0])
* ...
* next D elements store D values at (X[0],Y[0],Z[1])
* next D elements store D values at (X[1],Y[0],Z[1])
* next D elements store D values at (X[2],Y[0],Z[1])
* ...
* general form - D function values at (X[i],Y[j]) are stored
at F[D*(N*(M*K+J)+I)...D*(N*(M*K+J)+I)+D-1].
M,N,
L - grid size, M>=2, N>=2, L>=2
D - vector dimension, D>=1
OUTPUT PARAMETERS:
C - spline interpolant
-- ALGLIB PROJECT --
Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dbuildtrilinearv(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, const real_1d_array &z, const ae_int_t l, const real_1d_array &f, const ae_int_t d, spline3dinterpolant &c)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline3dbuildtrilinearv(const_cast<alglib_impl::ae_vector*>(x.c_ptr()), n, const_cast<alglib_impl::ae_vector*>(y.c_ptr()), m, const_cast<alglib_impl::ae_vector*>(z.c_ptr()), l, const_cast<alglib_impl::ae_vector*>(f.c_ptr()), d, const_cast<alglib_impl::spline3dinterpolant*>(c.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine calculates bilinear or bicubic vector-valued spline at the
given point (X,Y,Z).
INPUT PARAMETERS:
C - spline interpolant.
X, Y,
Z - point
F - output buffer, possibly preallocated array. In case array size
is large enough to store result, it is not reallocated. Array
which is too short will be reallocated
OUTPUT PARAMETERS:
F - array[D] (or larger) which stores function values
-- ALGLIB PROJECT --
Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dcalcvbuf(const spline3dinterpolant &c, const double x, const double y, const double z, real_1d_array &f)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline3dcalcvbuf(const_cast<alglib_impl::spline3dinterpolant*>(c.c_ptr()), x, y, z, const_cast<alglib_impl::ae_vector*>(f.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine calculates trilinear or tricubic vector-valued spline at the
given point (X,Y,Z).
INPUT PARAMETERS:
C - spline interpolant.
X, Y,
Z - point
OUTPUT PARAMETERS:
F - array[D] which stores function values. F is out-parameter and
it is reallocated after call to this function. In case you
want to reuse previously allocated F, you may use
Spline2DCalcVBuf(), which reallocates F only when it is too
small.
-- ALGLIB PROJECT --
Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dcalcv(const spline3dinterpolant &c, const double x, const double y, const double z, real_1d_array &f)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline3dcalcv(const_cast<alglib_impl::spline3dinterpolant*>(c.c_ptr()), x, y, z, const_cast<alglib_impl::ae_vector*>(f.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
/*************************************************************************
This subroutine unpacks tri-dimensional spline into the coefficients table
INPUT PARAMETERS:
C - spline interpolant.
Result:
N - grid size (X)
M - grid size (Y)
L - grid size (Z)
D - number of components
SType- spline type. Currently, only one spline type is supported:
trilinear spline, as indicated by SType=1.
Tbl - spline coefficients: [0..(N-1)*(M-1)*(L-1)*D-1, 0..13].
For T=0..D-1 (component index), I = 0...N-2 (x index),
J=0..M-2 (y index), K=0..L-2 (z index):
Q := T + I*D + J*D*(N-1) + K*D*(N-1)*(M-1),
Q-th row stores decomposition for T-th component of the
vector-valued function
Tbl[Q,0] = X[i]
Tbl[Q,1] = X[i+1]
Tbl[Q,2] = Y[j]
Tbl[Q,3] = Y[j+1]
Tbl[Q,4] = Z[k]
Tbl[Q,5] = Z[k+1]
Tbl[Q,6] = C000
Tbl[Q,7] = C100
Tbl[Q,8] = C010
Tbl[Q,9] = C110
Tbl[Q,10]= C001
Tbl[Q,11]= C101
Tbl[Q,12]= C011
Tbl[Q,13]= C111
On each grid square spline is equals to:
S(x) = SUM(c[i,j,k]*(x^i)*(y^j)*(z^k), i=0..1, j=0..1, k=0..1)
t = x-x[j]
u = y-y[i]
v = z-z[k]
NOTE: format of Tbl is given for SType=1. Future versions of
ALGLIB can use different formats for different values of
SType.
-- ALGLIB PROJECT --
Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dunpackv(const spline3dinterpolant &c, ae_int_t &n, ae_int_t &m, ae_int_t &l, ae_int_t &d, ae_int_t &stype, real_2d_array &tbl)
{
alglib_impl::ae_state _alglib_env_state;
alglib_impl::ae_state_init(&_alglib_env_state);
try
{
alglib_impl::spline3dunpackv(const_cast<alglib_impl::spline3dinterpolant*>(c.c_ptr()), &n, &m, &l, &d, &stype, const_cast<alglib_impl::ae_matrix*>(tbl.c_ptr()), &_alglib_env_state);
alglib_impl::ae_state_clear(&_alglib_env_state);
return;
}
catch(alglib_impl::ae_error_type)
{
throw ap_error(_alglib_env_state.error_msg);
}
}
}
/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS IMPLEMENTATION OF COMPUTATIONAL CORE
//
/////////////////////////////////////////////////////////////////////////
namespace alglib_impl
{
static double idwint_idwqfactor = 1.5;
static ae_int_t idwint_idwkmin = 5;
static double idwint_idwcalcq(idwinterpolant* z,
/* Real */ ae_vector* x,
ae_int_t k,
ae_state *_state);
static void idwint_idwinit1(ae_int_t n,
ae_int_t nx,
ae_int_t d,
ae_int_t nq,
ae_int_t nw,
idwinterpolant* z,
ae_state *_state);
static void idwint_idwinternalsolver(/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
/* Real */ ae_matrix* fmatrix,
/* Real */ ae_vector* temp,
ae_int_t n,
ae_int_t m,
ae_int_t* info,
/* Real */ ae_vector* x,
double* taskrcond,
ae_state *_state);
static void ratint_barycentricnormalize(barycentricinterpolant* b,
ae_state *_state);
static void spline1d_spline1dgriddiffcubicinternal(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_int_t boundltype,
double boundl,
ae_int_t boundrtype,
double boundr,
/* Real */ ae_vector* d,
/* Real */ ae_vector* a1,
/* Real */ ae_vector* a2,
/* Real */ ae_vector* a3,
/* Real */ ae_vector* b,
/* Real */ ae_vector* dt,
ae_state *_state);
static void spline1d_heapsortpoints(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_state *_state);
static void spline1d_heapsortppoints(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Integer */ ae_vector* p,
ae_int_t n,
ae_state *_state);
static void spline1d_solvetridiagonal(/* Real */ ae_vector* a,
/* Real */ ae_vector* b,
/* Real */ ae_vector* c,
/* Real */ ae_vector* d,
ae_int_t n,
/* Real */ ae_vector* x,
ae_state *_state);
static void spline1d_solvecyclictridiagonal(/* Real */ ae_vector* a,
/* Real */ ae_vector* b,
/* Real */ ae_vector* c,
/* Real */ ae_vector* d,
ae_int_t n,
/* Real */ ae_vector* x,
ae_state *_state);
static double spline1d_diffthreepoint(double t,
double x0,
double f0,
double x1,
double f1,
double x2,
double f2,
ae_state *_state);
static void spline1d_hermitecalc(double p0,
double m0,
double p1,
double m1,
double t,
double* s,
double* ds,
ae_state *_state);
static double spline1d_rescaleval(double a0,
double b0,
double a1,
double b1,
double t,
ae_state *_state);
static void lsfit_spline1dfitinternal(ae_int_t st,
/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
ae_int_t n,
/* Real */ ae_vector* xc,
/* Real */ ae_vector* yc,
/* Integer */ ae_vector* dc,
ae_int_t k,
ae_int_t m,
ae_int_t* info,
spline1dinterpolant* s,
spline1dfitreport* rep,
ae_state *_state);
static void lsfit_lsfitlinearinternal(/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
/* Real */ ae_matrix* fmatrix,
ae_int_t n,
ae_int_t m,
ae_int_t* info,
/* Real */ ae_vector* c,
lsfitreport* rep,
ae_state *_state);
static void lsfit_lsfitclearrequestfields(lsfitstate* state,
ae_state *_state);
static void lsfit_barycentriccalcbasis(barycentricinterpolant* b,
double t,
/* Real */ ae_vector* y,
ae_state *_state);
static void lsfit_internalchebyshevfit(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
ae_int_t n,
/* Real */ ae_vector* xc,
/* Real */ ae_vector* yc,
/* Integer */ ae_vector* dc,
ae_int_t k,
ae_int_t m,
ae_int_t* info,
/* Real */ ae_vector* c,
lsfitreport* rep,
ae_state *_state);
static void lsfit_barycentricfitwcfixedd(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
ae_int_t n,
/* Real */ ae_vector* xc,
/* Real */ ae_vector* yc,
/* Integer */ ae_vector* dc,
ae_int_t k,
ae_int_t m,
ae_int_t d,
ae_int_t* info,
barycentricinterpolant* b,
barycentricfitreport* rep,
ae_state *_state);
static void lsfit_clearreport(lsfitreport* rep, ae_state *_state);
static void lsfit_estimateerrors(/* Real */ ae_matrix* f1,
/* Real */ ae_vector* f0,
/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
/* Real */ ae_vector* x,
/* Real */ ae_vector* s,
ae_int_t n,
ae_int_t k,
lsfitreport* rep,
/* Real */ ae_matrix* z,
ae_int_t zkind,
ae_state *_state);
static void pspline_pspline2par(/* Real */ ae_matrix* xy,
ae_int_t n,
ae_int_t pt,
/* Real */ ae_vector* p,
ae_state *_state);
static void pspline_pspline3par(/* Real */ ae_matrix* xy,
ae_int_t n,
ae_int_t pt,
/* Real */ ae_vector* p,
ae_state *_state);
static double rbf_eps = 1.0E-6;
static ae_int_t rbf_mxnx = 3;
static double rbf_rbffarradius = 6;
static double rbf_rbfnearradius = 2.1;
static double rbf_rbfmlradius = 3;
static ae_int_t rbf_rbffirstversion = 0;
static void rbf_rbfgridpoints(rbfmodel* s, ae_state *_state);
static void rbf_rbfradnn(rbfmodel* s,
double q,
double z,
ae_state *_state);
static ae_bool rbf_buildlinearmodel(/* Real */ ae_matrix* x,
/* Real */ ae_matrix* y,
ae_int_t n,
ae_int_t ny,
ae_int_t modeltype,
/* Real */ ae_matrix* v,
ae_state *_state);
static void rbf_buildrbfmodellsqr(/* Real */ ae_matrix* x,
/* Real */ ae_matrix* y,
/* Real */ ae_matrix* xc,
/* Real */ ae_vector* r,
ae_int_t n,
ae_int_t nc,
ae_int_t ny,
kdtree* pointstree,
kdtree* centerstree,
double epsort,
double epserr,
ae_int_t maxits,
ae_int_t* gnnz,
ae_int_t* snnz,
/* Real */ ae_matrix* w,
ae_int_t* info,
ae_int_t* iterationscount,
ae_int_t* nmv,
ae_state *_state);
static void rbf_buildrbfmlayersmodellsqr(/* Real */ ae_matrix* x,
/* Real */ ae_matrix* y,
/* Real */ ae_matrix* xc,
double rval,
/* Real */ ae_vector* r,
ae_int_t n,
ae_int_t* nc,
ae_int_t ny,
ae_int_t nlayers,
kdtree* centerstree,
double epsort,
double epserr,
ae_int_t maxits,
double lambdav,
ae_int_t* annz,
/* Real */ ae_matrix* w,
ae_int_t* info,
ae_int_t* iterationscount,
ae_int_t* nmv,
ae_state *_state);
static void spline2d_bicubiccalcderivatives(/* Real */ ae_matrix* a,
/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t m,
ae_int_t n,
/* Real */ ae_matrix* dx,
/* Real */ ae_matrix* dy,
/* Real */ ae_matrix* dxy,
ae_state *_state);
static void spline3d_spline3ddiff(spline3dinterpolant* c,
double x,
double y,
double z,
double* f,
double* fx,
double* fy,
double* fxy,
ae_state *_state);
/*************************************************************************
IDW interpolation
INPUT PARAMETERS:
Z - IDW interpolant built with one of model building
subroutines.
X - array[0..NX-1], interpolation point
Result:
IDW interpolant Z(X)
-- ALGLIB --
Copyright 02.03.2010 by Bochkanov Sergey
*************************************************************************/
double idwcalc(idwinterpolant* z,
/* Real */ ae_vector* x,
ae_state *_state)
{
ae_int_t i;
ae_int_t k;
double r;
double s;
double w;
double v1;
double v2;
double d0;
double di;
double result;
/*
* these initializers are not really necessary,
* but without them compiler complains about uninitialized locals
*/
k = 0;
/*
* Query
*/
if( z->modeltype==0 )
{
/*
* NQ/NW-based model
*/
k = kdtreequeryknn(&z->tree, x, z->nw, ae_true, _state);
kdtreequeryresultsdistances(&z->tree, &z->rbuf, _state);
kdtreequeryresultstags(&z->tree, &z->tbuf, _state);
}
if( z->modeltype==1 )
{
/*
* R-based model
*/
k = kdtreequeryrnn(&z->tree, x, z->r, ae_true, _state);
kdtreequeryresultsdistances(&z->tree, &z->rbuf, _state);
kdtreequeryresultstags(&z->tree, &z->tbuf, _state);
if( k<idwint_idwkmin )
{
/*
* we need at least IDWKMin points
*/
k = kdtreequeryknn(&z->tree, x, idwint_idwkmin, ae_true, _state);
kdtreequeryresultsdistances(&z->tree, &z->rbuf, _state);
kdtreequeryresultstags(&z->tree, &z->tbuf, _state);
}
}
/*
* initialize weights for linear/quadratic members calculation.
*
* NOTE 1: weights are calculated using NORMALIZED modified
* Shepard's formula. Original formula gives w(i) = sqr((R-di)/(R*di)),
* where di is i-th distance, R is max(di). Modified formula have
* following form:
* w_mod(i) = 1, if di=d0
* w_mod(i) = w(i)/w(0), if di<>d0
*
* NOTE 2: self-match is USED for this query
*
* NOTE 3: last point almost always gain zero weight, but it MUST
* be used for fitting because sometimes it will gain NON-ZERO
* weight - for example, when all distances are equal.
*/
r = z->rbuf.ptr.p_double[k-1];
d0 = z->rbuf.ptr.p_double[0];
result = 0;
s = 0;
for(i=0; i<=k-1; i++)
{
di = z->rbuf.ptr.p_double[i];
if( ae_fp_eq(di,d0) )
{
/*
* distance is equal to shortest, set it 1.0
* without explicitly calculating (which would give
* us same result, but 'll expose us to the risk of
* division by zero).
*/
w = 1;
}
else
{
/*
* use normalized formula
*/
v1 = (r-di)/(r-d0);
v2 = d0/di;
w = ae_sqr(v1*v2, _state);
}
result = result+w*idwint_idwcalcq(z, x, z->tbuf.ptr.p_int[i], _state);
s = s+w;
}
result = result/s;
return result;
}
/*************************************************************************
IDW interpolant using modified Shepard method for uniform point
distributions.
INPUT PARAMETERS:
XY - X and Y values, array[0..N-1,0..NX].
First NX columns contain X-values, last column contain
Y-values.
N - number of nodes, N>0.
NX - space dimension, NX>=1.
D - nodal function type, either:
* 0 constant model. Just for demonstration only, worst
model ever.
* 1 linear model, least squares fitting. Simpe model for
datasets too small for quadratic models
* 2 quadratic model, least squares fitting. Best model
available (if your dataset is large enough).
* -1 "fast" linear model, use with caution!!! It is
significantly faster than linear/quadratic and better
than constant model. But it is less robust (especially
in the presence of noise).
NQ - number of points used to calculate nodal functions (ignored
for constant models). NQ should be LARGER than:
* max(1.5*(1+NX),2^NX+1) for linear model,
* max(3/4*(NX+2)*(NX+1),2^NX+1) for quadratic model.
Values less than this threshold will be silently increased.
NW - number of points used to calculate weights and to interpolate.
Required: >=2^NX+1, values less than this threshold will be
silently increased.
Recommended value: about 2*NQ
OUTPUT PARAMETERS:
Z - IDW interpolant.
NOTES:
* best results are obtained with quadratic models, worst - with constant
models
* when N is large, NQ and NW must be significantly smaller than N both
to obtain optimal performance and to obtain optimal accuracy. In 2 or
3-dimensional tasks NQ=15 and NW=25 are good values to start with.
* NQ and NW may be greater than N. In such cases they will be
automatically decreased.
* this subroutine is always succeeds (as long as correct parameters are
passed).
* see 'Multivariate Interpolation of Large Sets of Scattered Data' by
Robert J. Renka for more information on this algorithm.
* this subroutine assumes that point distribution is uniform at the small
scales. If it isn't - for example, points are concentrated along
"lines", but "lines" distribution is uniform at the larger scale - then
you should use IDWBuildModifiedShepardR()
-- ALGLIB PROJECT --
Copyright 02.03.2010 by Bochkanov Sergey
*************************************************************************/
void idwbuildmodifiedshepard(/* Real */ ae_matrix* xy,
ae_int_t n,
ae_int_t nx,
ae_int_t d,
ae_int_t nq,
ae_int_t nw,
idwinterpolant* z,
ae_state *_state)
{
ae_frame _frame_block;
ae_int_t i;
ae_int_t j;
ae_int_t k;
ae_int_t j2;
ae_int_t j3;
double v;
double r;
double s;
double d0;
double di;
double v1;
double v2;
ae_int_t nc;
ae_int_t offs;
ae_vector x;
ae_vector qrbuf;
ae_matrix qxybuf;
ae_vector y;
ae_matrix fmatrix;
ae_vector w;
ae_vector qsol;
ae_vector temp;
ae_vector tags;
ae_int_t info;
double taskrcond;
ae_frame_make(_state, &_frame_block);
_idwinterpolant_clear(z);
ae_vector_init(&x, 0, DT_REAL, _state, ae_true);
ae_vector_init(&qrbuf, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&qxybuf, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&y, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&fmatrix, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&w, 0, DT_REAL, _state, ae_true);
ae_vector_init(&qsol, 0, DT_REAL, _state, ae_true);
ae_vector_init(&temp, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tags, 0, DT_INT, _state, ae_true);
/*
* these initializers are not really necessary,
* but without them compiler complains about uninitialized locals
*/
nc = 0;
/*
* assertions
*/
ae_assert(n>0, "IDWBuildModifiedShepard: N<=0!", _state);
ae_assert(nx>=1, "IDWBuildModifiedShepard: NX<1!", _state);
ae_assert(d>=-1&&d<=2, "IDWBuildModifiedShepard: D<>-1 and D<>0 and D<>1 and D<>2!", _state);
/*
* Correct parameters if needed
*/
if( d==1 )
{
nq = ae_maxint(nq, ae_iceil(idwint_idwqfactor*(1+nx), _state)+1, _state);
nq = ae_maxint(nq, ae_round(ae_pow(2, nx, _state), _state)+1, _state);
}
if( d==2 )
{
nq = ae_maxint(nq, ae_iceil(idwint_idwqfactor*(nx+2)*(nx+1)/2, _state)+1, _state);
nq = ae_maxint(nq, ae_round(ae_pow(2, nx, _state), _state)+1, _state);
}
nw = ae_maxint(nw, ae_round(ae_pow(2, nx, _state), _state)+1, _state);
nq = ae_minint(nq, n, _state);
nw = ae_minint(nw, n, _state);
/*
* primary initialization of Z
*/
idwint_idwinit1(n, nx, d, nq, nw, z, _state);
z->modeltype = 0;
/*
* Create KD-tree
*/
ae_vector_set_length(&tags, n, _state);
for(i=0; i<=n-1; i++)
{
tags.ptr.p_int[i] = i;
}
kdtreebuildtagged(xy, &tags, n, nx, 1, 2, &z->tree, _state);
/*
* build nodal functions
*/
ae_vector_set_length(&temp, nq+1, _state);
ae_vector_set_length(&x, nx, _state);
ae_vector_set_length(&qrbuf, nq, _state);
ae_matrix_set_length(&qxybuf, nq, nx+1, _state);
if( d==-1 )
{
ae_vector_set_length(&w, nq, _state);
}
if( d==1 )
{
ae_vector_set_length(&y, nq, _state);
ae_vector_set_length(&w, nq, _state);
ae_vector_set_length(&qsol, nx, _state);
/*
* NX for linear members,
* 1 for temporary storage
*/
ae_matrix_set_length(&fmatrix, nq, nx+1, _state);
}
if( d==2 )
{
ae_vector_set_length(&y, nq, _state);
ae_vector_set_length(&w, nq, _state);
ae_vector_set_length(&qsol, nx+ae_round(nx*(nx+1)*0.5, _state), _state);
/*
* NX for linear members,
* Round(NX*(NX+1)*0.5) for quadratic model,
* 1 for temporary storage
*/
ae_matrix_set_length(&fmatrix, nq, nx+ae_round(nx*(nx+1)*0.5, _state)+1, _state);
}
for(i=0; i<=n-1; i++)
{
/*
* Initialize center and function value.
* If D=0 it is all what we need
*/
ae_v_move(&z->q.ptr.pp_double[i][0], 1, &xy->ptr.pp_double[i][0], 1, ae_v_len(0,nx));
if( d==0 )
{
continue;
}
/*
* calculate weights for linear/quadratic members calculation.
*
* NOTE 1: weights are calculated using NORMALIZED modified
* Shepard's formula. Original formula is w(i) = sqr((R-di)/(R*di)),
* where di is i-th distance, R is max(di). Modified formula have
* following form:
* w_mod(i) = 1, if di=d0
* w_mod(i) = w(i)/w(0), if di<>d0
*
* NOTE 2: self-match is NOT used for this query
*
* NOTE 3: last point almost always gain zero weight, but it MUST
* be used for fitting because sometimes it will gain NON-ZERO
* weight - for example, when all distances are equal.
*/
ae_v_move(&x.ptr.p_double[0], 1, &xy->ptr.pp_double[i][0], 1, ae_v_len(0,nx-1));
k = kdtreequeryknn(&z->tree, &x, nq, ae_false, _state);
kdtreequeryresultsxy(&z->tree, &qxybuf, _state);
kdtreequeryresultsdistances(&z->tree, &qrbuf, _state);
r = qrbuf.ptr.p_double[k-1];
d0 = qrbuf.ptr.p_double[0];
for(j=0; j<=k-1; j++)
{
di = qrbuf.ptr.p_double[j];
if( ae_fp_eq(di,d0) )
{
/*
* distance is equal to shortest, set it 1.0
* without explicitly calculating (which would give
* us same result, but 'll expose us to the risk of
* division by zero).
*/
w.ptr.p_double[j] = 1;
}
else
{
/*
* use normalized formula
*/
v1 = (r-di)/(r-d0);
v2 = d0/di;
w.ptr.p_double[j] = ae_sqr(v1*v2, _state);
}
}
/*
* calculate linear/quadratic members
*/
if( d==-1 )
{
/*
* "Fast" linear nodal function calculated using
* inverse distance weighting
*/
for(j=0; j<=nx-1; j++)
{
x.ptr.p_double[j] = 0;
}
s = 0;
for(j=0; j<=k-1; j++)
{
/*
* calculate J-th inverse distance weighted gradient:
* grad_k = (y_j-y_k)*(x_j-x_k)/sqr(norm(x_j-x_k))
* grad = sum(wk*grad_k)/sum(w_k)
*/
v = 0;
for(j2=0; j2<=nx-1; j2++)
{
v = v+ae_sqr(qxybuf.ptr.pp_double[j][j2]-xy->ptr.pp_double[i][j2], _state);
}
/*
* Although x_j<>x_k, sqr(norm(x_j-x_k)) may be zero due to
* underflow. If it is, we assume than J-th gradient is zero
* (i.e. don't add anything)
*/
if( ae_fp_neq(v,0) )
{
for(j2=0; j2<=nx-1; j2++)
{
x.ptr.p_double[j2] = x.ptr.p_double[j2]+w.ptr.p_double[j]*(qxybuf.ptr.pp_double[j][nx]-xy->ptr.pp_double[i][nx])*(qxybuf.ptr.pp_double[j][j2]-xy->ptr.pp_double[i][j2])/v;
}
}
s = s+w.ptr.p_double[j];
}
for(j=0; j<=nx-1; j++)
{
z->q.ptr.pp_double[i][nx+1+j] = x.ptr.p_double[j]/s;
}
}
else
{
/*
* Least squares models: build
*/
if( d==1 )
{
/*
* Linear nodal function calculated using
* least squares fitting to its neighbors
*/
for(j=0; j<=k-1; j++)
{
for(j2=0; j2<=nx-1; j2++)
{
fmatrix.ptr.pp_double[j][j2] = qxybuf.ptr.pp_double[j][j2]-xy->ptr.pp_double[i][j2];
}
y.ptr.p_double[j] = qxybuf.ptr.pp_double[j][nx]-xy->ptr.pp_double[i][nx];
}
nc = nx;
}
if( d==2 )
{
/*
* Quadratic nodal function calculated using
* least squares fitting to its neighbors
*/
for(j=0; j<=k-1; j++)
{
offs = 0;
for(j2=0; j2<=nx-1; j2++)
{
fmatrix.ptr.pp_double[j][offs] = qxybuf.ptr.pp_double[j][j2]-xy->ptr.pp_double[i][j2];
offs = offs+1;
}
for(j2=0; j2<=nx-1; j2++)
{
for(j3=j2; j3<=nx-1; j3++)
{
fmatrix.ptr.pp_double[j][offs] = (qxybuf.ptr.pp_double[j][j2]-xy->ptr.pp_double[i][j2])*(qxybuf.ptr.pp_double[j][j3]-xy->ptr.pp_double[i][j3]);
offs = offs+1;
}
}
y.ptr.p_double[j] = qxybuf.ptr.pp_double[j][nx]-xy->ptr.pp_double[i][nx];
}
nc = nx+ae_round(nx*(nx+1)*0.5, _state);
}
idwint_idwinternalsolver(&y, &w, &fmatrix, &temp, k, nc, &info, &qsol, &taskrcond, _state);
/*
* Least squares models: copy results
*/
if( info>0 )
{
/*
* LLS task is solved, copy results
*/
z->debugworstrcond = ae_minreal(z->debugworstrcond, taskrcond, _state);
z->debugbestrcond = ae_maxreal(z->debugbestrcond, taskrcond, _state);
for(j=0; j<=nc-1; j++)
{
z->q.ptr.pp_double[i][nx+1+j] = qsol.ptr.p_double[j];
}
}
else
{
/*
* Solver failure, very strange, but we will use
* zero values to handle it.
*/
z->debugsolverfailures = z->debugsolverfailures+1;
for(j=0; j<=nc-1; j++)
{
z->q.ptr.pp_double[i][nx+1+j] = 0;
}
}
}
}
ae_frame_leave(_state);
}
/*************************************************************************
IDW interpolant using modified Shepard method for non-uniform datasets.
This type of model uses constant nodal functions and interpolates using
all nodes which are closer than user-specified radius R. It may be used
when points distribution is non-uniform at the small scale, but it is at
the distances as large as R.
INPUT PARAMETERS:
XY - X and Y values, array[0..N-1,0..NX].
First NX columns contain X-values, last column contain
Y-values.
N - number of nodes, N>0.
NX - space dimension, NX>=1.
R - radius, R>0
OUTPUT PARAMETERS:
Z - IDW interpolant.
NOTES:
* if there is less than IDWKMin points within R-ball, algorithm selects
IDWKMin closest ones, so that continuity properties of interpolant are
preserved even far from points.
-- ALGLIB PROJECT --
Copyright 11.04.2010 by Bochkanov Sergey
*************************************************************************/
void idwbuildmodifiedshepardr(/* Real */ ae_matrix* xy,
ae_int_t n,
ae_int_t nx,
double r,
idwinterpolant* z,
ae_state *_state)
{
ae_frame _frame_block;
ae_int_t i;
ae_vector tags;
ae_frame_make(_state, &_frame_block);
_idwinterpolant_clear(z);
ae_vector_init(&tags, 0, DT_INT, _state, ae_true);
/*
* assertions
*/
ae_assert(n>0, "IDWBuildModifiedShepardR: N<=0!", _state);
ae_assert(nx>=1, "IDWBuildModifiedShepardR: NX<1!", _state);
ae_assert(ae_fp_greater(r,0), "IDWBuildModifiedShepardR: R<=0!", _state);
/*
* primary initialization of Z
*/
idwint_idwinit1(n, nx, 0, 0, n, z, _state);
z->modeltype = 1;
z->r = r;
/*
* Create KD-tree
*/
ae_vector_set_length(&tags, n, _state);
for(i=0; i<=n-1; i++)
{
tags.ptr.p_int[i] = i;
}
kdtreebuildtagged(xy, &tags, n, nx, 1, 2, &z->tree, _state);
/*
* build nodal functions
*/
for(i=0; i<=n-1; i++)
{
ae_v_move(&z->q.ptr.pp_double[i][0], 1, &xy->ptr.pp_double[i][0], 1, ae_v_len(0,nx));
}
ae_frame_leave(_state);
}
/*************************************************************************
IDW model for noisy data.
This subroutine may be used to handle noisy data, i.e. data with noise in
OUTPUT values. It differs from IDWBuildModifiedShepard() in the following
aspects:
* nodal functions are not constrained to pass through nodes: Qi(xi)<>yi,
i.e. we have fitting instead of interpolation.
* weights which are used during least squares fitting stage are all equal
to 1.0 (independently of distance)
* "fast"-linear or constant nodal functions are not supported (either not
robust enough or too rigid)
This problem require far more complex tuning than interpolation problems.
Below you can find some recommendations regarding this problem:
* focus on tuning NQ; it controls noise reduction. As for NW, you can just
make it equal to 2*NQ.
* you can use cross-validation to determine optimal NQ.
* optimal NQ is a result of complex tradeoff between noise level (more
noise = larger NQ required) and underlying function complexity (given
fixed N, larger NQ means smoothing of compex features in the data). For
example, NQ=N will reduce noise to the minimum level possible, but you
will end up with just constant/linear/quadratic (depending on D) least
squares model for the whole dataset.
INPUT PARAMETERS:
XY - X and Y values, array[0..N-1,0..NX].
First NX columns contain X-values, last column contain
Y-values.
N - number of nodes, N>0.
NX - space dimension, NX>=1.
D - nodal function degree, either:
* 1 linear model, least squares fitting. Simpe model for
datasets too small for quadratic models (or for very
noisy problems).
* 2 quadratic model, least squares fitting. Best model
available (if your dataset is large enough).
NQ - number of points used to calculate nodal functions. NQ should
be significantly larger than 1.5 times the number of
coefficients in a nodal function to overcome effects of noise:
* larger than 1.5*(1+NX) for linear model,
* larger than 3/4*(NX+2)*(NX+1) for quadratic model.
Values less than this threshold will be silently increased.
NW - number of points used to calculate weights and to interpolate.
Required: >=2^NX+1, values less than this threshold will be
silently increased.
Recommended value: about 2*NQ or larger
OUTPUT PARAMETERS:
Z - IDW interpolant.
NOTES:
* best results are obtained with quadratic models, linear models are not
recommended to use unless you are pretty sure that it is what you want
* this subroutine is always succeeds (as long as correct parameters are
passed).
* see 'Multivariate Interpolation of Large Sets of Scattered Data' by
Robert J. Renka for more information on this algorithm.
-- ALGLIB PROJECT --
Copyright 02.03.2010 by Bochkanov Sergey
*************************************************************************/
void idwbuildnoisy(/* Real */ ae_matrix* xy,
ae_int_t n,
ae_int_t nx,
ae_int_t d,
ae_int_t nq,
ae_int_t nw,
idwinterpolant* z,
ae_state *_state)
{
ae_frame _frame_block;
ae_int_t i;
ae_int_t j;
ae_int_t k;
ae_int_t j2;
ae_int_t j3;
double v;
ae_int_t nc;
ae_int_t offs;
double taskrcond;
ae_vector x;
ae_vector qrbuf;
ae_matrix qxybuf;
ae_vector y;
ae_vector w;
ae_matrix fmatrix;
ae_vector qsol;
ae_vector tags;
ae_vector temp;
ae_int_t info;
ae_frame_make(_state, &_frame_block);
_idwinterpolant_clear(z);
ae_vector_init(&x, 0, DT_REAL, _state, ae_true);
ae_vector_init(&qrbuf, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&qxybuf, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&y, 0, DT_REAL, _state, ae_true);
ae_vector_init(&w, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&fmatrix, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&qsol, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tags, 0, DT_INT, _state, ae_true);
ae_vector_init(&temp, 0, DT_REAL, _state, ae_true);
/*
* these initializers are not really necessary,
* but without them compiler complains about uninitialized locals
*/
nc = 0;
/*
* assertions
*/
ae_assert(n>0, "IDWBuildNoisy: N<=0!", _state);
ae_assert(nx>=1, "IDWBuildNoisy: NX<1!", _state);
ae_assert(d>=1&&d<=2, "IDWBuildNoisy: D<>1 and D<>2!", _state);
/*
* Correct parameters if needed
*/
if( d==1 )
{
nq = ae_maxint(nq, ae_iceil(idwint_idwqfactor*(1+nx), _state)+1, _state);
}
if( d==2 )
{
nq = ae_maxint(nq, ae_iceil(idwint_idwqfactor*(nx+2)*(nx+1)/2, _state)+1, _state);
}
nw = ae_maxint(nw, ae_round(ae_pow(2, nx, _state), _state)+1, _state);
nq = ae_minint(nq, n, _state);
nw = ae_minint(nw, n, _state);
/*
* primary initialization of Z
*/
idwint_idwinit1(n, nx, d, nq, nw, z, _state);
z->modeltype = 0;
/*
* Create KD-tree
*/
ae_vector_set_length(&tags, n, _state);
for(i=0; i<=n-1; i++)
{
tags.ptr.p_int[i] = i;
}
kdtreebuildtagged(xy, &tags, n, nx, 1, 2, &z->tree, _state);
/*
* build nodal functions
* (special algorithm for noisy data is used)
*/
ae_vector_set_length(&temp, nq+1, _state);
ae_vector_set_length(&x, nx, _state);
ae_vector_set_length(&qrbuf, nq, _state);
ae_matrix_set_length(&qxybuf, nq, nx+1, _state);
if( d==1 )
{
ae_vector_set_length(&y, nq, _state);
ae_vector_set_length(&w, nq, _state);
ae_vector_set_length(&qsol, 1+nx, _state);
/*
* 1 for constant member,
* NX for linear members,
* 1 for temporary storage
*/
ae_matrix_set_length(&fmatrix, nq, 1+nx+1, _state);
}
if( d==2 )
{
ae_vector_set_length(&y, nq, _state);
ae_vector_set_length(&w, nq, _state);
ae_vector_set_length(&qsol, 1+nx+ae_round(nx*(nx+1)*0.5, _state), _state);
/*
* 1 for constant member,
* NX for linear members,
* Round(NX*(NX+1)*0.5) for quadratic model,
* 1 for temporary storage
*/
ae_matrix_set_length(&fmatrix, nq, 1+nx+ae_round(nx*(nx+1)*0.5, _state)+1, _state);
}
for(i=0; i<=n-1; i++)
{
/*
* Initialize center.
*/
ae_v_move(&z->q.ptr.pp_double[i][0], 1, &xy->ptr.pp_double[i][0], 1, ae_v_len(0,nx-1));
/*
* Calculate linear/quadratic members
* using least squares fit
* NOTE 1: all weight are equal to 1.0
* NOTE 2: self-match is USED for this query
*/
ae_v_move(&x.ptr.p_double[0], 1, &xy->ptr.pp_double[i][0], 1, ae_v_len(0,nx-1));
k = kdtreequeryknn(&z->tree, &x, nq, ae_true, _state);
kdtreequeryresultsxy(&z->tree, &qxybuf, _state);
kdtreequeryresultsdistances(&z->tree, &qrbuf, _state);
if( d==1 )
{
/*
* Linear nodal function calculated using
* least squares fitting to its neighbors
*/
for(j=0; j<=k-1; j++)
{
fmatrix.ptr.pp_double[j][0] = 1.0;
for(j2=0; j2<=nx-1; j2++)
{
fmatrix.ptr.pp_double[j][1+j2] = qxybuf.ptr.pp_double[j][j2]-xy->ptr.pp_double[i][j2];
}
y.ptr.p_double[j] = qxybuf.ptr.pp_double[j][nx];
w.ptr.p_double[j] = 1;
}
nc = 1+nx;
}
if( d==2 )
{
/*
* Quadratic nodal function calculated using
* least squares fitting to its neighbors
*/
for(j=0; j<=k-1; j++)
{
fmatrix.ptr.pp_double[j][0] = 1;
offs = 1;
for(j2=0; j2<=nx-1; j2++)
{
fmatrix.ptr.pp_double[j][offs] = qxybuf.ptr.pp_double[j][j2]-xy->ptr.pp_double[i][j2];
offs = offs+1;
}
for(j2=0; j2<=nx-1; j2++)
{
for(j3=j2; j3<=nx-1; j3++)
{
fmatrix.ptr.pp_double[j][offs] = (qxybuf.ptr.pp_double[j][j2]-xy->ptr.pp_double[i][j2])*(qxybuf.ptr.pp_double[j][j3]-xy->ptr.pp_double[i][j3]);
offs = offs+1;
}
}
y.ptr.p_double[j] = qxybuf.ptr.pp_double[j][nx];
w.ptr.p_double[j] = 1;
}
nc = 1+nx+ae_round(nx*(nx+1)*0.5, _state);
}
idwint_idwinternalsolver(&y, &w, &fmatrix, &temp, k, nc, &info, &qsol, &taskrcond, _state);
/*
* Least squares models: copy results
*/
if( info>0 )
{
/*
* LLS task is solved, copy results
*/
z->debugworstrcond = ae_minreal(z->debugworstrcond, taskrcond, _state);
z->debugbestrcond = ae_maxreal(z->debugbestrcond, taskrcond, _state);
for(j=0; j<=nc-1; j++)
{
z->q.ptr.pp_double[i][nx+j] = qsol.ptr.p_double[j];
}
}
else
{
/*
* Solver failure, very strange, but we will use
* zero values to handle it.
*/
z->debugsolverfailures = z->debugsolverfailures+1;
v = 0;
for(j=0; j<=k-1; j++)
{
v = v+qxybuf.ptr.pp_double[j][nx];
}
z->q.ptr.pp_double[i][nx] = v/k;
for(j=0; j<=nc-2; j++)
{
z->q.ptr.pp_double[i][nx+1+j] = 0;
}
}
}
ae_frame_leave(_state);
}
/*************************************************************************
Internal subroutine: K-th nodal function calculation
-- ALGLIB --
Copyright 02.03.2010 by Bochkanov Sergey
*************************************************************************/
static double idwint_idwcalcq(idwinterpolant* z,
/* Real */ ae_vector* x,
ae_int_t k,
ae_state *_state)
{
ae_int_t nx;
ae_int_t i;
ae_int_t j;
ae_int_t offs;
double result;
nx = z->nx;
/*
* constant member
*/
result = z->q.ptr.pp_double[k][nx];
/*
* linear members
*/
if( z->d>=1 )
{
for(i=0; i<=nx-1; i++)
{
result = result+z->q.ptr.pp_double[k][nx+1+i]*(x->ptr.p_double[i]-z->q.ptr.pp_double[k][i]);
}
}
/*
* quadratic members
*/
if( z->d>=2 )
{
offs = nx+1+nx;
for(i=0; i<=nx-1; i++)
{
for(j=i; j<=nx-1; j++)
{
result = result+z->q.ptr.pp_double[k][offs]*(x->ptr.p_double[i]-z->q.ptr.pp_double[k][i])*(x->ptr.p_double[j]-z->q.ptr.pp_double[k][j]);
offs = offs+1;
}
}
}
return result;
}
/*************************************************************************
Initialization of internal structures.
It assumes correctness of all parameters.
-- ALGLIB --
Copyright 02.03.2010 by Bochkanov Sergey
*************************************************************************/
static void idwint_idwinit1(ae_int_t n,
ae_int_t nx,
ae_int_t d,
ae_int_t nq,
ae_int_t nw,
idwinterpolant* z,
ae_state *_state)
{
z->debugsolverfailures = 0;
z->debugworstrcond = 1.0;
z->debugbestrcond = 0;
z->n = n;
z->nx = nx;
z->d = 0;
if( d==1 )
{
z->d = 1;
}
if( d==2 )
{
z->d = 2;
}
if( d==-1 )
{
z->d = 1;
}
z->nw = nw;
if( d==-1 )
{
ae_matrix_set_length(&z->q, n, nx+1+nx, _state);
}
if( d==0 )
{
ae_matrix_set_length(&z->q, n, nx+1, _state);
}
if( d==1 )
{
ae_matrix_set_length(&z->q, n, nx+1+nx, _state);
}
if( d==2 )
{
ae_matrix_set_length(&z->q, n, nx+1+nx+ae_round(nx*(nx+1)*0.5, _state), _state);
}
ae_vector_set_length(&z->tbuf, nw, _state);
ae_vector_set_length(&z->rbuf, nw, _state);
ae_matrix_set_length(&z->xybuf, nw, nx+1, _state);
ae_vector_set_length(&z->xbuf, nx, _state);
}
/*************************************************************************
Linear least squares solver for small tasks.
Works faster than standard ALGLIB solver in non-degenerate cases (due to
absense of internal allocations and optimized row/colums). In degenerate
cases it calls standard solver, which results in small performance penalty
associated with preliminary steps.
INPUT PARAMETERS:
Y array[0..N-1]
W array[0..N-1]
FMatrix array[0..N-1,0..M], have additional column for temporary
values
Temp array[0..N]
*************************************************************************/
static void idwint_idwinternalsolver(/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
/* Real */ ae_matrix* fmatrix,
/* Real */ ae_vector* temp,
ae_int_t n,
ae_int_t m,
ae_int_t* info,
/* Real */ ae_vector* x,
double* taskrcond,
ae_state *_state)
{
ae_frame _frame_block;
ae_int_t i;
ae_int_t j;
double v;
double tau;
ae_vector b;
densesolverlsreport srep;
ae_frame_make(_state, &_frame_block);
*info = 0;
ae_vector_init(&b, 0, DT_REAL, _state, ae_true);
_densesolverlsreport_init(&srep, _state, ae_true);
/*
* set up info
*/
*info = 1;
/*
* prepare matrix
*/
for(i=0; i<=n-1; i++)
{
fmatrix->ptr.pp_double[i][m] = y->ptr.p_double[i];
v = w->ptr.p_double[i];
ae_v_muld(&fmatrix->ptr.pp_double[i][0], 1, ae_v_len(0,m), v);
}
/*
* use either fast algorithm or general algorithm
*/
if( m<=n )
{
/*
* QR decomposition
* We assume that M<=N (we would have called LSFit() otherwise)
*/
for(i=0; i<=m-1; i++)
{
if( i<n-1 )
{
ae_v_move(&temp->ptr.p_double[1], 1, &fmatrix->ptr.pp_double[i][i], fmatrix->stride, ae_v_len(1,n-i));
generatereflection(temp, n-i, &tau, _state);
fmatrix->ptr.pp_double[i][i] = temp->ptr.p_double[1];
temp->ptr.p_double[1] = 1;
for(j=i+1; j<=m; j++)
{
v = ae_v_dotproduct(&fmatrix->ptr.pp_double[i][j], fmatrix->stride, &temp->ptr.p_double[1], 1, ae_v_len(i,n-1));
v = tau*v;
ae_v_subd(&fmatrix->ptr.pp_double[i][j], fmatrix->stride, &temp->ptr.p_double[1], 1, ae_v_len(i,n-1), v);
}
}
}
/*
* Check condition number
*/
*taskrcond = rmatrixtrrcondinf(fmatrix, m, ae_true, ae_false, _state);
/*
* use either fast algorithm for non-degenerate cases
* or slow algorithm for degenerate cases
*/
if( ae_fp_greater(*taskrcond,10000*n*ae_machineepsilon) )
{
/*
* solve triangular system R*x = FMatrix[0:M-1,M]
* using fast algorithm, then exit
*/
x->ptr.p_double[m-1] = fmatrix->ptr.pp_double[m-1][m]/fmatrix->ptr.pp_double[m-1][m-1];
for(i=m-2; i>=0; i--)
{
v = ae_v_dotproduct(&fmatrix->ptr.pp_double[i][i+1], 1, &x->ptr.p_double[i+1], 1, ae_v_len(i+1,m-1));
x->ptr.p_double[i] = (fmatrix->ptr.pp_double[i][m]-v)/fmatrix->ptr.pp_double[i][i];
}
}
else
{
/*
* use more general algorithm
*/
ae_vector_set_length(&b, m, _state);
for(i=0; i<=m-1; i++)
{
for(j=0; j<=i-1; j++)
{
fmatrix->ptr.pp_double[i][j] = 0.0;
}
b.ptr.p_double[i] = fmatrix->ptr.pp_double[i][m];
}
rmatrixsolvels(fmatrix, m, m, &b, 10000*ae_machineepsilon, info, &srep, x, _state);
}
}
else
{
/*
* use more general algorithm
*/
ae_vector_set_length(&b, n, _state);
for(i=0; i<=n-1; i++)
{
b.ptr.p_double[i] = fmatrix->ptr.pp_double[i][m];
}
rmatrixsolvels(fmatrix, n, m, &b, 10000*ae_machineepsilon, info, &srep, x, _state);
*taskrcond = srep.r2;
}
ae_frame_leave(_state);
}
ae_bool _idwinterpolant_init(void* _p, ae_state *_state, ae_bool make_automatic)
{
idwinterpolant *p = (idwinterpolant*)_p;
ae_touch_ptr((void*)p);
if( !_kdtree_init(&p->tree, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init(&p->q, 0, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->xbuf, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->tbuf, 0, DT_INT, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->rbuf, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init(&p->xybuf, 0, 0, DT_REAL, _state, make_automatic) )
return ae_false;
return ae_true;
}
ae_bool _idwinterpolant_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic)
{
idwinterpolant *dst = (idwinterpolant*)_dst;
idwinterpolant *src = (idwinterpolant*)_src;
dst->n = src->n;
dst->nx = src->nx;
dst->d = src->d;
dst->r = src->r;
dst->nw = src->nw;
if( !_kdtree_init_copy(&dst->tree, &src->tree, _state, make_automatic) )
return ae_false;
dst->modeltype = src->modeltype;
if( !ae_matrix_init_copy(&dst->q, &src->q, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->xbuf, &src->xbuf, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->tbuf, &src->tbuf, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->rbuf, &src->rbuf, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init_copy(&dst->xybuf, &src->xybuf, _state, make_automatic) )
return ae_false;
dst->debugsolverfailures = src->debugsolverfailures;
dst->debugworstrcond = src->debugworstrcond;
dst->debugbestrcond = src->debugbestrcond;
return ae_true;
}
void _idwinterpolant_clear(void* _p)
{
idwinterpolant *p = (idwinterpolant*)_p;
ae_touch_ptr((void*)p);
_kdtree_clear(&p->tree);
ae_matrix_clear(&p->q);
ae_vector_clear(&p->xbuf);
ae_vector_clear(&p->tbuf);
ae_vector_clear(&p->rbuf);
ae_matrix_clear(&p->xybuf);
}
void _idwinterpolant_destroy(void* _p)
{
idwinterpolant *p = (idwinterpolant*)_p;
ae_touch_ptr((void*)p);
_kdtree_destroy(&p->tree);
ae_matrix_destroy(&p->q);
ae_vector_destroy(&p->xbuf);
ae_vector_destroy(&p->tbuf);
ae_vector_destroy(&p->rbuf);
ae_matrix_destroy(&p->xybuf);
}
/*************************************************************************
Rational interpolation using barycentric formula
F(t) = SUM(i=0,n-1,w[i]*f[i]/(t-x[i])) / SUM(i=0,n-1,w[i]/(t-x[i]))
Input parameters:
B - barycentric interpolant built with one of model building
subroutines.
T - interpolation point
Result:
barycentric interpolant F(t)
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
double barycentriccalc(barycentricinterpolant* b,
double t,
ae_state *_state)
{
double s1;
double s2;
double s;
double v;
ae_int_t i;
double result;
ae_assert(!ae_isinf(t, _state), "BarycentricCalc: infinite T!", _state);
/*
* special case: NaN
*/
if( ae_isnan(t, _state) )
{
result = _state->v_nan;
return result;
}
/*
* special case: N=1
*/
if( b->n==1 )
{
result = b->sy*b->y.ptr.p_double[0];
return result;
}
/*
* Here we assume that task is normalized, i.e.:
* 1. abs(Y[i])<=1
* 2. abs(W[i])<=1
* 3. X[] is ordered
*/
s = ae_fabs(t-b->x.ptr.p_double[0], _state);
for(i=0; i<=b->n-1; i++)
{
v = b->x.ptr.p_double[i];
if( ae_fp_eq(v,t) )
{
result = b->sy*b->y.ptr.p_double[i];
return result;
}
v = ae_fabs(t-v, _state);
if( ae_fp_less(v,s) )
{
s = v;
}
}
s1 = 0;
s2 = 0;
for(i=0; i<=b->n-1; i++)
{
v = s/(t-b->x.ptr.p_double[i]);
v = v*b->w.ptr.p_double[i];
s1 = s1+v*b->y.ptr.p_double[i];
s2 = s2+v;
}
result = b->sy*s1/s2;
return result;
}
/*************************************************************************
Differentiation of barycentric interpolant: first derivative.
Algorithm used in this subroutine is very robust and should not fail until
provided with values too close to MaxRealNumber (usually MaxRealNumber/N
or greater will overflow).
INPUT PARAMETERS:
B - barycentric interpolant built with one of model building
subroutines.
T - interpolation point
OUTPUT PARAMETERS:
F - barycentric interpolant at T
DF - first derivative
NOTE
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentricdiff1(barycentricinterpolant* b,
double t,
double* f,
double* df,
ae_state *_state)
{
double v;
double vv;
ae_int_t i;
ae_int_t k;
double n0;
double n1;
double d0;
double d1;
double s0;
double s1;
double xk;
double xi;
double xmin;
double xmax;
double xscale1;
double xoffs1;
double xscale2;
double xoffs2;
double xprev;
*f = 0;
*df = 0;
ae_assert(!ae_isinf(t, _state), "BarycentricDiff1: infinite T!", _state);
/*
* special case: NaN
*/
if( ae_isnan(t, _state) )
{
*f = _state->v_nan;
*df = _state->v_nan;
return;
}
/*
* special case: N=1
*/
if( b->n==1 )
{
*f = b->sy*b->y.ptr.p_double[0];
*df = 0;
return;
}
if( ae_fp_eq(b->sy,0) )
{
*f = 0;
*df = 0;
return;
}
ae_assert(ae_fp_greater(b->sy,0), "BarycentricDiff1: internal error", _state);
/*
* We assume than N>1 and B.SY>0. Find:
* 1. pivot point (X[i] closest to T)
* 2. width of interval containing X[i]
*/
v = ae_fabs(b->x.ptr.p_double[0]-t, _state);
k = 0;
xmin = b->x.ptr.p_double[0];
xmax = b->x.ptr.p_double[0];
for(i=1; i<=b->n-1; i++)
{
vv = b->x.ptr.p_double[i];
if( ae_fp_less(ae_fabs(vv-t, _state),v) )
{
v = ae_fabs(vv-t, _state);
k = i;
}
xmin = ae_minreal(xmin, vv, _state);
xmax = ae_maxreal(xmax, vv, _state);
}
/*
* pivot point found, calculate dNumerator and dDenominator
*/
xscale1 = 1/(xmax-xmin);
xoffs1 = -xmin/(xmax-xmin)+1;
xscale2 = 2;
xoffs2 = -3;
t = t*xscale1+xoffs1;
t = t*xscale2+xoffs2;
xk = b->x.ptr.p_double[k];
xk = xk*xscale1+xoffs1;
xk = xk*xscale2+xoffs2;
v = t-xk;
n0 = 0;
n1 = 0;
d0 = 0;
d1 = 0;
xprev = -2;
for(i=0; i<=b->n-1; i++)
{
xi = b->x.ptr.p_double[i];
xi = xi*xscale1+xoffs1;
xi = xi*xscale2+xoffs2;
ae_assert(ae_fp_greater(xi,xprev), "BarycentricDiff1: points are too close!", _state);
xprev = xi;
if( i!=k )
{
vv = ae_sqr(t-xi, _state);
s0 = (t-xk)/(t-xi);
s1 = (xk-xi)/vv;
}
else
{
s0 = 1;
s1 = 0;
}
vv = b->w.ptr.p_double[i]*b->y.ptr.p_double[i];
n0 = n0+s0*vv;
n1 = n1+s1*vv;
vv = b->w.ptr.p_double[i];
d0 = d0+s0*vv;
d1 = d1+s1*vv;
}
*f = b->sy*n0/d0;
*df = (n1*d0-n0*d1)/ae_sqr(d0, _state);
if( ae_fp_neq(*df,0) )
{
*df = ae_sign(*df, _state)*ae_exp(ae_log(ae_fabs(*df, _state), _state)+ae_log(b->sy, _state)+ae_log(xscale1, _state)+ae_log(xscale2, _state), _state);
}
}
/*************************************************************************
Differentiation of barycentric interpolant: first/second derivatives.
INPUT PARAMETERS:
B - barycentric interpolant built with one of model building
subroutines.
T - interpolation point
OUTPUT PARAMETERS:
F - barycentric interpolant at T
DF - first derivative
D2F - second derivative
NOTE: this algorithm may fail due to overflow/underflor if used on data
whose values are close to MaxRealNumber or MinRealNumber. Use more robust
BarycentricDiff1() subroutine in such cases.
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentricdiff2(barycentricinterpolant* b,
double t,
double* f,
double* df,
double* d2f,
ae_state *_state)
{
double v;
double vv;
ae_int_t i;
ae_int_t k;
double n0;
double n1;
double n2;
double d0;
double d1;
double d2;
double s0;
double s1;
double s2;
double xk;
double xi;
*f = 0;
*df = 0;
*d2f = 0;
ae_assert(!ae_isinf(t, _state), "BarycentricDiff1: infinite T!", _state);
/*
* special case: NaN
*/
if( ae_isnan(t, _state) )
{
*f = _state->v_nan;
*df = _state->v_nan;
*d2f = _state->v_nan;
return;
}
/*
* special case: N=1
*/
if( b->n==1 )
{
*f = b->sy*b->y.ptr.p_double[0];
*df = 0;
*d2f = 0;
return;
}
if( ae_fp_eq(b->sy,0) )
{
*f = 0;
*df = 0;
*d2f = 0;
return;
}
/*
* We assume than N>1 and B.SY>0. Find:
* 1. pivot point (X[i] closest to T)
* 2. width of interval containing X[i]
*/
ae_assert(ae_fp_greater(b->sy,0), "BarycentricDiff: internal error", _state);
*f = 0;
*df = 0;
*d2f = 0;
v = ae_fabs(b->x.ptr.p_double[0]-t, _state);
k = 0;
for(i=1; i<=b->n-1; i++)
{
vv = b->x.ptr.p_double[i];
if( ae_fp_less(ae_fabs(vv-t, _state),v) )
{
v = ae_fabs(vv-t, _state);
k = i;
}
}
/*
* pivot point found, calculate dNumerator and dDenominator
*/
xk = b->x.ptr.p_double[k];
v = t-xk;
n0 = 0;
n1 = 0;
n2 = 0;
d0 = 0;
d1 = 0;
d2 = 0;
for(i=0; i<=b->n-1; i++)
{
if( i!=k )
{
xi = b->x.ptr.p_double[i];
vv = ae_sqr(t-xi, _state);
s0 = (t-xk)/(t-xi);
s1 = (xk-xi)/vv;
s2 = -2*(xk-xi)/(vv*(t-xi));
}
else
{
s0 = 1;
s1 = 0;
s2 = 0;
}
vv = b->w.ptr.p_double[i]*b->y.ptr.p_double[i];
n0 = n0+s0*vv;
n1 = n1+s1*vv;
n2 = n2+s2*vv;
vv = b->w.ptr.p_double[i];
d0 = d0+s0*vv;
d1 = d1+s1*vv;
d2 = d2+s2*vv;
}
*f = b->sy*n0/d0;
*df = b->sy*(n1*d0-n0*d1)/ae_sqr(d0, _state);
*d2f = b->sy*((n2*d0-n0*d2)*ae_sqr(d0, _state)-(n1*d0-n0*d1)*2*d0*d1)/ae_sqr(ae_sqr(d0, _state), _state);
}
/*************************************************************************
This subroutine performs linear transformation of the argument.
INPUT PARAMETERS:
B - rational interpolant in barycentric form
CA, CB - transformation coefficients: x = CA*t + CB
OUTPUT PARAMETERS:
B - transformed interpolant with X replaced by T
-- ALGLIB PROJECT --
Copyright 19.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentriclintransx(barycentricinterpolant* b,
double ca,
double cb,
ae_state *_state)
{
ae_int_t i;
ae_int_t j;
double v;
/*
* special case, replace by constant F(CB)
*/
if( ae_fp_eq(ca,0) )
{
b->sy = barycentriccalc(b, cb, _state);
v = 1;
for(i=0; i<=b->n-1; i++)
{
b->y.ptr.p_double[i] = 1;
b->w.ptr.p_double[i] = v;
v = -v;
}
return;
}
/*
* general case: CA<>0
*/
for(i=0; i<=b->n-1; i++)
{
b->x.ptr.p_double[i] = (b->x.ptr.p_double[i]-cb)/ca;
}
if( ae_fp_less(ca,0) )
{
for(i=0; i<=b->n-1; i++)
{
if( i<b->n-1-i )
{
j = b->n-1-i;
v = b->x.ptr.p_double[i];
b->x.ptr.p_double[i] = b->x.ptr.p_double[j];
b->x.ptr.p_double[j] = v;
v = b->y.ptr.p_double[i];
b->y.ptr.p_double[i] = b->y.ptr.p_double[j];
b->y.ptr.p_double[j] = v;
v = b->w.ptr.p_double[i];
b->w.ptr.p_double[i] = b->w.ptr.p_double[j];
b->w.ptr.p_double[j] = v;
}
else
{
break;
}
}
}
}
/*************************************************************************
This subroutine performs linear transformation of the barycentric
interpolant.
INPUT PARAMETERS:
B - rational interpolant in barycentric form
CA, CB - transformation coefficients: B2(x) = CA*B(x) + CB
OUTPUT PARAMETERS:
B - transformed interpolant
-- ALGLIB PROJECT --
Copyright 19.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentriclintransy(barycentricinterpolant* b,
double ca,
double cb,
ae_state *_state)
{
ae_int_t i;
double v;
for(i=0; i<=b->n-1; i++)
{
b->y.ptr.p_double[i] = ca*b->sy*b->y.ptr.p_double[i]+cb;
}
b->sy = 0;
for(i=0; i<=b->n-1; i++)
{
b->sy = ae_maxreal(b->sy, ae_fabs(b->y.ptr.p_double[i], _state), _state);
}
if( ae_fp_greater(b->sy,0) )
{
v = 1/b->sy;
ae_v_muld(&b->y.ptr.p_double[0], 1, ae_v_len(0,b->n-1), v);
}
}
/*************************************************************************
Extracts X/Y/W arrays from rational interpolant
INPUT PARAMETERS:
B - barycentric interpolant
OUTPUT PARAMETERS:
N - nodes count, N>0
X - interpolation nodes, array[0..N-1]
F - function values, array[0..N-1]
W - barycentric weights, array[0..N-1]
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentricunpack(barycentricinterpolant* b,
ae_int_t* n,
/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
ae_state *_state)
{
double v;
*n = 0;
ae_vector_clear(x);
ae_vector_clear(y);
ae_vector_clear(w);
*n = b->n;
ae_vector_set_length(x, *n, _state);
ae_vector_set_length(y, *n, _state);
ae_vector_set_length(w, *n, _state);
v = b->sy;
ae_v_move(&x->ptr.p_double[0], 1, &b->x.ptr.p_double[0], 1, ae_v_len(0,*n-1));
ae_v_moved(&y->ptr.p_double[0], 1, &b->y.ptr.p_double[0], 1, ae_v_len(0,*n-1), v);
ae_v_move(&w->ptr.p_double[0], 1, &b->w.ptr.p_double[0], 1, ae_v_len(0,*n-1));
}
/*************************************************************************
Rational interpolant from X/Y/W arrays
F(t) = SUM(i=0,n-1,w[i]*f[i]/(t-x[i])) / SUM(i=0,n-1,w[i]/(t-x[i]))
INPUT PARAMETERS:
X - interpolation nodes, array[0..N-1]
F - function values, array[0..N-1]
W - barycentric weights, array[0..N-1]
N - nodes count, N>0
OUTPUT PARAMETERS:
B - barycentric interpolant built from (X, Y, W)
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentricbuildxyw(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
ae_int_t n,
barycentricinterpolant* b,
ae_state *_state)
{
_barycentricinterpolant_clear(b);
ae_assert(n>0, "BarycentricBuildXYW: incorrect N!", _state);
/*
* fill X/Y/W
*/
ae_vector_set_length(&b->x, n, _state);
ae_vector_set_length(&b->y, n, _state);
ae_vector_set_length(&b->w, n, _state);
ae_v_move(&b->x.ptr.p_double[0], 1, &x->ptr.p_double[0], 1, ae_v_len(0,n-1));
ae_v_move(&b->y.ptr.p_double[0], 1, &y->ptr.p_double[0], 1, ae_v_len(0,n-1));
ae_v_move(&b->w.ptr.p_double[0], 1, &w->ptr.p_double[0], 1, ae_v_len(0,n-1));
b->n = n;
/*
* Normalize
*/
ratint_barycentricnormalize(b, _state);
}
/*************************************************************************
Rational interpolant without poles
The subroutine constructs the rational interpolating function without real
poles (see 'Barycentric rational interpolation with no poles and high
rates of approximation', Michael S. Floater. and Kai Hormann, for more
information on this subject).
Input parameters:
X - interpolation nodes, array[0..N-1].
Y - function values, array[0..N-1].
N - number of nodes, N>0.
D - order of the interpolation scheme, 0 <= D <= N-1.
D<0 will cause an error.
D>=N it will be replaced with D=N-1.
if you don't know what D to choose, use small value about 3-5.
Output parameters:
B - barycentric interpolant.
Note:
this algorithm always succeeds and calculates the weights with close
to machine precision.
-- ALGLIB PROJECT --
Copyright 17.06.2007 by Bochkanov Sergey
*************************************************************************/
void barycentricbuildfloaterhormann(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_int_t d,
barycentricinterpolant* b,
ae_state *_state)
{
ae_frame _frame_block;
double s0;
double s;
double v;
ae_int_t i;
ae_int_t j;
ae_int_t k;
ae_vector perm;
ae_vector wtemp;
ae_vector sortrbuf;
ae_vector sortrbuf2;
ae_frame_make(_state, &_frame_block);
_barycentricinterpolant_clear(b);
ae_vector_init(&perm, 0, DT_INT, _state, ae_true);
ae_vector_init(&wtemp, 0, DT_REAL, _state, ae_true);
ae_vector_init(&sortrbuf, 0, DT_REAL, _state, ae_true);
ae_vector_init(&sortrbuf2, 0, DT_REAL, _state, ae_true);
ae_assert(n>0, "BarycentricFloaterHormann: N<=0!", _state);
ae_assert(d>=0, "BarycentricFloaterHormann: incorrect D!", _state);
/*
* Prepare
*/
if( d>n-1 )
{
d = n-1;
}
b->n = n;
/*
* special case: N=1
*/
if( n==1 )
{
ae_vector_set_length(&b->x, n, _state);
ae_vector_set_length(&b->y, n, _state);
ae_vector_set_length(&b->w, n, _state);
b->x.ptr.p_double[0] = x->ptr.p_double[0];
b->y.ptr.p_double[0] = y->ptr.p_double[0];
b->w.ptr.p_double[0] = 1;
ratint_barycentricnormalize(b, _state);
ae_frame_leave(_state);
return;
}
/*
* Fill X/Y
*/
ae_vector_set_length(&b->x, n, _state);
ae_vector_set_length(&b->y, n, _state);
ae_v_move(&b->x.ptr.p_double[0], 1, &x->ptr.p_double[0], 1, ae_v_len(0,n-1));
ae_v_move(&b->y.ptr.p_double[0], 1, &y->ptr.p_double[0], 1, ae_v_len(0,n-1));
tagsortfastr(&b->x, &b->y, &sortrbuf, &sortrbuf2, n, _state);
/*
* Calculate Wk
*/
ae_vector_set_length(&b->w, n, _state);
s0 = 1;
for(k=1; k<=d; k++)
{
s0 = -s0;
}
for(k=0; k<=n-1; k++)
{
/*
* Wk
*/
s = 0;
for(i=ae_maxint(k-d, 0, _state); i<=ae_minint(k, n-1-d, _state); i++)
{
v = 1;
for(j=i; j<=i+d; j++)
{
if( j!=k )
{
v = v/ae_fabs(b->x.ptr.p_double[k]-b->x.ptr.p_double[j], _state);
}
}
s = s+v;
}
b->w.ptr.p_double[k] = s0*s;
/*
* Next S0
*/
s0 = -s0;
}
/*
* Normalize
*/
ratint_barycentricnormalize(b, _state);
ae_frame_leave(_state);
}
/*************************************************************************
Copying of the barycentric interpolant (for internal use only)
INPUT PARAMETERS:
B - barycentric interpolant
OUTPUT PARAMETERS:
B2 - copy(B1)
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentriccopy(barycentricinterpolant* b,
barycentricinterpolant* b2,
ae_state *_state)
{
_barycentricinterpolant_clear(b2);
b2->n = b->n;
b2->sy = b->sy;
ae_vector_set_length(&b2->x, b2->n, _state);
ae_vector_set_length(&b2->y, b2->n, _state);
ae_vector_set_length(&b2->w, b2->n, _state);
ae_v_move(&b2->x.ptr.p_double[0], 1, &b->x.ptr.p_double[0], 1, ae_v_len(0,b2->n-1));
ae_v_move(&b2->y.ptr.p_double[0], 1, &b->y.ptr.p_double[0], 1, ae_v_len(0,b2->n-1));
ae_v_move(&b2->w.ptr.p_double[0], 1, &b->w.ptr.p_double[0], 1, ae_v_len(0,b2->n-1));
}
/*************************************************************************
Normalization of barycentric interpolant:
* B.N, B.X, B.Y and B.W are initialized
* B.SY is NOT initialized
* Y[] is normalized, scaling coefficient is stored in B.SY
* W[] is normalized, no scaling coefficient is stored
* X[] is sorted
Internal subroutine.
*************************************************************************/
static void ratint_barycentricnormalize(barycentricinterpolant* b,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector p1;
ae_vector p2;
ae_int_t i;
ae_int_t j;
ae_int_t j2;
double v;
ae_frame_make(_state, &_frame_block);
ae_vector_init(&p1, 0, DT_INT, _state, ae_true);
ae_vector_init(&p2, 0, DT_INT, _state, ae_true);
/*
* Normalize task: |Y|<=1, |W|<=1, sort X[]
*/
b->sy = 0;
for(i=0; i<=b->n-1; i++)
{
b->sy = ae_maxreal(b->sy, ae_fabs(b->y.ptr.p_double[i], _state), _state);
}
if( ae_fp_greater(b->sy,0)&&ae_fp_greater(ae_fabs(b->sy-1, _state),10*ae_machineepsilon) )
{
v = 1/b->sy;
ae_v_muld(&b->y.ptr.p_double[0], 1, ae_v_len(0,b->n-1), v);
}
v = 0;
for(i=0; i<=b->n-1; i++)
{
v = ae_maxreal(v, ae_fabs(b->w.ptr.p_double[i], _state), _state);
}
if( ae_fp_greater(v,0)&&ae_fp_greater(ae_fabs(v-1, _state),10*ae_machineepsilon) )
{
v = 1/v;
ae_v_muld(&b->w.ptr.p_double[0], 1, ae_v_len(0,b->n-1), v);
}
for(i=0; i<=b->n-2; i++)
{
if( ae_fp_less(b->x.ptr.p_double[i+1],b->x.ptr.p_double[i]) )
{
tagsort(&b->x, b->n, &p1, &p2, _state);
for(j=0; j<=b->n-1; j++)
{
j2 = p2.ptr.p_int[j];
v = b->y.ptr.p_double[j];
b->y.ptr.p_double[j] = b->y.ptr.p_double[j2];
b->y.ptr.p_double[j2] = v;
v = b->w.ptr.p_double[j];
b->w.ptr.p_double[j] = b->w.ptr.p_double[j2];
b->w.ptr.p_double[j2] = v;
}
break;
}
}
ae_frame_leave(_state);
}
ae_bool _barycentricinterpolant_init(void* _p, ae_state *_state, ae_bool make_automatic)
{
barycentricinterpolant *p = (barycentricinterpolant*)_p;
ae_touch_ptr((void*)p);
if( !ae_vector_init(&p->x, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->y, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->w, 0, DT_REAL, _state, make_automatic) )
return ae_false;
return ae_true;
}
ae_bool _barycentricinterpolant_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic)
{
barycentricinterpolant *dst = (barycentricinterpolant*)_dst;
barycentricinterpolant *src = (barycentricinterpolant*)_src;
dst->n = src->n;
dst->sy = src->sy;
if( !ae_vector_init_copy(&dst->x, &src->x, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->y, &src->y, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->w, &src->w, _state, make_automatic) )
return ae_false;
return ae_true;
}
void _barycentricinterpolant_clear(void* _p)
{
barycentricinterpolant *p = (barycentricinterpolant*)_p;
ae_touch_ptr((void*)p);
ae_vector_clear(&p->x);
ae_vector_clear(&p->y);
ae_vector_clear(&p->w);
}
void _barycentricinterpolant_destroy(void* _p)
{
barycentricinterpolant *p = (barycentricinterpolant*)_p;
ae_touch_ptr((void*)p);
ae_vector_destroy(&p->x);
ae_vector_destroy(&p->y);
ae_vector_destroy(&p->w);
}
/*************************************************************************
Conversion from barycentric representation to Chebyshev basis.
This function has O(N^2) complexity.
INPUT PARAMETERS:
P - polynomial in barycentric form
A,B - base interval for Chebyshev polynomials (see below)
A<>B
OUTPUT PARAMETERS
T - coefficients of Chebyshev representation;
P(x) = sum { T[i]*Ti(2*(x-A)/(B-A)-1), i=0..N-1 },
where Ti - I-th Chebyshev polynomial.
NOTES:
barycentric interpolant passed as P may be either polynomial obtained
from polynomial interpolation/ fitting or rational function which is
NOT polynomial. We can't distinguish between these two cases, and this
algorithm just tries to work assuming that P IS a polynomial. If not,
algorithm will return results, but they won't have any meaning.
-- ALGLIB --
Copyright 30.09.2010 by Bochkanov Sergey
*************************************************************************/
void polynomialbar2cheb(barycentricinterpolant* p,
double a,
double b,
/* Real */ ae_vector* t,
ae_state *_state)
{
ae_frame _frame_block;
ae_int_t i;
ae_int_t k;
ae_vector vp;
ae_vector vx;
ae_vector tk;
ae_vector tk1;
double v;
ae_frame_make(_state, &_frame_block);
ae_vector_clear(t);
ae_vector_init(&vp, 0, DT_REAL, _state, ae_true);
ae_vector_init(&vx, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tk, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tk1, 0, DT_REAL, _state, ae_true);
ae_assert(ae_isfinite(a, _state), "PolynomialBar2Cheb: A is not finite!", _state);
ae_assert(ae_isfinite(b, _state), "PolynomialBar2Cheb: B is not finite!", _state);
ae_assert(ae_fp_neq(a,b), "PolynomialBar2Cheb: A=B!", _state);
ae_assert(p->n>0, "PolynomialBar2Cheb: P is not correctly initialized barycentric interpolant!", _state);
/*
* Calculate function values on a Chebyshev grid
*/
ae_vector_set_length(&vp, p->n, _state);
ae_vector_set_length(&vx, p->n, _state);
for(i=0; i<=p->n-1; i++)
{
vx.ptr.p_double[i] = ae_cos(ae_pi*(i+0.5)/p->n, _state);
vp.ptr.p_double[i] = barycentriccalc(p, 0.5*(vx.ptr.p_double[i]+1)*(b-a)+a, _state);
}
/*
* T[0]
*/
ae_vector_set_length(t, p->n, _state);
v = 0;
for(i=0; i<=p->n-1; i++)
{
v = v+vp.ptr.p_double[i];
}
t->ptr.p_double[0] = v/p->n;
/*
* other T's.
*
* NOTES:
* 1. TK stores T{k} on VX, TK1 stores T{k-1} on VX
* 2. we can do same calculations with fast DCT, but it
* * adds dependencies
* * still leaves us with O(N^2) algorithm because
* preparation of function values is O(N^2) process
*/
if( p->n>1 )
{
ae_vector_set_length(&tk, p->n, _state);
ae_vector_set_length(&tk1, p->n, _state);
for(i=0; i<=p->n-1; i++)
{
tk.ptr.p_double[i] = vx.ptr.p_double[i];
tk1.ptr.p_double[i] = 1;
}
for(k=1; k<=p->n-1; k++)
{
/*
* calculate discrete product of function vector and TK
*/
v = ae_v_dotproduct(&tk.ptr.p_double[0], 1, &vp.ptr.p_double[0], 1, ae_v_len(0,p->n-1));
t->ptr.p_double[k] = v/(0.5*p->n);
/*
* Update TK and TK1
*/
for(i=0; i<=p->n-1; i++)
{
v = 2*vx.ptr.p_double[i]*tk.ptr.p_double[i]-tk1.ptr.p_double[i];
tk1.ptr.p_double[i] = tk.ptr.p_double[i];
tk.ptr.p_double[i] = v;
}
}
}
ae_frame_leave(_state);
}
/*************************************************************************
Conversion from Chebyshev basis to barycentric representation.
This function has O(N^2) complexity.
INPUT PARAMETERS:
T - coefficients of Chebyshev representation;
P(x) = sum { T[i]*Ti(2*(x-A)/(B-A)-1), i=0..N },
where Ti - I-th Chebyshev polynomial.
N - number of coefficients:
* if given, only leading N elements of T are used
* if not given, automatically determined from size of T
A,B - base interval for Chebyshev polynomials (see above)
A<B
OUTPUT PARAMETERS
P - polynomial in barycentric form
-- ALGLIB --
Copyright 30.09.2010 by Bochkanov Sergey
*************************************************************************/
void polynomialcheb2bar(/* Real */ ae_vector* t,
ae_int_t n,
double a,
double b,
barycentricinterpolant* p,
ae_state *_state)
{
ae_frame _frame_block;
ae_int_t i;
ae_int_t k;
ae_vector y;
double tk;
double tk1;
double vx;
double vy;
double v;
ae_frame_make(_state, &_frame_block);
_barycentricinterpolant_clear(p);
ae_vector_init(&y, 0, DT_REAL, _state, ae_true);
ae_assert(ae_isfinite(a, _state), "PolynomialBar2Cheb: A is not finite!", _state);
ae_assert(ae_isfinite(b, _state), "PolynomialBar2Cheb: B is not finite!", _state);
ae_assert(ae_fp_neq(a,b), "PolynomialBar2Cheb: A=B!", _state);
ae_assert(n>=1, "PolynomialBar2Cheb: N<1", _state);
ae_assert(t->cnt>=n, "PolynomialBar2Cheb: Length(T)<N", _state);
ae_assert(isfinitevector(t, n, _state), "PolynomialBar2Cheb: T[] contains INF or NAN", _state);
/*
* Calculate function values on a Chebyshev grid spanning [-1,+1]
*/
ae_vector_set_length(&y, n, _state);
for(i=0; i<=n-1; i++)
{
/*
* Calculate value on a grid spanning [-1,+1]
*/
vx = ae_cos(ae_pi*(i+0.5)/n, _state);
vy = t->ptr.p_double[0];
tk1 = 1;
tk = vx;
for(k=1; k<=n-1; k++)
{
vy = vy+t->ptr.p_double[k]*tk;
v = 2*vx*tk-tk1;
tk1 = tk;
tk = v;
}
y.ptr.p_double[i] = vy;
}
/*
* Build barycentric interpolant, map grid from [-1,+1] to [A,B]
*/
polynomialbuildcheb1(a, b, &y, n, p, _state);
ae_frame_leave(_state);
}
/*************************************************************************
Conversion from barycentric representation to power basis.
This function has O(N^2) complexity.
INPUT PARAMETERS:
P - polynomial in barycentric form
C - offset (see below); 0.0 is used as default value.
S - scale (see below); 1.0 is used as default value. S<>0.
OUTPUT PARAMETERS
A - coefficients, P(x) = sum { A[i]*((X-C)/S)^i, i=0..N-1 }
N - number of coefficients (polynomial degree plus 1)
NOTES:
1. this function accepts offset and scale, which can be set to improve
numerical properties of polynomial. For example, if P was obtained as
result of interpolation on [-1,+1], you can set C=0 and S=1 and
represent P as sum of 1, x, x^2, x^3 and so on. In most cases you it
is exactly what you need.
However, if your interpolation model was built on [999,1001], you will
see significant growth of numerical errors when using {1, x, x^2, x^3}
as basis. Representing P as sum of 1, (x-1000), (x-1000)^2, (x-1000)^3
will be better option. Such representation can be obtained by using
1000.0 as offset C and 1.0 as scale S.
2. power basis is ill-conditioned and tricks described above can't solve
this problem completely. This function will return coefficients in
any case, but for N>8 they will become unreliable. However, N's
less than 5 are pretty safe.
3. barycentric interpolant passed as P may be either polynomial obtained
from polynomial interpolation/ fitting or rational function which is
NOT polynomial. We can't distinguish between these two cases, and this
algorithm just tries to work assuming that P IS a polynomial. If not,
algorithm will return results, but they won't have any meaning.
-- ALGLIB --
Copyright 30.09.2010 by Bochkanov Sergey
*************************************************************************/
void polynomialbar2pow(barycentricinterpolant* p,
double c,
double s,
/* Real */ ae_vector* a,
ae_state *_state)
{
ae_frame _frame_block;
ae_int_t i;
ae_int_t k;
double e;
double d;
ae_vector vp;
ae_vector vx;
ae_vector tk;
ae_vector tk1;
ae_vector t;
double v;
ae_frame_make(_state, &_frame_block);
ae_vector_clear(a);
ae_vector_init(&vp, 0, DT_REAL, _state, ae_true);
ae_vector_init(&vx, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tk, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tk1, 0, DT_REAL, _state, ae_true);
ae_vector_init(&t, 0, DT_REAL, _state, ae_true);
ae_assert(ae_isfinite(c, _state), "PolynomialBar2Pow: C is not finite!", _state);
ae_assert(ae_isfinite(s, _state), "PolynomialBar2Pow: S is not finite!", _state);
ae_assert(ae_fp_neq(s,0), "PolynomialBar2Pow: S=0!", _state);
ae_assert(p->n>0, "PolynomialBar2Pow: P is not correctly initialized barycentric interpolant!", _state);
/*
* Calculate function values on a Chebyshev grid
*/
ae_vector_set_length(&vp, p->n, _state);
ae_vector_set_length(&vx, p->n, _state);
for(i=0; i<=p->n-1; i++)
{
vx.ptr.p_double[i] = ae_cos(ae_pi*(i+0.5)/p->n, _state);
vp.ptr.p_double[i] = barycentriccalc(p, s*vx.ptr.p_double[i]+c, _state);
}
/*
* T[0]
*/
ae_vector_set_length(&t, p->n, _state);
v = 0;
for(i=0; i<=p->n-1; i++)
{
v = v+vp.ptr.p_double[i];
}
t.ptr.p_double[0] = v/p->n;
/*
* other T's.
*
* NOTES:
* 1. TK stores T{k} on VX, TK1 stores T{k-1} on VX
* 2. we can do same calculations with fast DCT, but it
* * adds dependencies
* * still leaves us with O(N^2) algorithm because
* preparation of function values is O(N^2) process
*/
if( p->n>1 )
{
ae_vector_set_length(&tk, p->n, _state);
ae_vector_set_length(&tk1, p->n, _state);
for(i=0; i<=p->n-1; i++)
{
tk.ptr.p_double[i] = vx.ptr.p_double[i];
tk1.ptr.p_double[i] = 1;
}
for(k=1; k<=p->n-1; k++)
{
/*
* calculate discrete product of function vector and TK
*/
v = ae_v_dotproduct(&tk.ptr.p_double[0], 1, &vp.ptr.p_double[0], 1, ae_v_len(0,p->n-1));
t.ptr.p_double[k] = v/(0.5*p->n);
/*
* Update TK and TK1
*/
for(i=0; i<=p->n-1; i++)
{
v = 2*vx.ptr.p_double[i]*tk.ptr.p_double[i]-tk1.ptr.p_double[i];
tk1.ptr.p_double[i] = tk.ptr.p_double[i];
tk.ptr.p_double[i] = v;
}
}
}
/*
* Convert from Chebyshev basis to power basis
*/
ae_vector_set_length(a, p->n, _state);
for(i=0; i<=p->n-1; i++)
{
a->ptr.p_double[i] = 0;
}
d = 0;
for(i=0; i<=p->n-1; i++)
{
for(k=i; k<=p->n-1; k++)
{
e = a->ptr.p_double[k];
a->ptr.p_double[k] = 0;
if( i<=1&&k==i )
{
a->ptr.p_double[k] = 1;
}
else
{
if( i!=0 )
{
a->ptr.p_double[k] = 2*d;
}
if( k>i+1 )
{
a->ptr.p_double[k] = a->ptr.p_double[k]-a->ptr.p_double[k-2];
}
}
d = e;
}
d = a->ptr.p_double[i];
e = 0;
k = i;
while(k<=p->n-1)
{
e = e+a->ptr.p_double[k]*t.ptr.p_double[k];
k = k+2;
}
a->ptr.p_double[i] = e;
}
ae_frame_leave(_state);
}
/*************************************************************************
Conversion from power basis to barycentric representation.
This function has O(N^2) complexity.
INPUT PARAMETERS:
A - coefficients, P(x) = sum { A[i]*((X-C)/S)^i, i=0..N-1 }
N - number of coefficients (polynomial degree plus 1)
* if given, only leading N elements of A are used
* if not given, automatically determined from size of A
C - offset (see below); 0.0 is used as default value.
S - scale (see below); 1.0 is used as default value. S<>0.
OUTPUT PARAMETERS
P - polynomial in barycentric form
NOTES:
1. this function accepts offset and scale, which can be set to improve
numerical properties of polynomial. For example, if you interpolate on
[-1,+1], you can set C=0 and S=1 and convert from sum of 1, x, x^2,
x^3 and so on. In most cases you it is exactly what you need.
However, if your interpolation model was built on [999,1001], you will
see significant growth of numerical errors when using {1, x, x^2, x^3}
as input basis. Converting from sum of 1, (x-1000), (x-1000)^2,
(x-1000)^3 will be better option (you have to specify 1000.0 as offset
C and 1.0 as scale S).
2. power basis is ill-conditioned and tricks described above can't solve
this problem completely. This function will return barycentric model
in any case, but for N>8 accuracy well degrade. However, N's less than
5 are pretty safe.
-- ALGLIB --
Copyright 30.09.2010 by Bochkanov Sergey
*************************************************************************/
void polynomialpow2bar(/* Real */ ae_vector* a,
ae_int_t n,
double c,
double s,
barycentricinterpolant* p,
ae_state *_state)
{
ae_frame _frame_block;
ae_int_t i;
ae_int_t k;
ae_vector y;
double vx;
double vy;
double px;
ae_frame_make(_state, &_frame_block);
_barycentricinterpolant_clear(p);
ae_vector_init(&y, 0, DT_REAL, _state, ae_true);
ae_assert(ae_isfinite(c, _state), "PolynomialPow2Bar: C is not finite!", _state);
ae_assert(ae_isfinite(s, _state), "PolynomialPow2Bar: S is not finite!", _state);
ae_assert(ae_fp_neq(s,0), "PolynomialPow2Bar: S is zero!", _state);
ae_assert(n>=1, "PolynomialPow2Bar: N<1", _state);
ae_assert(a->cnt>=n, "PolynomialPow2Bar: Length(A)<N", _state);
ae_assert(isfinitevector(a, n, _state), "PolynomialPow2Bar: A[] contains INF or NAN", _state);
/*
* Calculate function values on a Chebyshev grid spanning [-1,+1]
*/
ae_vector_set_length(&y, n, _state);
for(i=0; i<=n-1; i++)
{
/*
* Calculate value on a grid spanning [-1,+1]
*/
vx = ae_cos(ae_pi*(i+0.5)/n, _state);
vy = a->ptr.p_double[0];
px = vx;
for(k=1; k<=n-1; k++)
{
vy = vy+px*a->ptr.p_double[k];
px = px*vx;
}
y.ptr.p_double[i] = vy;
}
/*
* Build barycentric interpolant, map grid from [-1,+1] to [A,B]
*/
polynomialbuildcheb1(c-s, c+s, &y, n, p, _state);
ae_frame_leave(_state);
}
/*************************************************************************
Lagrange intepolant: generation of the model on the general grid.
This function has O(N^2) complexity.
INPUT PARAMETERS:
X - abscissas, array[0..N-1]
Y - function values, array[0..N-1]
N - number of points, N>=1
OUTPUT PARAMETERS
P - barycentric model which represents Lagrange interpolant
(see ratint unit info and BarycentricCalc() description for
more information).
-- ALGLIB --
Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialbuild(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
barycentricinterpolant* p,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _x;
ae_vector _y;
ae_int_t j;
ae_int_t k;
ae_vector w;
double b;
double a;
double v;
double mx;
ae_vector sortrbuf;
ae_vector sortrbuf2;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_x, x, _state, ae_true);
x = &_x;
ae_vector_init_copy(&_y, y, _state, ae_true);
y = &_y;
_barycentricinterpolant_clear(p);
ae_vector_init(&w, 0, DT_REAL, _state, ae_true);
ae_vector_init(&sortrbuf, 0, DT_REAL, _state, ae_true);
ae_vector_init(&sortrbuf2, 0, DT_REAL, _state, ae_true);
ae_assert(n>0, "PolynomialBuild: N<=0!", _state);
ae_assert(x->cnt>=n, "PolynomialBuild: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "PolynomialBuild: Length(Y)<N!", _state);
ae_assert(isfinitevector(x, n, _state), "PolynomialBuild: X contains infinite or NaN values!", _state);
ae_assert(isfinitevector(y, n, _state), "PolynomialBuild: Y contains infinite or NaN values!", _state);
tagsortfastr(x, y, &sortrbuf, &sortrbuf2, n, _state);
ae_assert(aredistinct(x, n, _state), "PolynomialBuild: at least two consequent points are too close!", _state);
/*
* calculate W[j]
* multi-pass algorithm is used to avoid overflow
*/
ae_vector_set_length(&w, n, _state);
a = x->ptr.p_double[0];
b = x->ptr.p_double[0];
for(j=0; j<=n-1; j++)
{
w.ptr.p_double[j] = 1;
a = ae_minreal(a, x->ptr.p_double[j], _state);
b = ae_maxreal(b, x->ptr.p_double[j], _state);
}
for(k=0; k<=n-1; k++)
{
/*
* W[K] is used instead of 0.0 because
* cycle on J does not touch K-th element
* and we MUST get maximum from ALL elements
*/
mx = ae_fabs(w.ptr.p_double[k], _state);
for(j=0; j<=n-1; j++)
{
if( j!=k )
{
v = (b-a)/(x->ptr.p_double[j]-x->ptr.p_double[k]);
w.ptr.p_double[j] = w.ptr.p_double[j]*v;
mx = ae_maxreal(mx, ae_fabs(w.ptr.p_double[j], _state), _state);
}
}
if( k%5==0 )
{
/*
* every 5-th run we renormalize W[]
*/
v = 1/mx;
ae_v_muld(&w.ptr.p_double[0], 1, ae_v_len(0,n-1), v);
}
}
barycentricbuildxyw(x, y, &w, n, p, _state);
ae_frame_leave(_state);
}
/*************************************************************************
Lagrange intepolant: generation of the model on equidistant grid.
This function has O(N) complexity.
INPUT PARAMETERS:
A - left boundary of [A,B]
B - right boundary of [A,B]
Y - function values at the nodes, array[0..N-1]
N - number of points, N>=1
for N=1 a constant model is constructed.
OUTPUT PARAMETERS
P - barycentric model which represents Lagrange interpolant
(see ratint unit info and BarycentricCalc() description for
more information).
-- ALGLIB --
Copyright 03.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialbuildeqdist(double a,
double b,
/* Real */ ae_vector* y,
ae_int_t n,
barycentricinterpolant* p,
ae_state *_state)
{
ae_frame _frame_block;
ae_int_t i;
ae_vector w;
ae_vector x;
double v;
ae_frame_make(_state, &_frame_block);
_barycentricinterpolant_clear(p);
ae_vector_init(&w, 0, DT_REAL, _state, ae_true);
ae_vector_init(&x, 0, DT_REAL, _state, ae_true);
ae_assert(n>0, "PolynomialBuildEqDist: N<=0!", _state);
ae_assert(y->cnt>=n, "PolynomialBuildEqDist: Length(Y)<N!", _state);
ae_assert(ae_isfinite(a, _state), "PolynomialBuildEqDist: A is infinite or NaN!", _state);
ae_assert(ae_isfinite(b, _state), "PolynomialBuildEqDist: B is infinite or NaN!", _state);
ae_assert(isfinitevector(y, n, _state), "PolynomialBuildEqDist: Y contains infinite or NaN values!", _state);
ae_assert(ae_fp_neq(b,a), "PolynomialBuildEqDist: B=A!", _state);
ae_assert(ae_fp_neq(a+(b-a)/n,a), "PolynomialBuildEqDist: B is too close to A!", _state);
/*
* Special case: N=1
*/
if( n==1 )
{
ae_vector_set_length(&x, 1, _state);
ae_vector_set_length(&w, 1, _state);
x.ptr.p_double[0] = 0.5*(b+a);
w.ptr.p_double[0] = 1;
barycentricbuildxyw(&x, y, &w, 1, p, _state);
ae_frame_leave(_state);
return;
}
/*
* general case
*/
ae_vector_set_length(&x, n, _state);
ae_vector_set_length(&w, n, _state);
v = 1;
for(i=0; i<=n-1; i++)
{
w.ptr.p_double[i] = v;
x.ptr.p_double[i] = a+(b-a)*i/(n-1);
v = -v*(n-1-i);
v = v/(i+1);
}
barycentricbuildxyw(&x, y, &w, n, p, _state);
ae_frame_leave(_state);
}
/*************************************************************************
Lagrange intepolant on Chebyshev grid (first kind).
This function has O(N) complexity.
INPUT PARAMETERS:
A - left boundary of [A,B]
B - right boundary of [A,B]
Y - function values at the nodes, array[0..N-1],
Y[I] = Y(0.5*(B+A) + 0.5*(B-A)*Cos(PI*(2*i+1)/(2*n)))
N - number of points, N>=1
for N=1 a constant model is constructed.
OUTPUT PARAMETERS
P - barycentric model which represents Lagrange interpolant
(see ratint unit info and BarycentricCalc() description for
more information).
-- ALGLIB --
Copyright 03.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialbuildcheb1(double a,
double b,
/* Real */ ae_vector* y,
ae_int_t n,
barycentricinterpolant* p,
ae_state *_state)
{
ae_frame _frame_block;
ae_int_t i;
ae_vector w;
ae_vector x;
double v;
double t;
ae_frame_make(_state, &_frame_block);
_barycentricinterpolant_clear(p);
ae_vector_init(&w, 0, DT_REAL, _state, ae_true);
ae_vector_init(&x, 0, DT_REAL, _state, ae_true);
ae_assert(n>0, "PolynomialBuildCheb1: N<=0!", _state);
ae_assert(y->cnt>=n, "PolynomialBuildCheb1: Length(Y)<N!", _state);
ae_assert(ae_isfinite(a, _state), "PolynomialBuildCheb1: A is infinite or NaN!", _state);
ae_assert(ae_isfinite(b, _state), "PolynomialBuildCheb1: B is infinite or NaN!", _state);
ae_assert(isfinitevector(y, n, _state), "PolynomialBuildCheb1: Y contains infinite or NaN values!", _state);
ae_assert(ae_fp_neq(b,a), "PolynomialBuildCheb1: B=A!", _state);
/*
* Special case: N=1
*/
if( n==1 )
{
ae_vector_set_length(&x, 1, _state);
ae_vector_set_length(&w, 1, _state);
x.ptr.p_double[0] = 0.5*(b+a);
w.ptr.p_double[0] = 1;
barycentricbuildxyw(&x, y, &w, 1, p, _state);
ae_frame_leave(_state);
return;
}
/*
* general case
*/
ae_vector_set_length(&x, n, _state);
ae_vector_set_length(&w, n, _state);
v = 1;
for(i=0; i<=n-1; i++)
{
t = ae_tan(0.5*ae_pi*(2*i+1)/(2*n), _state);
w.ptr.p_double[i] = 2*v*t/(1+ae_sqr(t, _state));
x.ptr.p_double[i] = 0.5*(b+a)+0.5*(b-a)*(1-ae_sqr(t, _state))/(1+ae_sqr(t, _state));
v = -v;
}
barycentricbuildxyw(&x, y, &w, n, p, _state);
ae_frame_leave(_state);
}
/*************************************************************************
Lagrange intepolant on Chebyshev grid (second kind).
This function has O(N) complexity.
INPUT PARAMETERS:
A - left boundary of [A,B]
B - right boundary of [A,B]
Y - function values at the nodes, array[0..N-1],
Y[I] = Y(0.5*(B+A) + 0.5*(B-A)*Cos(PI*i/(n-1)))
N - number of points, N>=1
for N=1 a constant model is constructed.
OUTPUT PARAMETERS
P - barycentric model which represents Lagrange interpolant
(see ratint unit info and BarycentricCalc() description for
more information).
-- ALGLIB --
Copyright 03.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialbuildcheb2(double a,
double b,
/* Real */ ae_vector* y,
ae_int_t n,
barycentricinterpolant* p,
ae_state *_state)
{
ae_frame _frame_block;
ae_int_t i;
ae_vector w;
ae_vector x;
double v;
ae_frame_make(_state, &_frame_block);
_barycentricinterpolant_clear(p);
ae_vector_init(&w, 0, DT_REAL, _state, ae_true);
ae_vector_init(&x, 0, DT_REAL, _state, ae_true);
ae_assert(n>0, "PolynomialBuildCheb2: N<=0!", _state);
ae_assert(y->cnt>=n, "PolynomialBuildCheb2: Length(Y)<N!", _state);
ae_assert(ae_isfinite(a, _state), "PolynomialBuildCheb2: A is infinite or NaN!", _state);
ae_assert(ae_isfinite(b, _state), "PolynomialBuildCheb2: B is infinite or NaN!", _state);
ae_assert(ae_fp_neq(b,a), "PolynomialBuildCheb2: B=A!", _state);
ae_assert(isfinitevector(y, n, _state), "PolynomialBuildCheb2: Y contains infinite or NaN values!", _state);
/*
* Special case: N=1
*/
if( n==1 )
{
ae_vector_set_length(&x, 1, _state);
ae_vector_set_length(&w, 1, _state);
x.ptr.p_double[0] = 0.5*(b+a);
w.ptr.p_double[0] = 1;
barycentricbuildxyw(&x, y, &w, 1, p, _state);
ae_frame_leave(_state);
return;
}
/*
* general case
*/
ae_vector_set_length(&x, n, _state);
ae_vector_set_length(&w, n, _state);
v = 1;
for(i=0; i<=n-1; i++)
{
if( i==0||i==n-1 )
{
w.ptr.p_double[i] = v*0.5;
}
else
{
w.ptr.p_double[i] = v;
}
x.ptr.p_double[i] = 0.5*(b+a)+0.5*(b-a)*ae_cos(ae_pi*i/(n-1), _state);
v = -v;
}
barycentricbuildxyw(&x, y, &w, n, p, _state);
ae_frame_leave(_state);
}
/*************************************************************************
Fast equidistant polynomial interpolation function with O(N) complexity
INPUT PARAMETERS:
A - left boundary of [A,B]
B - right boundary of [A,B]
F - function values, array[0..N-1]
N - number of points on equidistant grid, N>=1
for N=1 a constant model is constructed.
T - position where P(x) is calculated
RESULT
value of the Lagrange interpolant at T
IMPORTANT
this function provides fast interface which is not overflow-safe
nor it is very precise.
the best option is to use PolynomialBuildEqDist()/BarycentricCalc()
subroutines unless you are pretty sure that your data will not result
in overflow.
-- ALGLIB --
Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
double polynomialcalceqdist(double a,
double b,
/* Real */ ae_vector* f,
ae_int_t n,
double t,
ae_state *_state)
{
double s1;
double s2;
double v;
double threshold;
double s;
double h;
ae_int_t i;
ae_int_t j;
double w;
double x;
double result;
ae_assert(n>0, "PolynomialCalcEqDist: N<=0!", _state);
ae_assert(f->cnt>=n, "PolynomialCalcEqDist: Length(F)<N!", _state);
ae_assert(ae_isfinite(a, _state), "PolynomialCalcEqDist: A is infinite or NaN!", _state);
ae_assert(ae_isfinite(b, _state), "PolynomialCalcEqDist: B is infinite or NaN!", _state);
ae_assert(isfinitevector(f, n, _state), "PolynomialCalcEqDist: F contains infinite or NaN values!", _state);
ae_assert(ae_fp_neq(b,a), "PolynomialCalcEqDist: B=A!", _state);
ae_assert(!ae_isinf(t, _state), "PolynomialCalcEqDist: T is infinite!", _state);
/*
* Special case: T is NAN
*/
if( ae_isnan(t, _state) )
{
result = _state->v_nan;
return result;
}
/*
* Special case: N=1
*/
if( n==1 )
{
result = f->ptr.p_double[0];
return result;
}
/*
* First, decide: should we use "safe" formula (guarded
* against overflow) or fast one?
*/
threshold = ae_sqrt(ae_minrealnumber, _state);
j = 0;
s = t-a;
for(i=1; i<=n-1; i++)
{
x = a+(double)i/(double)(n-1)*(b-a);
if( ae_fp_less(ae_fabs(t-x, _state),ae_fabs(s, _state)) )
{
s = t-x;
j = i;
}
}
if( ae_fp_eq(s,0) )
{
result = f->ptr.p_double[j];
return result;
}
if( ae_fp_greater(ae_fabs(s, _state),threshold) )
{
/*
* use fast formula
*/
j = -1;
s = 1.0;
}
/*
* Calculate using safe or fast barycentric formula
*/
s1 = 0;
s2 = 0;
w = 1.0;
h = (b-a)/(n-1);
for(i=0; i<=n-1; i++)
{
if( i!=j )
{
v = s*w/(t-(a+i*h));
s1 = s1+v*f->ptr.p_double[i];
s2 = s2+v;
}
else
{
v = w;
s1 = s1+v*f->ptr.p_double[i];
s2 = s2+v;
}
w = -w*(n-1-i);
w = w/(i+1);
}
result = s1/s2;
return result;
}
/*************************************************************************
Fast polynomial interpolation function on Chebyshev points (first kind)
with O(N) complexity.
INPUT PARAMETERS:
A - left boundary of [A,B]
B - right boundary of [A,B]
F - function values, array[0..N-1]
N - number of points on Chebyshev grid (first kind),
X[i] = 0.5*(B+A) + 0.5*(B-A)*Cos(PI*(2*i+1)/(2*n))
for N=1 a constant model is constructed.
T - position where P(x) is calculated
RESULT
value of the Lagrange interpolant at T
IMPORTANT
this function provides fast interface which is not overflow-safe
nor it is very precise.
the best option is to use PolIntBuildCheb1()/BarycentricCalc()
subroutines unless you are pretty sure that your data will not result
in overflow.
-- ALGLIB --
Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
double polynomialcalccheb1(double a,
double b,
/* Real */ ae_vector* f,
ae_int_t n,
double t,
ae_state *_state)
{
double s1;
double s2;
double v;
double threshold;
double s;
ae_int_t i;
ae_int_t j;
double a0;
double delta;
double alpha;
double beta;
double ca;
double sa;
double tempc;
double temps;
double x;
double w;
double p1;
double result;
ae_assert(n>0, "PolynomialCalcCheb1: N<=0!", _state);
ae_assert(f->cnt>=n, "PolynomialCalcCheb1: Length(F)<N!", _state);
ae_assert(ae_isfinite(a, _state), "PolynomialCalcCheb1: A is infinite or NaN!", _state);
ae_assert(ae_isfinite(b, _state), "PolynomialCalcCheb1: B is infinite or NaN!", _state);
ae_assert(isfinitevector(f, n, _state), "PolynomialCalcCheb1: F contains infinite or NaN values!", _state);
ae_assert(ae_fp_neq(b,a), "PolynomialCalcCheb1: B=A!", _state);
ae_assert(!ae_isinf(t, _state), "PolynomialCalcCheb1: T is infinite!", _state);
/*
* Special case: T is NAN
*/
if( ae_isnan(t, _state) )
{
result = _state->v_nan;
return result;
}
/*
* Special case: N=1
*/
if( n==1 )
{
result = f->ptr.p_double[0];
return result;
}
/*
* Prepare information for the recurrence formula
* used to calculate sin(pi*(2j+1)/(2n+2)) and
* cos(pi*(2j+1)/(2n+2)):
*
* A0 = pi/(2n+2)
* Delta = pi/(n+1)
* Alpha = 2 sin^2 (Delta/2)
* Beta = sin(Delta)
*
* so that sin(..) = sin(A0+j*delta) and cos(..) = cos(A0+j*delta).
* Then we use
*
* sin(x+delta) = sin(x) - (alpha*sin(x) - beta*cos(x))
* cos(x+delta) = cos(x) - (alpha*cos(x) - beta*sin(x))
*
* to repeatedly calculate sin(..) and cos(..).
*/
threshold = ae_sqrt(ae_minrealnumber, _state);
t = (t-0.5*(a+b))/(0.5*(b-a));
a0 = ae_pi/(2*(n-1)+2);
delta = 2*ae_pi/(2*(n-1)+2);
alpha = 2*ae_sqr(ae_sin(delta/2, _state), _state);
beta = ae_sin(delta, _state);
/*
* First, decide: should we use "safe" formula (guarded
* against overflow) or fast one?
*/
ca = ae_cos(a0, _state);
sa = ae_sin(a0, _state);
j = 0;
x = ca;
s = t-x;
for(i=1; i<=n-1; i++)
{
/*
* Next X[i]
*/
temps = sa-(alpha*sa-beta*ca);
tempc = ca-(alpha*ca+beta*sa);
sa = temps;
ca = tempc;
x = ca;
/*
* Use X[i]
*/
if( ae_fp_less(ae_fabs(t-x, _state),ae_fabs(s, _state)) )
{
s = t-x;
j = i;
}
}
if( ae_fp_eq(s,0) )
{
result = f->ptr.p_double[j];
return result;
}
if( ae_fp_greater(ae_fabs(s, _state),threshold) )
{
/*
* use fast formula
*/
j = -1;
s = 1.0;
}
/*
* Calculate using safe or fast barycentric formula
*/
s1 = 0;
s2 = 0;
ca = ae_cos(a0, _state);
sa = ae_sin(a0, _state);
p1 = 1.0;
for(i=0; i<=n-1; i++)
{
/*
* Calculate X[i], W[i]
*/
x = ca;
w = p1*sa;
/*
* Proceed
*/
if( i!=j )
{
v = s*w/(t-x);
s1 = s1+v*f->ptr.p_double[i];
s2 = s2+v;
}
else
{
v = w;
s1 = s1+v*f->ptr.p_double[i];
s2 = s2+v;
}
/*
* Next CA, SA, P1
*/
temps = sa-(alpha*sa-beta*ca);
tempc = ca-(alpha*ca+beta*sa);
sa = temps;
ca = tempc;
p1 = -p1;
}
result = s1/s2;
return result;
}
/*************************************************************************
Fast polynomial interpolation function on Chebyshev points (second kind)
with O(N) complexity.
INPUT PARAMETERS:
A - left boundary of [A,B]
B - right boundary of [A,B]
F - function values, array[0..N-1]
N - number of points on Chebyshev grid (second kind),
X[i] = 0.5*(B+A) + 0.5*(B-A)*Cos(PI*i/(n-1))
for N=1 a constant model is constructed.
T - position where P(x) is calculated
RESULT
value of the Lagrange interpolant at T
IMPORTANT
this function provides fast interface which is not overflow-safe
nor it is very precise.
the best option is to use PolIntBuildCheb2()/BarycentricCalc()
subroutines unless you are pretty sure that your data will not result
in overflow.
-- ALGLIB --
Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
double polynomialcalccheb2(double a,
double b,
/* Real */ ae_vector* f,
ae_int_t n,
double t,
ae_state *_state)
{
double s1;
double s2;
double v;
double threshold;
double s;
ae_int_t i;
ae_int_t j;
double a0;
double delta;
double alpha;
double beta;
double ca;
double sa;
double tempc;
double temps;
double x;
double w;
double p1;
double result;
ae_assert(n>0, "PolynomialCalcCheb2: N<=0!", _state);
ae_assert(f->cnt>=n, "PolynomialCalcCheb2: Length(F)<N!", _state);
ae_assert(ae_isfinite(a, _state), "PolynomialCalcCheb2: A is infinite or NaN!", _state);
ae_assert(ae_isfinite(b, _state), "PolynomialCalcCheb2: B is infinite or NaN!", _state);
ae_assert(ae_fp_neq(b,a), "PolynomialCalcCheb2: B=A!", _state);
ae_assert(isfinitevector(f, n, _state), "PolynomialCalcCheb2: F contains infinite or NaN values!", _state);
ae_assert(!ae_isinf(t, _state), "PolynomialCalcEqDist: T is infinite!", _state);
/*
* Special case: T is NAN
*/
if( ae_isnan(t, _state) )
{
result = _state->v_nan;
return result;
}
/*
* Special case: N=1
*/
if( n==1 )
{
result = f->ptr.p_double[0];
return result;
}
/*
* Prepare information for the recurrence formula
* used to calculate sin(pi*i/n) and
* cos(pi*i/n):
*
* A0 = 0
* Delta = pi/n
* Alpha = 2 sin^2 (Delta/2)
* Beta = sin(Delta)
*
* so that sin(..) = sin(A0+j*delta) and cos(..) = cos(A0+j*delta).
* Then we use
*
* sin(x+delta) = sin(x) - (alpha*sin(x) - beta*cos(x))
* cos(x+delta) = cos(x) - (alpha*cos(x) - beta*sin(x))
*
* to repeatedly calculate sin(..) and cos(..).
*/
threshold = ae_sqrt(ae_minrealnumber, _state);
t = (t-0.5*(a+b))/(0.5*(b-a));
a0 = 0.0;
delta = ae_pi/(n-1);
alpha = 2*ae_sqr(ae_sin(delta/2, _state), _state);
beta = ae_sin(delta, _state);
/*
* First, decide: should we use "safe" formula (guarded
* against overflow) or fast one?
*/
ca = ae_cos(a0, _state);
sa = ae_sin(a0, _state);
j = 0;
x = ca;
s = t-x;
for(i=1; i<=n-1; i++)
{
/*
* Next X[i]
*/
temps = sa-(alpha*sa-beta*ca);
tempc = ca-(alpha*ca+beta*sa);
sa = temps;
ca = tempc;
x = ca;
/*
* Use X[i]
*/
if( ae_fp_less(ae_fabs(t-x, _state),ae_fabs(s, _state)) )
{
s = t-x;
j = i;
}
}
if( ae_fp_eq(s,0) )
{
result = f->ptr.p_double[j];
return result;
}
if( ae_fp_greater(ae_fabs(s, _state),threshold) )
{
/*
* use fast formula
*/
j = -1;
s = 1.0;
}
/*
* Calculate using safe or fast barycentric formula
*/
s1 = 0;
s2 = 0;
ca = ae_cos(a0, _state);
sa = ae_sin(a0, _state);
p1 = 1.0;
for(i=0; i<=n-1; i++)
{
/*
* Calculate X[i], W[i]
*/
x = ca;
if( i==0||i==n-1 )
{
w = 0.5*p1;
}
else
{
w = 1.0*p1;
}
/*
* Proceed
*/
if( i!=j )
{
v = s*w/(t-x);
s1 = s1+v*f->ptr.p_double[i];
s2 = s2+v;
}
else
{
v = w;
s1 = s1+v*f->ptr.p_double[i];
s2 = s2+v;
}
/*
* Next CA, SA, P1
*/
temps = sa-(alpha*sa-beta*ca);
tempc = ca-(alpha*ca+beta*sa);
sa = temps;
ca = tempc;
p1 = -p1;
}
result = s1/s2;
return result;
}
/*************************************************************************
This subroutine builds linear spline interpolant
INPUT PARAMETERS:
X - spline nodes, array[0..N-1]
Y - function values, array[0..N-1]
N - points count (optional):
* N>=2
* if given, only first N points are used to build spline
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
OUTPUT PARAMETERS:
C - spline interpolant
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
-- ALGLIB PROJECT --
Copyright 24.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildlinear(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
spline1dinterpolant* c,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _x;
ae_vector _y;
ae_int_t i;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_x, x, _state, ae_true);
x = &_x;
ae_vector_init_copy(&_y, y, _state, ae_true);
y = &_y;
_spline1dinterpolant_clear(c);
ae_assert(n>1, "Spline1DBuildLinear: N<2!", _state);
ae_assert(x->cnt>=n, "Spline1DBuildLinear: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "Spline1DBuildLinear: Length(Y)<N!", _state);
/*
* check and sort points
*/
ae_assert(isfinitevector(x, n, _state), "Spline1DBuildLinear: X contains infinite or NAN values!", _state);
ae_assert(isfinitevector(y, n, _state), "Spline1DBuildLinear: Y contains infinite or NAN values!", _state);
spline1d_heapsortpoints(x, y, n, _state);
ae_assert(aredistinct(x, n, _state), "Spline1DBuildLinear: at least two consequent points are too close!", _state);
/*
* Build
*/
c->periodic = ae_false;
c->n = n;
c->k = 3;
c->continuity = 0;
ae_vector_set_length(&c->x, n, _state);
ae_vector_set_length(&c->c, 4*(n-1)+2, _state);
for(i=0; i<=n-1; i++)
{
c->x.ptr.p_double[i] = x->ptr.p_double[i];
}
for(i=0; i<=n-2; i++)
{
c->c.ptr.p_double[4*i+0] = y->ptr.p_double[i];
c->c.ptr.p_double[4*i+1] = (y->ptr.p_double[i+1]-y->ptr.p_double[i])/(x->ptr.p_double[i+1]-x->ptr.p_double[i]);
c->c.ptr.p_double[4*i+2] = 0;
c->c.ptr.p_double[4*i+3] = 0;
}
c->c.ptr.p_double[4*(n-1)+0] = y->ptr.p_double[n-1];
c->c.ptr.p_double[4*(n-1)+1] = c->c.ptr.p_double[4*(n-2)+1];
ae_frame_leave(_state);
}
/*************************************************************************
This subroutine builds cubic spline interpolant.
INPUT PARAMETERS:
X - spline nodes, array[0..N-1].
Y - function values, array[0..N-1].
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points are used to build spline
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundLType - boundary condition type for the left boundary
BoundL - left boundary condition (first or second derivative,
depending on the BoundLType)
BoundRType - boundary condition type for the right boundary
BoundR - right boundary condition (first or second derivative,
depending on the BoundRType)
OUTPUT PARAMETERS:
C - spline interpolant
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
SETTING BOUNDARY VALUES:
The BoundLType/BoundRType parameters can have the following values:
* -1, which corresonds to the periodic (cyclic) boundary conditions.
In this case:
* both BoundLType and BoundRType must be equal to -1.
* BoundL/BoundR are ignored
* Y[last] is ignored (it is assumed to be equal to Y[first]).
* 0, which corresponds to the parabolically terminated spline
(BoundL and/or BoundR are ignored).
* 1, which corresponds to the first derivative boundary condition
* 2, which corresponds to the second derivative boundary condition
* by default, BoundType=0 is used
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildcubic(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_int_t boundltype,
double boundl,
ae_int_t boundrtype,
double boundr,
spline1dinterpolant* c,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _x;
ae_vector _y;
ae_vector a1;
ae_vector a2;
ae_vector a3;
ae_vector b;
ae_vector dt;
ae_vector d;
ae_vector p;
ae_int_t ylen;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_x, x, _state, ae_true);
x = &_x;
ae_vector_init_copy(&_y, y, _state, ae_true);
y = &_y;
_spline1dinterpolant_clear(c);
ae_vector_init(&a1, 0, DT_REAL, _state, ae_true);
ae_vector_init(&a2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&a3, 0, DT_REAL, _state, ae_true);
ae_vector_init(&b, 0, DT_REAL, _state, ae_true);
ae_vector_init(&dt, 0, DT_REAL, _state, ae_true);
ae_vector_init(&d, 0, DT_REAL, _state, ae_true);
ae_vector_init(&p, 0, DT_INT, _state, ae_true);
/*
* check correctness of boundary conditions
*/
ae_assert(((boundltype==-1||boundltype==0)||boundltype==1)||boundltype==2, "Spline1DBuildCubic: incorrect BoundLType!", _state);
ae_assert(((boundrtype==-1||boundrtype==0)||boundrtype==1)||boundrtype==2, "Spline1DBuildCubic: incorrect BoundRType!", _state);
ae_assert((boundrtype==-1&&boundltype==-1)||(boundrtype!=-1&&boundltype!=-1), "Spline1DBuildCubic: incorrect BoundLType/BoundRType!", _state);
if( boundltype==1||boundltype==2 )
{
ae_assert(ae_isfinite(boundl, _state), "Spline1DBuildCubic: BoundL is infinite or NAN!", _state);
}
if( boundrtype==1||boundrtype==2 )
{
ae_assert(ae_isfinite(boundr, _state), "Spline1DBuildCubic: BoundR is infinite or NAN!", _state);
}
/*
* check lengths of arguments
*/
ae_assert(n>=2, "Spline1DBuildCubic: N<2!", _state);
ae_assert(x->cnt>=n, "Spline1DBuildCubic: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "Spline1DBuildCubic: Length(Y)<N!", _state);
/*
* check and sort points
*/
ylen = n;
if( boundltype==-1 )
{
ylen = n-1;
}
ae_assert(isfinitevector(x, n, _state), "Spline1DBuildCubic: X contains infinite or NAN values!", _state);
ae_assert(isfinitevector(y, ylen, _state), "Spline1DBuildCubic: Y contains infinite or NAN values!", _state);
spline1d_heapsortppoints(x, y, &p, n, _state);
ae_assert(aredistinct(x, n, _state), "Spline1DBuildCubic: at least two consequent points are too close!", _state);
/*
* Now we've checked and preordered everything,
* so we can call internal function to calculate derivatives,
* and then build Hermite spline using these derivatives
*/
if( boundltype==-1||boundrtype==-1 )
{
y->ptr.p_double[n-1] = y->ptr.p_double[0];
}
spline1d_spline1dgriddiffcubicinternal(x, y, n, boundltype, boundl, boundrtype, boundr, &d, &a1, &a2, &a3, &b, &dt, _state);
spline1dbuildhermite(x, y, &d, n, c, _state);
c->periodic = boundltype==-1||boundrtype==-1;
c->continuity = 2;
ae_frame_leave(_state);
}
/*************************************************************************
This function solves following problem: given table y[] of function values
at nodes x[], it calculates and returns table of function derivatives d[]
(calculated at the same nodes x[]).
This function yields same result as Spline1DBuildCubic() call followed by
sequence of Spline1DDiff() calls, but it can be several times faster when
called for ordered X[] and X2[].
INPUT PARAMETERS:
X - spline nodes
Y - function values
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points are used
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundLType - boundary condition type for the left boundary
BoundL - left boundary condition (first or second derivative,
depending on the BoundLType)
BoundRType - boundary condition type for the right boundary
BoundR - right boundary condition (first or second derivative,
depending on the BoundRType)
OUTPUT PARAMETERS:
D - derivative values at X[]
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
Derivative values are correctly reordered on return, so D[I] is always
equal to S'(X[I]) independently of points order.
SETTING BOUNDARY VALUES:
The BoundLType/BoundRType parameters can have the following values:
* -1, which corresonds to the periodic (cyclic) boundary conditions.
In this case:
* both BoundLType and BoundRType must be equal to -1.
* BoundL/BoundR are ignored
* Y[last] is ignored (it is assumed to be equal to Y[first]).
* 0, which corresponds to the parabolically terminated spline
(BoundL and/or BoundR are ignored).
* 1, which corresponds to the first derivative boundary condition
* 2, which corresponds to the second derivative boundary condition
* by default, BoundType=0 is used
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dgriddiffcubic(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_int_t boundltype,
double boundl,
ae_int_t boundrtype,
double boundr,
/* Real */ ae_vector* d,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _x;
ae_vector _y;
ae_vector a1;
ae_vector a2;
ae_vector a3;
ae_vector b;
ae_vector dt;
ae_vector p;
ae_int_t i;
ae_int_t ylen;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_x, x, _state, ae_true);
x = &_x;
ae_vector_init_copy(&_y, y, _state, ae_true);
y = &_y;
ae_vector_clear(d);
ae_vector_init(&a1, 0, DT_REAL, _state, ae_true);
ae_vector_init(&a2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&a3, 0, DT_REAL, _state, ae_true);
ae_vector_init(&b, 0, DT_REAL, _state, ae_true);
ae_vector_init(&dt, 0, DT_REAL, _state, ae_true);
ae_vector_init(&p, 0, DT_INT, _state, ae_true);
/*
* check correctness of boundary conditions
*/
ae_assert(((boundltype==-1||boundltype==0)||boundltype==1)||boundltype==2, "Spline1DGridDiffCubic: incorrect BoundLType!", _state);
ae_assert(((boundrtype==-1||boundrtype==0)||boundrtype==1)||boundrtype==2, "Spline1DGridDiffCubic: incorrect BoundRType!", _state);
ae_assert((boundrtype==-1&&boundltype==-1)||(boundrtype!=-1&&boundltype!=-1), "Spline1DGridDiffCubic: incorrect BoundLType/BoundRType!", _state);
if( boundltype==1||boundltype==2 )
{
ae_assert(ae_isfinite(boundl, _state), "Spline1DGridDiffCubic: BoundL is infinite or NAN!", _state);
}
if( boundrtype==1||boundrtype==2 )
{
ae_assert(ae_isfinite(boundr, _state), "Spline1DGridDiffCubic: BoundR is infinite or NAN!", _state);
}
/*
* check lengths of arguments
*/
ae_assert(n>=2, "Spline1DGridDiffCubic: N<2!", _state);
ae_assert(x->cnt>=n, "Spline1DGridDiffCubic: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "Spline1DGridDiffCubic: Length(Y)<N!", _state);
/*
* check and sort points
*/
ylen = n;
if( boundltype==-1 )
{
ylen = n-1;
}
ae_assert(isfinitevector(x, n, _state), "Spline1DGridDiffCubic: X contains infinite or NAN values!", _state);
ae_assert(isfinitevector(y, ylen, _state), "Spline1DGridDiffCubic: Y contains infinite or NAN values!", _state);
spline1d_heapsortppoints(x, y, &p, n, _state);
ae_assert(aredistinct(x, n, _state), "Spline1DGridDiffCubic: at least two consequent points are too close!", _state);
/*
* Now we've checked and preordered everything,
* so we can call internal function.
*/
spline1d_spline1dgriddiffcubicinternal(x, y, n, boundltype, boundl, boundrtype, boundr, d, &a1, &a2, &a3, &b, &dt, _state);
/*
* Remember that HeapSortPPoints() call?
* Now we have to reorder them back.
*/
if( dt.cnt<n )
{
ae_vector_set_length(&dt, n, _state);
}
for(i=0; i<=n-1; i++)
{
dt.ptr.p_double[p.ptr.p_int[i]] = d->ptr.p_double[i];
}
ae_v_move(&d->ptr.p_double[0], 1, &dt.ptr.p_double[0], 1, ae_v_len(0,n-1));
ae_frame_leave(_state);
}
/*************************************************************************
This function solves following problem: given table y[] of function values
at nodes x[], it calculates and returns tables of first and second
function derivatives d1[] and d2[] (calculated at the same nodes x[]).
This function yields same result as Spline1DBuildCubic() call followed by
sequence of Spline1DDiff() calls, but it can be several times faster when
called for ordered X[] and X2[].
INPUT PARAMETERS:
X - spline nodes
Y - function values
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points are used
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundLType - boundary condition type for the left boundary
BoundL - left boundary condition (first or second derivative,
depending on the BoundLType)
BoundRType - boundary condition type for the right boundary
BoundR - right boundary condition (first or second derivative,
depending on the BoundRType)
OUTPUT PARAMETERS:
D1 - S' values at X[]
D2 - S'' values at X[]
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
Derivative values are correctly reordered on return, so D[I] is always
equal to S'(X[I]) independently of points order.
SETTING BOUNDARY VALUES:
The BoundLType/BoundRType parameters can have the following values:
* -1, which corresonds to the periodic (cyclic) boundary conditions.
In this case:
* both BoundLType and BoundRType must be equal to -1.
* BoundL/BoundR are ignored
* Y[last] is ignored (it is assumed to be equal to Y[first]).
* 0, which corresponds to the parabolically terminated spline
(BoundL and/or BoundR are ignored).
* 1, which corresponds to the first derivative boundary condition
* 2, which corresponds to the second derivative boundary condition
* by default, BoundType=0 is used
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dgriddiff2cubic(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_int_t boundltype,
double boundl,
ae_int_t boundrtype,
double boundr,
/* Real */ ae_vector* d1,
/* Real */ ae_vector* d2,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _x;
ae_vector _y;
ae_vector a1;
ae_vector a2;
ae_vector a3;
ae_vector b;
ae_vector dt;
ae_vector p;
ae_int_t i;
ae_int_t ylen;
double delta;
double delta2;
double delta3;
double s2;
double s3;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_x, x, _state, ae_true);
x = &_x;
ae_vector_init_copy(&_y, y, _state, ae_true);
y = &_y;
ae_vector_clear(d1);
ae_vector_clear(d2);
ae_vector_init(&a1, 0, DT_REAL, _state, ae_true);
ae_vector_init(&a2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&a3, 0, DT_REAL, _state, ae_true);
ae_vector_init(&b, 0, DT_REAL, _state, ae_true);
ae_vector_init(&dt, 0, DT_REAL, _state, ae_true);
ae_vector_init(&p, 0, DT_INT, _state, ae_true);
/*
* check correctness of boundary conditions
*/
ae_assert(((boundltype==-1||boundltype==0)||boundltype==1)||boundltype==2, "Spline1DGridDiff2Cubic: incorrect BoundLType!", _state);
ae_assert(((boundrtype==-1||boundrtype==0)||boundrtype==1)||boundrtype==2, "Spline1DGridDiff2Cubic: incorrect BoundRType!", _state);
ae_assert((boundrtype==-1&&boundltype==-1)||(boundrtype!=-1&&boundltype!=-1), "Spline1DGridDiff2Cubic: incorrect BoundLType/BoundRType!", _state);
if( boundltype==1||boundltype==2 )
{
ae_assert(ae_isfinite(boundl, _state), "Spline1DGridDiff2Cubic: BoundL is infinite or NAN!", _state);
}
if( boundrtype==1||boundrtype==2 )
{
ae_assert(ae_isfinite(boundr, _state), "Spline1DGridDiff2Cubic: BoundR is infinite or NAN!", _state);
}
/*
* check lengths of arguments
*/
ae_assert(n>=2, "Spline1DGridDiff2Cubic: N<2!", _state);
ae_assert(x->cnt>=n, "Spline1DGridDiff2Cubic: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "Spline1DGridDiff2Cubic: Length(Y)<N!", _state);
/*
* check and sort points
*/
ylen = n;
if( boundltype==-1 )
{
ylen = n-1;
}
ae_assert(isfinitevector(x, n, _state), "Spline1DGridDiff2Cubic: X contains infinite or NAN values!", _state);
ae_assert(isfinitevector(y, ylen, _state), "Spline1DGridDiff2Cubic: Y contains infinite or NAN values!", _state);
spline1d_heapsortppoints(x, y, &p, n, _state);
ae_assert(aredistinct(x, n, _state), "Spline1DGridDiff2Cubic: at least two consequent points are too close!", _state);
/*
* Now we've checked and preordered everything,
* so we can call internal function.
*
* After this call we will calculate second derivatives
* (manually, by converting to the power basis)
*/
spline1d_spline1dgriddiffcubicinternal(x, y, n, boundltype, boundl, boundrtype, boundr, d1, &a1, &a2, &a3, &b, &dt, _state);
ae_vector_set_length(d2, n, _state);
delta = 0;
s2 = 0;
s3 = 0;
for(i=0; i<=n-2; i++)
{
/*
* We convert from Hermite basis to the power basis.
* Si is coefficient before x^i.
*
* Inside this cycle we need just S2,
* because we calculate S'' exactly at spline node,
* (only x^2 matters at x=0), but after iterations
* will be over, we will need other coefficients
* to calculate spline value at the last node.
*/
delta = x->ptr.p_double[i+1]-x->ptr.p_double[i];
delta2 = ae_sqr(delta, _state);
delta3 = delta*delta2;
s2 = (3*(y->ptr.p_double[i+1]-y->ptr.p_double[i])-2*d1->ptr.p_double[i]*delta-d1->ptr.p_double[i+1]*delta)/delta2;
s3 = (2*(y->ptr.p_double[i]-y->ptr.p_double[i+1])+d1->ptr.p_double[i]*delta+d1->ptr.p_double[i+1]*delta)/delta3;
d2->ptr.p_double[i] = 2*s2;
}
d2->ptr.p_double[n-1] = 2*s2+6*s3*delta;
/*
* Remember that HeapSortPPoints() call?
* Now we have to reorder them back.
*/
if( dt.cnt<n )
{
ae_vector_set_length(&dt, n, _state);
}
for(i=0; i<=n-1; i++)
{
dt.ptr.p_double[p.ptr.p_int[i]] = d1->ptr.p_double[i];
}
ae_v_move(&d1->ptr.p_double[0], 1, &dt.ptr.p_double[0], 1, ae_v_len(0,n-1));
for(i=0; i<=n-1; i++)
{
dt.ptr.p_double[p.ptr.p_int[i]] = d2->ptr.p_double[i];
}
ae_v_move(&d2->ptr.p_double[0], 1, &dt.ptr.p_double[0], 1, ae_v_len(0,n-1));
ae_frame_leave(_state);
}
/*************************************************************************
This function solves following problem: given table y[] of function values
at old nodes x[] and new nodes x2[], it calculates and returns table of
function values y2[] (calculated at x2[]).
This function yields same result as Spline1DBuildCubic() call followed by
sequence of Spline1DDiff() calls, but it can be several times faster when
called for ordered X[] and X2[].
INPUT PARAMETERS:
X - old spline nodes
Y - function values
X2 - new spline nodes
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points from X/Y are used
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundLType - boundary condition type for the left boundary
BoundL - left boundary condition (first or second derivative,
depending on the BoundLType)
BoundRType - boundary condition type for the right boundary
BoundR - right boundary condition (first or second derivative,
depending on the BoundRType)
N2 - new points count:
* N2>=2
* if given, only first N2 points from X2 are used
* if not given, automatically detected from X2 size
OUTPUT PARAMETERS:
F2 - function values at X2[]
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
Function values are correctly reordered on return, so F2[I] is always
equal to S(X2[I]) independently of points order.
SETTING BOUNDARY VALUES:
The BoundLType/BoundRType parameters can have the following values:
* -1, which corresonds to the periodic (cyclic) boundary conditions.
In this case:
* both BoundLType and BoundRType must be equal to -1.
* BoundL/BoundR are ignored
* Y[last] is ignored (it is assumed to be equal to Y[first]).
* 0, which corresponds to the parabolically terminated spline
(BoundL and/or BoundR are ignored).
* 1, which corresponds to the first derivative boundary condition
* 2, which corresponds to the second derivative boundary condition
* by default, BoundType=0 is used
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dconvcubic(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_int_t boundltype,
double boundl,
ae_int_t boundrtype,
double boundr,
/* Real */ ae_vector* x2,
ae_int_t n2,
/* Real */ ae_vector* y2,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _x;
ae_vector _y;
ae_vector _x2;
ae_vector a1;
ae_vector a2;
ae_vector a3;
ae_vector b;
ae_vector d;
ae_vector dt;
ae_vector d1;
ae_vector d2;
ae_vector p;
ae_vector p2;
ae_int_t i;
ae_int_t ylen;
double t;
double t2;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_x, x, _state, ae_true);
x = &_x;
ae_vector_init_copy(&_y, y, _state, ae_true);
y = &_y;
ae_vector_init_copy(&_x2, x2, _state, ae_true);
x2 = &_x2;
ae_vector_clear(y2);
ae_vector_init(&a1, 0, DT_REAL, _state, ae_true);
ae_vector_init(&a2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&a3, 0, DT_REAL, _state, ae_true);
ae_vector_init(&b, 0, DT_REAL, _state, ae_true);
ae_vector_init(&d, 0, DT_REAL, _state, ae_true);
ae_vector_init(&dt, 0, DT_REAL, _state, ae_true);
ae_vector_init(&d1, 0, DT_REAL, _state, ae_true);
ae_vector_init(&d2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&p, 0, DT_INT, _state, ae_true);
ae_vector_init(&p2, 0, DT_INT, _state, ae_true);
/*
* check correctness of boundary conditions
*/
ae_assert(((boundltype==-1||boundltype==0)||boundltype==1)||boundltype==2, "Spline1DConvCubic: incorrect BoundLType!", _state);
ae_assert(((boundrtype==-1||boundrtype==0)||boundrtype==1)||boundrtype==2, "Spline1DConvCubic: incorrect BoundRType!", _state);
ae_assert((boundrtype==-1&&boundltype==-1)||(boundrtype!=-1&&boundltype!=-1), "Spline1DConvCubic: incorrect BoundLType/BoundRType!", _state);
if( boundltype==1||boundltype==2 )
{
ae_assert(ae_isfinite(boundl, _state), "Spline1DConvCubic: BoundL is infinite or NAN!", _state);
}
if( boundrtype==1||boundrtype==2 )
{
ae_assert(ae_isfinite(boundr, _state), "Spline1DConvCubic: BoundR is infinite or NAN!", _state);
}
/*
* check lengths of arguments
*/
ae_assert(n>=2, "Spline1DConvCubic: N<2!", _state);
ae_assert(x->cnt>=n, "Spline1DConvCubic: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "Spline1DConvCubic: Length(Y)<N!", _state);
ae_assert(n2>=2, "Spline1DConvCubic: N2<2!", _state);
ae_assert(x2->cnt>=n2, "Spline1DConvCubic: Length(X2)<N2!", _state);
/*
* check and sort X/Y
*/
ylen = n;
if( boundltype==-1 )
{
ylen = n-1;
}
ae_assert(isfinitevector(x, n, _state), "Spline1DConvCubic: X contains infinite or NAN values!", _state);
ae_assert(isfinitevector(y, ylen, _state), "Spline1DConvCubic: Y contains infinite or NAN values!", _state);
ae_assert(isfinitevector(x2, n2, _state), "Spline1DConvCubic: X2 contains infinite or NAN values!", _state);
spline1d_heapsortppoints(x, y, &p, n, _state);
ae_assert(aredistinct(x, n, _state), "Spline1DConvCubic: at least two consequent points are too close!", _state);
/*
* set up DT (we will need it below)
*/
ae_vector_set_length(&dt, ae_maxint(n, n2, _state), _state);
/*
* sort X2:
* * use fake array DT because HeapSortPPoints() needs both integer AND real arrays
* * if we have periodic problem, wrap points
* * sort them, store permutation at P2
*/
if( boundrtype==-1&&boundltype==-1 )
{
for(i=0; i<=n2-1; i++)
{
t = x2->ptr.p_double[i];
apperiodicmap(&t, x->ptr.p_double[0], x->ptr.p_double[n-1], &t2, _state);
x2->ptr.p_double[i] = t;
}
}
spline1d_heapsortppoints(x2, &dt, &p2, n2, _state);
/*
* Now we've checked and preordered everything, so we:
* * call internal GridDiff() function to get Hermite form of spline
* * convert using internal Conv() function
* * convert Y2 back to original order
*/
spline1d_spline1dgriddiffcubicinternal(x, y, n, boundltype, boundl, boundrtype, boundr, &d, &a1, &a2, &a3, &b, &dt, _state);
spline1dconvdiffinternal(x, y, &d, n, x2, n2, y2, ae_true, &d1, ae_false, &d2, ae_false, _state);
ae_assert(dt.cnt>=n2, "Spline1DConvCubic: internal error!", _state);
for(i=0; i<=n2-1; i++)
{
dt.ptr.p_double[p2.ptr.p_int[i]] = y2->ptr.p_double[i];
}
ae_v_move(&y2->ptr.p_double[0], 1, &dt.ptr.p_double[0], 1, ae_v_len(0,n2-1));
ae_frame_leave(_state);
}
/*************************************************************************
This function solves following problem: given table y[] of function values
at old nodes x[] and new nodes x2[], it calculates and returns table of
function values y2[] and derivatives d2[] (calculated at x2[]).
This function yields same result as Spline1DBuildCubic() call followed by
sequence of Spline1DDiff() calls, but it can be several times faster when
called for ordered X[] and X2[].
INPUT PARAMETERS:
X - old spline nodes
Y - function values
X2 - new spline nodes
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points from X/Y are used
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundLType - boundary condition type for the left boundary
BoundL - left boundary condition (first or second derivative,
depending on the BoundLType)
BoundRType - boundary condition type for the right boundary
BoundR - right boundary condition (first or second derivative,
depending on the BoundRType)
N2 - new points count:
* N2>=2
* if given, only first N2 points from X2 are used
* if not given, automatically detected from X2 size
OUTPUT PARAMETERS:
F2 - function values at X2[]
D2 - first derivatives at X2[]
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
Function values are correctly reordered on return, so F2[I] is always
equal to S(X2[I]) independently of points order.
SETTING BOUNDARY VALUES:
The BoundLType/BoundRType parameters can have the following values:
* -1, which corresonds to the periodic (cyclic) boundary conditions.
In this case:
* both BoundLType and BoundRType must be equal to -1.
* BoundL/BoundR are ignored
* Y[last] is ignored (it is assumed to be equal to Y[first]).
* 0, which corresponds to the parabolically terminated spline
(BoundL and/or BoundR are ignored).
* 1, which corresponds to the first derivative boundary condition
* 2, which corresponds to the second derivative boundary condition
* by default, BoundType=0 is used
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dconvdiffcubic(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_int_t boundltype,
double boundl,
ae_int_t boundrtype,
double boundr,
/* Real */ ae_vector* x2,
ae_int_t n2,
/* Real */ ae_vector* y2,
/* Real */ ae_vector* d2,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _x;
ae_vector _y;
ae_vector _x2;
ae_vector a1;
ae_vector a2;
ae_vector a3;
ae_vector b;
ae_vector d;
ae_vector dt;
ae_vector rt1;
ae_vector p;
ae_vector p2;
ae_int_t i;
ae_int_t ylen;
double t;
double t2;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_x, x, _state, ae_true);
x = &_x;
ae_vector_init_copy(&_y, y, _state, ae_true);
y = &_y;
ae_vector_init_copy(&_x2, x2, _state, ae_true);
x2 = &_x2;
ae_vector_clear(y2);
ae_vector_clear(d2);
ae_vector_init(&a1, 0, DT_REAL, _state, ae_true);
ae_vector_init(&a2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&a3, 0, DT_REAL, _state, ae_true);
ae_vector_init(&b, 0, DT_REAL, _state, ae_true);
ae_vector_init(&d, 0, DT_REAL, _state, ae_true);
ae_vector_init(&dt, 0, DT_REAL, _state, ae_true);
ae_vector_init(&rt1, 0, DT_REAL, _state, ae_true);
ae_vector_init(&p, 0, DT_INT, _state, ae_true);
ae_vector_init(&p2, 0, DT_INT, _state, ae_true);
/*
* check correctness of boundary conditions
*/
ae_assert(((boundltype==-1||boundltype==0)||boundltype==1)||boundltype==2, "Spline1DConvDiffCubic: incorrect BoundLType!", _state);
ae_assert(((boundrtype==-1||boundrtype==0)||boundrtype==1)||boundrtype==2, "Spline1DConvDiffCubic: incorrect BoundRType!", _state);
ae_assert((boundrtype==-1&&boundltype==-1)||(boundrtype!=-1&&boundltype!=-1), "Spline1DConvDiffCubic: incorrect BoundLType/BoundRType!", _state);
if( boundltype==1||boundltype==2 )
{
ae_assert(ae_isfinite(boundl, _state), "Spline1DConvDiffCubic: BoundL is infinite or NAN!", _state);
}
if( boundrtype==1||boundrtype==2 )
{
ae_assert(ae_isfinite(boundr, _state), "Spline1DConvDiffCubic: BoundR is infinite or NAN!", _state);
}
/*
* check lengths of arguments
*/
ae_assert(n>=2, "Spline1DConvDiffCubic: N<2!", _state);
ae_assert(x->cnt>=n, "Spline1DConvDiffCubic: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "Spline1DConvDiffCubic: Length(Y)<N!", _state);
ae_assert(n2>=2, "Spline1DConvDiffCubic: N2<2!", _state);
ae_assert(x2->cnt>=n2, "Spline1DConvDiffCubic: Length(X2)<N2!", _state);
/*
* check and sort X/Y
*/
ylen = n;
if( boundltype==-1 )
{
ylen = n-1;
}
ae_assert(isfinitevector(x, n, _state), "Spline1DConvDiffCubic: X contains infinite or NAN values!", _state);
ae_assert(isfinitevector(y, ylen, _state), "Spline1DConvDiffCubic: Y contains infinite or NAN values!", _state);
ae_assert(isfinitevector(x2, n2, _state), "Spline1DConvDiffCubic: X2 contains infinite or NAN values!", _state);
spline1d_heapsortppoints(x, y, &p, n, _state);
ae_assert(aredistinct(x, n, _state), "Spline1DConvDiffCubic: at least two consequent points are too close!", _state);
/*
* set up DT (we will need it below)
*/
ae_vector_set_length(&dt, ae_maxint(n, n2, _state), _state);
/*
* sort X2:
* * use fake array DT because HeapSortPPoints() needs both integer AND real arrays
* * if we have periodic problem, wrap points
* * sort them, store permutation at P2
*/
if( boundrtype==-1&&boundltype==-1 )
{
for(i=0; i<=n2-1; i++)
{
t = x2->ptr.p_double[i];
apperiodicmap(&t, x->ptr.p_double[0], x->ptr.p_double[n-1], &t2, _state);
x2->ptr.p_double[i] = t;
}
}
spline1d_heapsortppoints(x2, &dt, &p2, n2, _state);
/*
* Now we've checked and preordered everything, so we:
* * call internal GridDiff() function to get Hermite form of spline
* * convert using internal Conv() function
* * convert Y2 back to original order
*/
spline1d_spline1dgriddiffcubicinternal(x, y, n, boundltype, boundl, boundrtype, boundr, &d, &a1, &a2, &a3, &b, &dt, _state);
spline1dconvdiffinternal(x, y, &d, n, x2, n2, y2, ae_true, d2, ae_true, &rt1, ae_false, _state);
ae_assert(dt.cnt>=n2, "Spline1DConvDiffCubic: internal error!", _state);
for(i=0; i<=n2-1; i++)
{
dt.ptr.p_double[p2.ptr.p_int[i]] = y2->ptr.p_double[i];
}
ae_v_move(&y2->ptr.p_double[0], 1, &dt.ptr.p_double[0], 1, ae_v_len(0,n2-1));
for(i=0; i<=n2-1; i++)
{
dt.ptr.p_double[p2.ptr.p_int[i]] = d2->ptr.p_double[i];
}
ae_v_move(&d2->ptr.p_double[0], 1, &dt.ptr.p_double[0], 1, ae_v_len(0,n2-1));
ae_frame_leave(_state);
}
/*************************************************************************
This function solves following problem: given table y[] of function values
at old nodes x[] and new nodes x2[], it calculates and returns table of
function values y2[], first and second derivatives d2[] and dd2[]
(calculated at x2[]).
This function yields same result as Spline1DBuildCubic() call followed by
sequence of Spline1DDiff() calls, but it can be several times faster when
called for ordered X[] and X2[].
INPUT PARAMETERS:
X - old spline nodes
Y - function values
X2 - new spline nodes
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points from X/Y are used
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundLType - boundary condition type for the left boundary
BoundL - left boundary condition (first or second derivative,
depending on the BoundLType)
BoundRType - boundary condition type for the right boundary
BoundR - right boundary condition (first or second derivative,
depending on the BoundRType)
N2 - new points count:
* N2>=2
* if given, only first N2 points from X2 are used
* if not given, automatically detected from X2 size
OUTPUT PARAMETERS:
F2 - function values at X2[]
D2 - first derivatives at X2[]
DD2 - second derivatives at X2[]
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
Function values are correctly reordered on return, so F2[I] is always
equal to S(X2[I]) independently of points order.
SETTING BOUNDARY VALUES:
The BoundLType/BoundRType parameters can have the following values:
* -1, which corresonds to the periodic (cyclic) boundary conditions.
In this case:
* both BoundLType and BoundRType must be equal to -1.
* BoundL/BoundR are ignored
* Y[last] is ignored (it is assumed to be equal to Y[first]).
* 0, which corresponds to the parabolically terminated spline
(BoundL and/or BoundR are ignored).
* 1, which corresponds to the first derivative boundary condition
* 2, which corresponds to the second derivative boundary condition
* by default, BoundType=0 is used
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dconvdiff2cubic(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_int_t boundltype,
double boundl,
ae_int_t boundrtype,
double boundr,
/* Real */ ae_vector* x2,
ae_int_t n2,
/* Real */ ae_vector* y2,
/* Real */ ae_vector* d2,
/* Real */ ae_vector* dd2,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _x;
ae_vector _y;
ae_vector _x2;
ae_vector a1;
ae_vector a2;
ae_vector a3;
ae_vector b;
ae_vector d;
ae_vector dt;
ae_vector p;
ae_vector p2;
ae_int_t i;
ae_int_t ylen;
double t;
double t2;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_x, x, _state, ae_true);
x = &_x;
ae_vector_init_copy(&_y, y, _state, ae_true);
y = &_y;
ae_vector_init_copy(&_x2, x2, _state, ae_true);
x2 = &_x2;
ae_vector_clear(y2);
ae_vector_clear(d2);
ae_vector_clear(dd2);
ae_vector_init(&a1, 0, DT_REAL, _state, ae_true);
ae_vector_init(&a2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&a3, 0, DT_REAL, _state, ae_true);
ae_vector_init(&b, 0, DT_REAL, _state, ae_true);
ae_vector_init(&d, 0, DT_REAL, _state, ae_true);
ae_vector_init(&dt, 0, DT_REAL, _state, ae_true);
ae_vector_init(&p, 0, DT_INT, _state, ae_true);
ae_vector_init(&p2, 0, DT_INT, _state, ae_true);
/*
* check correctness of boundary conditions
*/
ae_assert(((boundltype==-1||boundltype==0)||boundltype==1)||boundltype==2, "Spline1DConvDiff2Cubic: incorrect BoundLType!", _state);
ae_assert(((boundrtype==-1||boundrtype==0)||boundrtype==1)||boundrtype==2, "Spline1DConvDiff2Cubic: incorrect BoundRType!", _state);
ae_assert((boundrtype==-1&&boundltype==-1)||(boundrtype!=-1&&boundltype!=-1), "Spline1DConvDiff2Cubic: incorrect BoundLType/BoundRType!", _state);
if( boundltype==1||boundltype==2 )
{
ae_assert(ae_isfinite(boundl, _state), "Spline1DConvDiff2Cubic: BoundL is infinite or NAN!", _state);
}
if( boundrtype==1||boundrtype==2 )
{
ae_assert(ae_isfinite(boundr, _state), "Spline1DConvDiff2Cubic: BoundR is infinite or NAN!", _state);
}
/*
* check lengths of arguments
*/
ae_assert(n>=2, "Spline1DConvDiff2Cubic: N<2!", _state);
ae_assert(x->cnt>=n, "Spline1DConvDiff2Cubic: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "Spline1DConvDiff2Cubic: Length(Y)<N!", _state);
ae_assert(n2>=2, "Spline1DConvDiff2Cubic: N2<2!", _state);
ae_assert(x2->cnt>=n2, "Spline1DConvDiff2Cubic: Length(X2)<N2!", _state);
/*
* check and sort X/Y
*/
ylen = n;
if( boundltype==-1 )
{
ylen = n-1;
}
ae_assert(isfinitevector(x, n, _state), "Spline1DConvDiff2Cubic: X contains infinite or NAN values!", _state);
ae_assert(isfinitevector(y, ylen, _state), "Spline1DConvDiff2Cubic: Y contains infinite or NAN values!", _state);
ae_assert(isfinitevector(x2, n2, _state), "Spline1DConvDiff2Cubic: X2 contains infinite or NAN values!", _state);
spline1d_heapsortppoints(x, y, &p, n, _state);
ae_assert(aredistinct(x, n, _state), "Spline1DConvDiff2Cubic: at least two consequent points are too close!", _state);
/*
* set up DT (we will need it below)
*/
ae_vector_set_length(&dt, ae_maxint(n, n2, _state), _state);
/*
* sort X2:
* * use fake array DT because HeapSortPPoints() needs both integer AND real arrays
* * if we have periodic problem, wrap points
* * sort them, store permutation at P2
*/
if( boundrtype==-1&&boundltype==-1 )
{
for(i=0; i<=n2-1; i++)
{
t = x2->ptr.p_double[i];
apperiodicmap(&t, x->ptr.p_double[0], x->ptr.p_double[n-1], &t2, _state);
x2->ptr.p_double[i] = t;
}
}
spline1d_heapsortppoints(x2, &dt, &p2, n2, _state);
/*
* Now we've checked and preordered everything, so we:
* * call internal GridDiff() function to get Hermite form of spline
* * convert using internal Conv() function
* * convert Y2 back to original order
*/
spline1d_spline1dgriddiffcubicinternal(x, y, n, boundltype, boundl, boundrtype, boundr, &d, &a1, &a2, &a3, &b, &dt, _state);
spline1dconvdiffinternal(x, y, &d, n, x2, n2, y2, ae_true, d2, ae_true, dd2, ae_true, _state);
ae_assert(dt.cnt>=n2, "Spline1DConvDiff2Cubic: internal error!", _state);
for(i=0; i<=n2-1; i++)
{
dt.ptr.p_double[p2.ptr.p_int[i]] = y2->ptr.p_double[i];
}
ae_v_move(&y2->ptr.p_double[0], 1, &dt.ptr.p_double[0], 1, ae_v_len(0,n2-1));
for(i=0; i<=n2-1; i++)
{
dt.ptr.p_double[p2.ptr.p_int[i]] = d2->ptr.p_double[i];
}
ae_v_move(&d2->ptr.p_double[0], 1, &dt.ptr.p_double[0], 1, ae_v_len(0,n2-1));
for(i=0; i<=n2-1; i++)
{
dt.ptr.p_double[p2.ptr.p_int[i]] = dd2->ptr.p_double[i];
}
ae_v_move(&dd2->ptr.p_double[0], 1, &dt.ptr.p_double[0], 1, ae_v_len(0,n2-1));
ae_frame_leave(_state);
}
/*************************************************************************
This subroutine builds Catmull-Rom spline interpolant.
INPUT PARAMETERS:
X - spline nodes, array[0..N-1].
Y - function values, array[0..N-1].
OPTIONAL PARAMETERS:
N - points count:
* N>=2
* if given, only first N points are used to build spline
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
BoundType - boundary condition type:
* -1 for periodic boundary condition
* 0 for parabolically terminated spline (default)
Tension - tension parameter:
* tension=0 corresponds to classic Catmull-Rom spline (default)
* 0<tension<1 corresponds to more general form - cardinal spline
OUTPUT PARAMETERS:
C - spline interpolant
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:
Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal values for
the first and last points - it automatically forces them to be equal by
copying Y[first_point] (corresponds to the leftmost, minimal X[]) to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].
-- ALGLIB PROJECT --
Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildcatmullrom(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_int_t boundtype,
double tension,
spline1dinterpolant* c,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _x;
ae_vector _y;
ae_vector d;
ae_int_t i;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_x, x, _state, ae_true);
x = &_x;
ae_vector_init_copy(&_y, y, _state, ae_true);
y = &_y;
_spline1dinterpolant_clear(c);
ae_vector_init(&d, 0, DT_REAL, _state, ae_true);
ae_assert(n>=2, "Spline1DBuildCatmullRom: N<2!", _state);
ae_assert(boundtype==-1||boundtype==0, "Spline1DBuildCatmullRom: incorrect BoundType!", _state);
ae_assert(ae_fp_greater_eq(tension,0), "Spline1DBuildCatmullRom: Tension<0!", _state);
ae_assert(ae_fp_less_eq(tension,1), "Spline1DBuildCatmullRom: Tension>1!", _state);
ae_assert(x->cnt>=n, "Spline1DBuildCatmullRom: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "Spline1DBuildCatmullRom: Length(Y)<N!", _state);
/*
* check and sort points
*/
ae_assert(isfinitevector(x, n, _state), "Spline1DBuildCatmullRom: X contains infinite or NAN values!", _state);
ae_assert(isfinitevector(y, n, _state), "Spline1DBuildCatmullRom: Y contains infinite or NAN values!", _state);
spline1d_heapsortpoints(x, y, n, _state);
ae_assert(aredistinct(x, n, _state), "Spline1DBuildCatmullRom: at least two consequent points are too close!", _state);
/*
* Special cases:
* * N=2, parabolic terminated boundary condition on both ends
* * N=2, periodic boundary condition
*/
if( n==2&&boundtype==0 )
{
/*
* Just linear spline
*/
spline1dbuildlinear(x, y, n, c, _state);
ae_frame_leave(_state);
return;
}
if( n==2&&boundtype==-1 )
{
/*
* Same as cubic spline with periodic conditions
*/
spline1dbuildcubic(x, y, n, -1, 0.0, -1, 0.0, c, _state);
ae_frame_leave(_state);
return;
}
/*
* Periodic or non-periodic boundary conditions
*/
if( boundtype==-1 )
{
/*
* Periodic boundary conditions
*/
y->ptr.p_double[n-1] = y->ptr.p_double[0];
ae_vector_set_length(&d, n, _state);
d.ptr.p_double[0] = (y->ptr.p_double[1]-y->ptr.p_double[n-2])/(2*(x->ptr.p_double[1]-x->ptr.p_double[0]+x->ptr.p_double[n-1]-x->ptr.p_double[n-2]));
for(i=1; i<=n-2; i++)
{
d.ptr.p_double[i] = (1-tension)*(y->ptr.p_double[i+1]-y->ptr.p_double[i-1])/(x->ptr.p_double[i+1]-x->ptr.p_double[i-1]);
}
d.ptr.p_double[n-1] = d.ptr.p_double[0];
/*
* Now problem is reduced to the cubic Hermite spline
*/
spline1dbuildhermite(x, y, &d, n, c, _state);
c->periodic = ae_true;
}
else
{
/*
* Non-periodic boundary conditions
*/
ae_vector_set_length(&d, n, _state);
for(i=1; i<=n-2; i++)
{
d.ptr.p_double[i] = (1-tension)*(y->ptr.p_double[i+1]-y->ptr.p_double[i-1])/(x->ptr.p_double[i+1]-x->ptr.p_double[i-1]);
}
d.ptr.p_double[0] = 2*(y->ptr.p_double[1]-y->ptr.p_double[0])/(x->ptr.p_double[1]-x->ptr.p_double[0])-d.ptr.p_double[1];
d.ptr.p_double[n-1] = 2*(y->ptr.p_double[n-1]-y->ptr.p_double[n-2])/(x->ptr.p_double[n-1]-x->ptr.p_double[n-2])-d.ptr.p_double[n-2];
/*
* Now problem is reduced to the cubic Hermite spline
*/
spline1dbuildhermite(x, y, &d, n, c, _state);
}
ae_frame_leave(_state);
}
/*************************************************************************
This subroutine builds Hermite spline interpolant.
INPUT PARAMETERS:
X - spline nodes, array[0..N-1]
Y - function values, array[0..N-1]
D - derivatives, array[0..N-1]
N - points count (optional):
* N>=2
* if given, only first N points are used to build spline
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
OUTPUT PARAMETERS:
C - spline interpolant.
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
-- ALGLIB PROJECT --
Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildhermite(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* d,
ae_int_t n,
spline1dinterpolant* c,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _x;
ae_vector _y;
ae_vector _d;
ae_int_t i;
double delta;
double delta2;
double delta3;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_x, x, _state, ae_true);
x = &_x;
ae_vector_init_copy(&_y, y, _state, ae_true);
y = &_y;
ae_vector_init_copy(&_d, d, _state, ae_true);
d = &_d;
_spline1dinterpolant_clear(c);
ae_assert(n>=2, "Spline1DBuildHermite: N<2!", _state);
ae_assert(x->cnt>=n, "Spline1DBuildHermite: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "Spline1DBuildHermite: Length(Y)<N!", _state);
ae_assert(d->cnt>=n, "Spline1DBuildHermite: Length(D)<N!", _state);
/*
* check and sort points
*/
ae_assert(isfinitevector(x, n, _state), "Spline1DBuildHermite: X contains infinite or NAN values!", _state);
ae_assert(isfinitevector(y, n, _state), "Spline1DBuildHermite: Y contains infinite or NAN values!", _state);
ae_assert(isfinitevector(d, n, _state), "Spline1DBuildHermite: D contains infinite or NAN values!", _state);
heapsortdpoints(x, y, d, n, _state);
ae_assert(aredistinct(x, n, _state), "Spline1DBuildHermite: at least two consequent points are too close!", _state);
/*
* Build
*/
ae_vector_set_length(&c->x, n, _state);
ae_vector_set_length(&c->c, 4*(n-1)+2, _state);
c->periodic = ae_false;
c->k = 3;
c->n = n;
c->continuity = 1;
for(i=0; i<=n-1; i++)
{
c->x.ptr.p_double[i] = x->ptr.p_double[i];
}
for(i=0; i<=n-2; i++)
{
delta = x->ptr.p_double[i+1]-x->ptr.p_double[i];
delta2 = ae_sqr(delta, _state);
delta3 = delta*delta2;
c->c.ptr.p_double[4*i+0] = y->ptr.p_double[i];
c->c.ptr.p_double[4*i+1] = d->ptr.p_double[i];
c->c.ptr.p_double[4*i+2] = (3*(y->ptr.p_double[i+1]-y->ptr.p_double[i])-2*d->ptr.p_double[i]*delta-d->ptr.p_double[i+1]*delta)/delta2;
c->c.ptr.p_double[4*i+3] = (2*(y->ptr.p_double[i]-y->ptr.p_double[i+1])+d->ptr.p_double[i]*delta+d->ptr.p_double[i+1]*delta)/delta3;
}
c->c.ptr.p_double[4*(n-1)+0] = y->ptr.p_double[n-1];
c->c.ptr.p_double[4*(n-1)+1] = d->ptr.p_double[n-1];
ae_frame_leave(_state);
}
/*************************************************************************
This subroutine builds Akima spline interpolant
INPUT PARAMETERS:
X - spline nodes, array[0..N-1]
Y - function values, array[0..N-1]
N - points count (optional):
* N>=2
* if given, only first N points are used to build spline
* if not given, automatically detected from X/Y sizes
(len(X) must be equal to len(Y))
OUTPUT PARAMETERS:
C - spline interpolant
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
-- ALGLIB PROJECT --
Copyright 24.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildakima(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
spline1dinterpolant* c,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _x;
ae_vector _y;
ae_int_t i;
ae_vector d;
ae_vector w;
ae_vector diff;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_x, x, _state, ae_true);
x = &_x;
ae_vector_init_copy(&_y, y, _state, ae_true);
y = &_y;
_spline1dinterpolant_clear(c);
ae_vector_init(&d, 0, DT_REAL, _state, ae_true);
ae_vector_init(&w, 0, DT_REAL, _state, ae_true);
ae_vector_init(&diff, 0, DT_REAL, _state, ae_true);
ae_assert(n>=2, "Spline1DBuildAkima: N<2!", _state);
ae_assert(x->cnt>=n, "Spline1DBuildAkima: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "Spline1DBuildAkima: Length(Y)<N!", _state);
/*
* check and sort points
*/
ae_assert(isfinitevector(x, n, _state), "Spline1DBuildAkima: X contains infinite or NAN values!", _state);
ae_assert(isfinitevector(y, n, _state), "Spline1DBuildAkima: Y contains infinite or NAN values!", _state);
spline1d_heapsortpoints(x, y, n, _state);
ae_assert(aredistinct(x, n, _state), "Spline1DBuildAkima: at least two consequent points are too close!", _state);
/*
* Handle special cases: N=2, N=3, N=4
*/
if( n<=4 )
{
spline1dbuildcubic(x, y, n, 0, 0.0, 0, 0.0, c, _state);
ae_frame_leave(_state);
return;
}
/*
* Prepare W (weights), Diff (divided differences)
*/
ae_vector_set_length(&w, n-1, _state);
ae_vector_set_length(&diff, n-1, _state);
for(i=0; i<=n-2; i++)
{
diff.ptr.p_double[i] = (y->ptr.p_double[i+1]-y->ptr.p_double[i])/(x->ptr.p_double[i+1]-x->ptr.p_double[i]);
}
for(i=1; i<=n-2; i++)
{
w.ptr.p_double[i] = ae_fabs(diff.ptr.p_double[i]-diff.ptr.p_double[i-1], _state);
}
/*
* Prepare Hermite interpolation scheme
*/
ae_vector_set_length(&d, n, _state);
for(i=2; i<=n-3; i++)
{
if( ae_fp_neq(ae_fabs(w.ptr.p_double[i-1], _state)+ae_fabs(w.ptr.p_double[i+1], _state),0) )
{
d.ptr.p_double[i] = (w.ptr.p_double[i+1]*diff.ptr.p_double[i-1]+w.ptr.p_double[i-1]*diff.ptr.p_double[i])/(w.ptr.p_double[i+1]+w.ptr.p_double[i-1]);
}
else
{
d.ptr.p_double[i] = ((x->ptr.p_double[i+1]-x->ptr.p_double[i])*diff.ptr.p_double[i-1]+(x->ptr.p_double[i]-x->ptr.p_double[i-1])*diff.ptr.p_double[i])/(x->ptr.p_double[i+1]-x->ptr.p_double[i-1]);
}
}
d.ptr.p_double[0] = spline1d_diffthreepoint(x->ptr.p_double[0], x->ptr.p_double[0], y->ptr.p_double[0], x->ptr.p_double[1], y->ptr.p_double[1], x->ptr.p_double[2], y->ptr.p_double[2], _state);
d.ptr.p_double[1] = spline1d_diffthreepoint(x->ptr.p_double[1], x->ptr.p_double[0], y->ptr.p_double[0], x->ptr.p_double[1], y->ptr.p_double[1], x->ptr.p_double[2], y->ptr.p_double[2], _state);
d.ptr.p_double[n-2] = spline1d_diffthreepoint(x->ptr.p_double[n-2], x->ptr.p_double[n-3], y->ptr.p_double[n-3], x->ptr.p_double[n-2], y->ptr.p_double[n-2], x->ptr.p_double[n-1], y->ptr.p_double[n-1], _state);
d.ptr.p_double[n-1] = spline1d_diffthreepoint(x->ptr.p_double[n-1], x->ptr.p_double[n-3], y->ptr.p_double[n-3], x->ptr.p_double[n-2], y->ptr.p_double[n-2], x->ptr.p_double[n-1], y->ptr.p_double[n-1], _state);
/*
* Build Akima spline using Hermite interpolation scheme
*/
spline1dbuildhermite(x, y, &d, n, c, _state);
ae_frame_leave(_state);
}
/*************************************************************************
This subroutine calculates the value of the spline at the given point X.
INPUT PARAMETERS:
C - spline interpolant
X - point
Result:
S(x)
-- ALGLIB PROJECT --
Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
double spline1dcalc(spline1dinterpolant* c, double x, ae_state *_state)
{
ae_int_t l;
ae_int_t r;
ae_int_t m;
double t;
double result;
ae_assert(c->k==3, "Spline1DCalc: internal error", _state);
ae_assert(!ae_isinf(x, _state), "Spline1DCalc: infinite X!", _state);
/*
* special case: NaN
*/
if( ae_isnan(x, _state) )
{
result = _state->v_nan;
return result;
}
/*
* correct if periodic
*/
if( c->periodic )
{
apperiodicmap(&x, c->x.ptr.p_double[0], c->x.ptr.p_double[c->n-1], &t, _state);
}
/*
* Binary search in the [ x[0], ..., x[n-2] ] (x[n-1] is not included)
*/
l = 0;
r = c->n-2+1;
while(l!=r-1)
{
m = (l+r)/2;
if( c->x.ptr.p_double[m]>=x )
{
r = m;
}
else
{
l = m;
}
}
/*
* Interpolation
*/
x = x-c->x.ptr.p_double[l];
m = 4*l;
result = c->c.ptr.p_double[m]+x*(c->c.ptr.p_double[m+1]+x*(c->c.ptr.p_double[m+2]+x*c->c.ptr.p_double[m+3]));
return result;
}
/*************************************************************************
This subroutine differentiates the spline.
INPUT PARAMETERS:
C - spline interpolant.
X - point
Result:
S - S(x)
DS - S'(x)
D2S - S''(x)
-- ALGLIB PROJECT --
Copyright 24.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1ddiff(spline1dinterpolant* c,
double x,
double* s,
double* ds,
double* d2s,
ae_state *_state)
{
ae_int_t l;
ae_int_t r;
ae_int_t m;
double t;
*s = 0;
*ds = 0;
*d2s = 0;
ae_assert(c->k==3, "Spline1DDiff: internal error", _state);
ae_assert(!ae_isinf(x, _state), "Spline1DDiff: infinite X!", _state);
/*
* special case: NaN
*/
if( ae_isnan(x, _state) )
{
*s = _state->v_nan;
*ds = _state->v_nan;
*d2s = _state->v_nan;
return;
}
/*
* correct if periodic
*/
if( c->periodic )
{
apperiodicmap(&x, c->x.ptr.p_double[0], c->x.ptr.p_double[c->n-1], &t, _state);
}
/*
* Binary search
*/
l = 0;
r = c->n-2+1;
while(l!=r-1)
{
m = (l+r)/2;
if( c->x.ptr.p_double[m]>=x )
{
r = m;
}
else
{
l = m;
}
}
/*
* Differentiation
*/
x = x-c->x.ptr.p_double[l];
m = 4*l;
*s = c->c.ptr.p_double[m]+x*(c->c.ptr.p_double[m+1]+x*(c->c.ptr.p_double[m+2]+x*c->c.ptr.p_double[m+3]));
*ds = c->c.ptr.p_double[m+1]+2*x*c->c.ptr.p_double[m+2]+3*ae_sqr(x, _state)*c->c.ptr.p_double[m+3];
*d2s = 2*c->c.ptr.p_double[m+2]+6*x*c->c.ptr.p_double[m+3];
}
/*************************************************************************
This subroutine makes the copy of the spline.
INPUT PARAMETERS:
C - spline interpolant.
Result:
CC - spline copy
-- ALGLIB PROJECT --
Copyright 29.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dcopy(spline1dinterpolant* c,
spline1dinterpolant* cc,
ae_state *_state)
{
ae_int_t s;
_spline1dinterpolant_clear(cc);
cc->periodic = c->periodic;
cc->n = c->n;
cc->k = c->k;
cc->continuity = c->continuity;
ae_vector_set_length(&cc->x, cc->n, _state);
ae_v_move(&cc->x.ptr.p_double[0], 1, &c->x.ptr.p_double[0], 1, ae_v_len(0,cc->n-1));
s = c->c.cnt;
ae_vector_set_length(&cc->c, s, _state);
ae_v_move(&cc->c.ptr.p_double[0], 1, &c->c.ptr.p_double[0], 1, ae_v_len(0,s-1));
}
/*************************************************************************
This subroutine unpacks the spline into the coefficients table.
INPUT PARAMETERS:
C - spline interpolant.
X - point
OUTPUT PARAMETERS:
Tbl - coefficients table, unpacked format, array[0..N-2, 0..5].
For I = 0...N-2:
Tbl[I,0] = X[i]
Tbl[I,1] = X[i+1]
Tbl[I,2] = C0
Tbl[I,3] = C1
Tbl[I,4] = C2
Tbl[I,5] = C3
On [x[i], x[i+1]] spline is equals to:
S(x) = C0 + C1*t + C2*t^2 + C3*t^3
t = x-x[i]
NOTE:
You can rebuild spline with Spline1DBuildHermite() function, which
accepts as inputs function values and derivatives at nodes, which are
easy to calculate when you have coefficients.
-- ALGLIB PROJECT --
Copyright 29.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dunpack(spline1dinterpolant* c,
ae_int_t* n,
/* Real */ ae_matrix* tbl,
ae_state *_state)
{
ae_int_t i;
ae_int_t j;
*n = 0;
ae_matrix_clear(tbl);
ae_matrix_set_length(tbl, c->n-2+1, 2+c->k+1, _state);
*n = c->n;
/*
* Fill
*/
for(i=0; i<=*n-2; i++)
{
tbl->ptr.pp_double[i][0] = c->x.ptr.p_double[i];
tbl->ptr.pp_double[i][1] = c->x.ptr.p_double[i+1];
for(j=0; j<=c->k; j++)
{
tbl->ptr.pp_double[i][2+j] = c->c.ptr.p_double[(c->k+1)*i+j];
}
}
}
/*************************************************************************
This subroutine performs linear transformation of the spline argument.
INPUT PARAMETERS:
C - spline interpolant.
A, B- transformation coefficients: x = A*t + B
Result:
C - transformed spline
-- ALGLIB PROJECT --
Copyright 30.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dlintransx(spline1dinterpolant* c,
double a,
double b,
ae_state *_state)
{
ae_frame _frame_block;
ae_int_t i;
ae_int_t n;
double v;
double dv;
double d2v;
ae_vector x;
ae_vector y;
ae_vector d;
ae_bool isperiodic;
ae_int_t contval;
ae_frame_make(_state, &_frame_block);
ae_vector_init(&x, 0, DT_REAL, _state, ae_true);
ae_vector_init(&y, 0, DT_REAL, _state, ae_true);
ae_vector_init(&d, 0, DT_REAL, _state, ae_true);
ae_assert(c->k==3, "Spline1DLinTransX: internal error", _state);
n = c->n;
ae_vector_set_length(&x, n, _state);
ae_vector_set_length(&y, n, _state);
ae_vector_set_length(&d, n, _state);
/*
* Unpack, X, Y, dY/dX.
* Scale and pack with Spline1DBuildHermite again.
*/
if( ae_fp_eq(a,0) )
{
/*
* Special case: A=0
*/
v = spline1dcalc(c, b, _state);
for(i=0; i<=n-1; i++)
{
x.ptr.p_double[i] = c->x.ptr.p_double[i];
y.ptr.p_double[i] = v;
d.ptr.p_double[i] = 0.0;
}
}
else
{
/*
* General case, A<>0
*/
for(i=0; i<=n-1; i++)
{
x.ptr.p_double[i] = c->x.ptr.p_double[i];
spline1ddiff(c, x.ptr.p_double[i], &v, &dv, &d2v, _state);
x.ptr.p_double[i] = (x.ptr.p_double[i]-b)/a;
y.ptr.p_double[i] = v;
d.ptr.p_double[i] = a*dv;
}
}
isperiodic = c->periodic;
contval = c->continuity;
if( contval>0 )
{
spline1dbuildhermite(&x, &y, &d, n, c, _state);
}
else
{
spline1dbuildlinear(&x, &y, n, c, _state);
}
c->periodic = isperiodic;
c->continuity = contval;
ae_frame_leave(_state);
}
/*************************************************************************
This subroutine performs linear transformation of the spline.
INPUT PARAMETERS:
C - spline interpolant.
A, B- transformation coefficients: S2(x) = A*S(x) + B
Result:
C - transformed spline
-- ALGLIB PROJECT --
Copyright 30.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dlintransy(spline1dinterpolant* c,
double a,
double b,
ae_state *_state)
{
ae_int_t i;
ae_int_t j;
ae_int_t n;
ae_assert(c->k==3, "Spline1DLinTransX: internal error", _state);
n = c->n;
for(i=0; i<=n-2; i++)
{
c->c.ptr.p_double[4*i] = a*c->c.ptr.p_double[4*i]+b;
for(j=1; j<=3; j++)
{
c->c.ptr.p_double[4*i+j] = a*c->c.ptr.p_double[4*i+j];
}
}
c->c.ptr.p_double[4*(n-1)+0] = a*c->c.ptr.p_double[4*(n-1)+0]+b;
c->c.ptr.p_double[4*(n-1)+1] = a*c->c.ptr.p_double[4*(n-1)+1];
}
/*************************************************************************
This subroutine integrates the spline.
INPUT PARAMETERS:
C - spline interpolant.
X - right bound of the integration interval [a, x],
here 'a' denotes min(x[])
Result:
integral(S(t)dt,a,x)
-- ALGLIB PROJECT --
Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
double spline1dintegrate(spline1dinterpolant* c,
double x,
ae_state *_state)
{
ae_int_t n;
ae_int_t i;
ae_int_t j;
ae_int_t l;
ae_int_t r;
ae_int_t m;
double w;
double v;
double t;
double intab;
double additionalterm;
double result;
n = c->n;
/*
* Periodic splines require special treatment. We make
* following transformation:
*
* integral(S(t)dt,A,X) = integral(S(t)dt,A,Z)+AdditionalTerm
*
* here X may lie outside of [A,B], Z lies strictly in [A,B],
* AdditionalTerm is equals to integral(S(t)dt,A,B) times some
* integer number (may be zero).
*/
if( c->periodic&&(ae_fp_less(x,c->x.ptr.p_double[0])||ae_fp_greater(x,c->x.ptr.p_double[c->n-1])) )
{
/*
* compute integral(S(x)dx,A,B)
*/
intab = 0;
for(i=0; i<=c->n-2; i++)
{
w = c->x.ptr.p_double[i+1]-c->x.ptr.p_double[i];
m = (c->k+1)*i;
intab = intab+c->c.ptr.p_double[m]*w;
v = w;
for(j=1; j<=c->k; j++)
{
v = v*w;
intab = intab+c->c.ptr.p_double[m+j]*v/(j+1);
}
}
/*
* map X into [A,B]
*/
apperiodicmap(&x, c->x.ptr.p_double[0], c->x.ptr.p_double[c->n-1], &t, _state);
additionalterm = t*intab;
}
else
{
additionalterm = 0;
}
/*
* Binary search in the [ x[0], ..., x[n-2] ] (x[n-1] is not included)
*/
l = 0;
r = n-2+1;
while(l!=r-1)
{
m = (l+r)/2;
if( ae_fp_greater_eq(c->x.ptr.p_double[m],x) )
{
r = m;
}
else
{
l = m;
}
}
/*
* Integration
*/
result = 0;
for(i=0; i<=l-1; i++)
{
w = c->x.ptr.p_double[i+1]-c->x.ptr.p_double[i];
m = (c->k+1)*i;
result = result+c->c.ptr.p_double[m]*w;
v = w;
for(j=1; j<=c->k; j++)
{
v = v*w;
result = result+c->c.ptr.p_double[m+j]*v/(j+1);
}
}
w = x-c->x.ptr.p_double[l];
m = (c->k+1)*l;
v = w;
result = result+c->c.ptr.p_double[m]*w;
for(j=1; j<=c->k; j++)
{
v = v*w;
result = result+c->c.ptr.p_double[m+j]*v/(j+1);
}
result = result+additionalterm;
return result;
}
/*************************************************************************
Internal version of Spline1DConvDiff
Converts from Hermite spline given by grid XOld to new grid X2
INPUT PARAMETERS:
XOld - old grid
YOld - values at old grid
DOld - first derivative at old grid
N - grid size
X2 - new grid
N2 - new grid size
Y - possibly preallocated output array
(reallocate if too small)
NeedY - do we need Y?
D1 - possibly preallocated output array
(reallocate if too small)
NeedD1 - do we need D1?
D2 - possibly preallocated output array
(reallocate if too small)
NeedD2 - do we need D1?
OUTPUT ARRAYS:
Y - values, if needed
D1 - first derivative, if needed
D2 - second derivative, if needed
-- ALGLIB PROJECT --
Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dconvdiffinternal(/* Real */ ae_vector* xold,
/* Real */ ae_vector* yold,
/* Real */ ae_vector* dold,
ae_int_t n,
/* Real */ ae_vector* x2,
ae_int_t n2,
/* Real */ ae_vector* y,
ae_bool needy,
/* Real */ ae_vector* d1,
ae_bool needd1,
/* Real */ ae_vector* d2,
ae_bool needd2,
ae_state *_state)
{
ae_int_t intervalindex;
ae_int_t pointindex;
ae_bool havetoadvance;
double c0;
double c1;
double c2;
double c3;
double a;
double b;
double w;
double w2;
double w3;
double fa;
double fb;
double da;
double db;
double t;
/*
* Prepare space
*/
if( needy&&y->cnt<n2 )
{
ae_vector_set_length(y, n2, _state);
}
if( needd1&&d1->cnt<n2 )
{
ae_vector_set_length(d1, n2, _state);
}
if( needd2&&d2->cnt<n2 )
{
ae_vector_set_length(d2, n2, _state);
}
/*
* These assignments aren't actually needed
* (variables are initialized in the loop below),
* but without them compiler will complain about uninitialized locals
*/
c0 = 0;
c1 = 0;
c2 = 0;
c3 = 0;
a = 0;
b = 0;
/*
* Cycle
*/
intervalindex = -1;
pointindex = 0;
for(;;)
{
/*
* are we ready to exit?
*/
if( pointindex>=n2 )
{
break;
}
t = x2->ptr.p_double[pointindex];
/*
* do we need to advance interval?
*/
havetoadvance = ae_false;
if( intervalindex==-1 )
{
havetoadvance = ae_true;
}
else
{
if( intervalindex<n-2 )
{
havetoadvance = ae_fp_greater_eq(t,b);
}
}
if( havetoadvance )
{
intervalindex = intervalindex+1;
a = xold->ptr.p_double[intervalindex];
b = xold->ptr.p_double[intervalindex+1];
w = b-a;
w2 = w*w;
w3 = w*w2;
fa = yold->ptr.p_double[intervalindex];
fb = yold->ptr.p_double[intervalindex+1];
da = dold->ptr.p_double[intervalindex];
db = dold->ptr.p_double[intervalindex+1];
c0 = fa;
c1 = da;
c2 = (3*(fb-fa)-2*da*w-db*w)/w2;
c3 = (2*(fa-fb)+da*w+db*w)/w3;
continue;
}
/*
* Calculate spline and its derivatives using power basis
*/
t = t-a;
if( needy )
{
y->ptr.p_double[pointindex] = c0+t*(c1+t*(c2+t*c3));
}
if( needd1 )
{
d1->ptr.p_double[pointindex] = c1+2*t*c2+3*t*t*c3;
}
if( needd2 )
{
d2->ptr.p_double[pointindex] = 2*c2+6*t*c3;
}
pointindex = pointindex+1;
}
}
/*************************************************************************
This function finds all roots and extrema of the spline S(x) defined at
[A,B] (interval which contains spline nodes).
It does not extrapolates function, so roots and extrema located outside
of [A,B] will not be found. It returns all isolated (including multiple)
roots and extrema.
INPUT PARAMETERS
C - spline interpolant
OUTPUT PARAMETERS
R - array[NR], contains roots of the spline.
In case there is no roots, this array has zero length.
NR - number of roots, >=0
DR - is set to True in case there is at least one interval
where spline is just a zero constant. Such degenerate
cases are not reported in the R/NR
E - array[NE], contains extrema (maximums/minimums) of
the spline. In case there is no extrema, this array
has zero length.
ET - array[NE], extrema types:
* ET[i]>0 in case I-th extrema is a minimum
* ET[i]<0 in case I-th extrema is a maximum
NE - number of extrema, >=0
DE - is set to True in case there is at least one interval
where spline is a constant. Such degenerate cases are
not reported in the E/NE.
NOTES:
1. This function does NOT report following kinds of roots:
* intervals where function is constantly zero
* roots which are outside of [A,B] (note: it CAN return A or B)
2. This function does NOT report following kinds of extrema:
* intervals where function is a constant
* extrema which are outside of (A,B) (note: it WON'T return A or B)
-- ALGLIB PROJECT --
Copyright 26.09.2011 by Bochkanov Sergey
*************************************************************************/
void spline1drootsandextrema(spline1dinterpolant* c,
/* Real */ ae_vector* r,
ae_int_t* nr,
ae_bool* dr,
/* Real */ ae_vector* e,
/* Integer */ ae_vector* et,
ae_int_t* ne,
ae_bool* de,
ae_state *_state)
{
ae_frame _frame_block;
double pl;
double ml;
double pll;
double pr;
double mr;
ae_vector tr;
ae_vector tmpr;
ae_vector tmpe;
ae_vector tmpet;
ae_vector tmpc;
double x0;
double x1;
double x2;
double ex0;
double ex1;
ae_int_t tne;
ae_int_t tnr;
ae_int_t i;
ae_int_t j;
ae_bool nstep;
ae_frame_make(_state, &_frame_block);
ae_vector_clear(r);
*nr = 0;
*dr = ae_false;
ae_vector_clear(e);
ae_vector_clear(et);
*ne = 0;
*de = ae_false;
ae_vector_init(&tr, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmpr, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmpe, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmpet, 0, DT_INT, _state, ae_true);
ae_vector_init(&tmpc, 0, DT_REAL, _state, ae_true);
/*
*exception handling
*/
ae_assert(c->k==3, "Spline1DRootsAndExtrema : incorrect parameter C.K!", _state);
ae_assert(c->continuity>=0, "Spline1DRootsAndExtrema : parameter C.Continuity must not be less than 0!", _state);
/*
*initialization of variable
*/
*nr = 0;
*ne = 0;
*dr = ae_false;
*de = ae_false;
nstep = ae_true;
/*
*consider case, when C.Continuty=0
*/
if( c->continuity==0 )
{
/*
*allocation for auxiliary arrays
*'TmpR ' - it stores a time value for roots
*'TmpE ' - it stores a time value for extremums
*'TmpET '- it stores a time value for extremums type
*/
rvectorsetlengthatleast(&tmpr, 3*(c->n-1), _state);
rvectorsetlengthatleast(&tmpe, 2*(c->n-1), _state);
ivectorsetlengthatleast(&tmpet, 2*(c->n-1), _state);
/*
*start calculating
*/
for(i=0; i<=c->n-2; i++)
{
/*
*initialization pL, mL, pR, mR
*/
pl = c->c.ptr.p_double[4*i];
ml = c->c.ptr.p_double[4*i+1];
pr = c->c.ptr.p_double[4*(i+1)];
mr = c->c.ptr.p_double[4*i+1]+2*c->c.ptr.p_double[4*i+2]*(c->x.ptr.p_double[i+1]-c->x.ptr.p_double[i])+3*c->c.ptr.p_double[4*i+3]*(c->x.ptr.p_double[i+1]-c->x.ptr.p_double[i])*(c->x.ptr.p_double[i+1]-c->x.ptr.p_double[i]);
/*
*pre-searching roots and extremums
*/
solvecubicpolinom(pl, ml, pr, mr, c->x.ptr.p_double[i], c->x.ptr.p_double[i+1], &x0, &x1, &x2, &ex0, &ex1, &tnr, &tne, &tr, _state);
*dr = *dr||tnr==-1;
*de = *de||tne==-1;
/*
*searching of roots
*/
if( tnr==1&&nstep )
{
/*
*is there roots?
*/
if( *nr>0 )
{
/*
*is a next root equal a previous root?
*if is't, then write new root
*/
if( ae_fp_neq(x0,tmpr.ptr.p_double[*nr-1]) )
{
tmpr.ptr.p_double[*nr] = x0;
*nr = *nr+1;
}
}
else
{
/*
*write a first root
*/
tmpr.ptr.p_double[*nr] = x0;
*nr = *nr+1;
}
}
else
{
/*
*case when function at a segment identically to zero
*then we have to clear a root, if the one located on a
*constant segment
*/
if( tnr==-1 )
{
/*
*safe state variable as constant
*/
if( nstep )
{
nstep = ae_false;
}
/*
*clear the root, if there is
*/
if( *nr>0 )
{
if( ae_fp_eq(c->x.ptr.p_double[i],tmpr.ptr.p_double[*nr-1]) )
{
*nr = *nr-1;
}
}
/*
*change state for 'DR'
*/
if( !*dr )
{
*dr = ae_true;
}
}
else
{
nstep = ae_true;
}
}
/*
*searching of extremums
*/
if( i>0 )
{
pll = c->c.ptr.p_double[4*(i-1)];
/*
*if pL=pLL or pL=pR then
*/
if( tne==-1 )
{
if( !*de )
{
*de = ae_true;
}
}
else
{
if( ae_fp_greater(pl,pll)&&ae_fp_greater(pl,pr) )
{
/*
*maximum
*/
tmpet.ptr.p_int[*ne] = -1;
tmpe.ptr.p_double[*ne] = c->x.ptr.p_double[i];
*ne = *ne+1;
}
else
{
if( ae_fp_less(pl,pll)&&ae_fp_less(pl,pr) )
{
/*
*minimum
*/
tmpet.ptr.p_int[*ne] = 1;
tmpe.ptr.p_double[*ne] = c->x.ptr.p_double[i];
*ne = *ne+1;
}
}
}
}
}
/*
*write final result
*/
rvectorsetlengthatleast(r, *nr, _state);
rvectorsetlengthatleast(e, *ne, _state);
ivectorsetlengthatleast(et, *ne, _state);
/*
*write roots
*/
for(i=0; i<=*nr-1; i++)
{
r->ptr.p_double[i] = tmpr.ptr.p_double[i];
}
/*
*write extremums and their types
*/
for(i=0; i<=*ne-1; i++)
{
e->ptr.p_double[i] = tmpe.ptr.p_double[i];
et->ptr.p_int[i] = tmpet.ptr.p_int[i];
}
}
else
{
/*
*case, when C.Continuity>=1
*'TmpR ' - it stores a time value for roots
*'TmpC' - it stores a time value for extremums and
*their function value (TmpC={EX0,F(EX0), EX1,F(EX1), ..., EXn,F(EXn)};)
*'TmpE' - it stores a time value for extremums only
*'TmpET'- it stores a time value for extremums type
*/
rvectorsetlengthatleast(&tmpr, 2*c->n-1, _state);
rvectorsetlengthatleast(&tmpc, 4*c->n, _state);
rvectorsetlengthatleast(&tmpe, 2*c->n, _state);
ivectorsetlengthatleast(&tmpet, 2*c->n, _state);
/*
*start calculating
*/
for(i=0; i<=c->n-2; i++)
{
/*
*we calculate pL,mL, pR,mR as Fi+1(F'i+1) at left border
*/
pl = c->c.ptr.p_double[4*i];
ml = c->c.ptr.p_double[4*i+1];
pr = c->c.ptr.p_double[4*(i+1)];
mr = c->c.ptr.p_double[4*(i+1)+1];
/*
*calculating roots and extremums at [X[i],X[i+1]]
*/
solvecubicpolinom(pl, ml, pr, mr, c->x.ptr.p_double[i], c->x.ptr.p_double[i+1], &x0, &x1, &x2, &ex0, &ex1, &tnr, &tne, &tr, _state);
/*
*searching roots
*/
if( tnr>0 )
{
/*
*re-init tR
*/
if( tnr>=1 )
{
tr.ptr.p_double[0] = x0;
}
if( tnr>=2 )
{
tr.ptr.p_double[1] = x1;
}
if( tnr==3 )
{
tr.ptr.p_double[2] = x2;
}
/*
*start root selection
*/
if( *nr>0 )
{
if( ae_fp_neq(tmpr.ptr.p_double[*nr-1],x0) )
{
/*
*previous segment was't constant identical zero
*/
if( nstep )
{
for(j=0; j<=tnr-1; j++)
{
tmpr.ptr.p_double[*nr+j] = tr.ptr.p_double[j];
}
*nr = *nr+tnr;
}
else
{
/*
*previous segment was constant identical zero
*and we must ignore [NR+j-1] root
*/
for(j=1; j<=tnr-1; j++)
{
tmpr.ptr.p_double[*nr+j-1] = tr.ptr.p_double[j];
}
*nr = *nr+tnr-1;
nstep = ae_true;
}
}
else
{
for(j=1; j<=tnr-1; j++)
{
tmpr.ptr.p_double[*nr+j-1] = tr.ptr.p_double[j];
}
*nr = *nr+tnr-1;
}
}
else
{
/*
*write first root
*/
for(j=0; j<=tnr-1; j++)
{
tmpr.ptr.p_double[*nr+j] = tr.ptr.p_double[j];
}
*nr = *nr+tnr;
}
}
else
{
if( tnr==-1 )
{
/*
*decrement 'NR' if at previous step was writen a root
*(previous segment identical zero)
*/
if( *nr>0&&nstep )
{
*nr = *nr-1;
}
/*
*previous segment is't constant
*/
if( nstep )
{
nstep = ae_false;
}
/*
*rewrite 'DR'
*/
if( !*dr )
{
*dr = ae_true;
}
}
}
/*
*searching extremums
*write all term like extremums
*/
if( tne==1 )
{
if( *ne>0 )
{
/*
*just ignore identical extremums
*because he must be one
*/
if( ae_fp_neq(tmpc.ptr.p_double[*ne-2],ex0) )
{
tmpc.ptr.p_double[*ne] = ex0;
tmpc.ptr.p_double[*ne+1] = c->c.ptr.p_double[4*i]+c->c.ptr.p_double[4*i+1]*(ex0-c->x.ptr.p_double[i])+c->c.ptr.p_double[4*i+2]*(ex0-c->x.ptr.p_double[i])*(ex0-c->x.ptr.p_double[i])+c->c.ptr.p_double[4*i+3]*(ex0-c->x.ptr.p_double[i])*(ex0-c->x.ptr.p_double[i])*(ex0-c->x.ptr.p_double[i]);
*ne = *ne+2;
}
}
else
{
/*
*write first extremum and it function value
*/
tmpc.ptr.p_double[*ne] = ex0;
tmpc.ptr.p_double[*ne+1] = c->c.ptr.p_double[4*i]+c->c.ptr.p_double[4*i+1]*(ex0-c->x.ptr.p_double[i])+c->c.ptr.p_double[4*i+2]*(ex0-c->x.ptr.p_double[i])*(ex0-c->x.ptr.p_double[i])+c->c.ptr.p_double[4*i+3]*(ex0-c->x.ptr.p_double[i])*(ex0-c->x.ptr.p_double[i])*(ex0-c->x.ptr.p_double[i]);
*ne = *ne+2;
}
}
else
{
if( tne==2 )
{
if( *ne>0 )
{
/*
*ignore identical extremum
*/
if( ae_fp_neq(tmpc.ptr.p_double[*ne-2],ex0) )
{
tmpc.ptr.p_double[*ne] = ex0;
tmpc.ptr.p_double[*ne+1] = c->c.ptr.p_double[4*i]+c->c.ptr.p_double[4*i+1]*(ex0-c->x.ptr.p_double[i])+c->c.ptr.p_double[4*i+2]*(ex0-c->x.ptr.p_double[i])*(ex0-c->x.ptr.p_double[i])+c->c.ptr.p_double[4*i+3]*(ex0-c->x.ptr.p_double[i])*(ex0-c->x.ptr.p_double[i])*(ex0-c->x.ptr.p_double[i]);
*ne = *ne+2;
}
}
else
{
/*
*write first extremum
*/
tmpc.ptr.p_double[*ne] = ex0;
tmpc.ptr.p_double[*ne+1] = c->c.ptr.p_double[4*i]+c->c.ptr.p_double[4*i+1]*(ex0-c->x.ptr.p_double[i])+c->c.ptr.p_double[4*i+2]*(ex0-c->x.ptr.p_double[i])*(ex0-c->x.ptr.p_double[i])+c->c.ptr.p_double[4*i+3]*(ex0-c->x.ptr.p_double[i])*(ex0-c->x.ptr.p_double[i])*(ex0-c->x.ptr.p_double[i]);
*ne = *ne+2;
}
/*
*write second extremum
*/
tmpc.ptr.p_double[*ne] = ex1;
tmpc.ptr.p_double[*ne+1] = c->c.ptr.p_double[4*i]+c->c.ptr.p_double[4*i+1]*(ex1-c->x.ptr.p_double[i])+c->c.ptr.p_double[4*i+2]*(ex1-c->x.ptr.p_double[i])*(ex1-c->x.ptr.p_double[i])+c->c.ptr.p_double[4*i+3]*(ex1-c->x.ptr.p_double[i])*(ex1-c->x.ptr.p_double[i])*(ex1-c->x.ptr.p_double[i]);
*ne = *ne+2;
}
else
{
if( tne==-1 )
{
if( !*de )
{
*de = ae_true;
}
}
}
}
}
/*
*checking of arrays
*get number of extremums (tNe=NE/2)
*initialize pL as value F0(X[0]) and
*initialize pR as value Fn-1(X[N])
*/
tne = *ne/2;
*ne = 0;
pl = c->c.ptr.p_double[0];
pr = c->c.ptr.p_double[4*(c->n-1)];
for(i=0; i<=tne-1; i++)
{
if( i>0&&i<tne-1 )
{
if( ae_fp_greater(tmpc.ptr.p_double[2*i+1],tmpc.ptr.p_double[2*(i-1)+1])&&ae_fp_greater(tmpc.ptr.p_double[2*i+1],tmpc.ptr.p_double[2*(i+1)+1]) )
{
/*
*maximum
*/
tmpe.ptr.p_double[*ne] = tmpc.ptr.p_double[2*i];
tmpet.ptr.p_int[*ne] = -1;
*ne = *ne+1;
}
else
{
if( ae_fp_less(tmpc.ptr.p_double[2*i+1],tmpc.ptr.p_double[2*(i-1)+1])&&ae_fp_less(tmpc.ptr.p_double[2*i+1],tmpc.ptr.p_double[2*(i+1)+1]) )
{
/*
*minimum
*/
tmpe.ptr.p_double[*ne] = tmpc.ptr.p_double[2*i];
tmpet.ptr.p_int[*ne] = 1;
*ne = *ne+1;
}
}
}
else
{
if( i==0 )
{
if( ae_fp_neq(tmpc.ptr.p_double[2*i],c->x.ptr.p_double[0]) )
{
if( ae_fp_greater(tmpc.ptr.p_double[2*i+1],pl)&&ae_fp_greater(tmpc.ptr.p_double[2*i+1],tmpc.ptr.p_double[2*(i+1)+1]) )
{
/*
*maximum
*/
tmpe.ptr.p_double[*ne] = tmpc.ptr.p_double[2*i];
tmpet.ptr.p_int[*ne] = -1;
*ne = *ne+1;
}
else
{
if( ae_fp_less(tmpc.ptr.p_double[2*i+1],pl)&&ae_fp_less(tmpc.ptr.p_double[2*i+1],tmpc.ptr.p_double[2*(i+1)+1]) )
{
/*
*minimum
*/
tmpe.ptr.p_double[*ne] = tmpc.ptr.p_double[2*i];
tmpet.ptr.p_int[*ne] = 1;
*ne = *ne+1;
}
}
}
}
else
{
if( i==tne-1 )
{
if( ae_fp_neq(tmpc.ptr.p_double[2*i],c->x.ptr.p_double[c->n-1]) )
{
if( ae_fp_greater(tmpc.ptr.p_double[2*i+1],tmpc.ptr.p_double[2*(i-1)+1])&&ae_fp_greater(tmpc.ptr.p_double[2*i+1],pr) )
{
/*
*maximum
*/
tmpe.ptr.p_double[*ne] = tmpc.ptr.p_double[2*i];
tmpet.ptr.p_int[*ne] = -1;
*ne = *ne+1;
}
else
{
if( ae_fp_less(tmpc.ptr.p_double[2*i+1],tmpc.ptr.p_double[2*(i-1)+1])&&ae_fp_less(tmpc.ptr.p_double[2*i+1],pr) )
{
/*
*minimum
*/
tmpe.ptr.p_double[*ne] = tmpc.ptr.p_double[2*i];
tmpet.ptr.p_int[*ne] = 1;
*ne = *ne+1;
}
}
}
}
}
}
}
/*
*final results
*allocate R, E, ET
*/
rvectorsetlengthatleast(r, *nr, _state);
rvectorsetlengthatleast(e, *ne, _state);
ivectorsetlengthatleast(et, *ne, _state);
/*
*write result for extremus and their types
*/
for(i=0; i<=*ne-1; i++)
{
e->ptr.p_double[i] = tmpe.ptr.p_double[i];
et->ptr.p_int[i] = tmpet.ptr.p_int[i];
}
/*
*write result for roots
*/
for(i=0; i<=*nr-1; i++)
{
r->ptr.p_double[i] = tmpr.ptr.p_double[i];
}
}
ae_frame_leave(_state);
}
/*************************************************************************
Internal subroutine. Heap sort.
*************************************************************************/
void heapsortdpoints(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* d,
ae_int_t n,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector rbuf;
ae_vector ibuf;
ae_vector rbuf2;
ae_vector ibuf2;
ae_int_t i;
ae_frame_make(_state, &_frame_block);
ae_vector_init(&rbuf, 0, DT_REAL, _state, ae_true);
ae_vector_init(&ibuf, 0, DT_INT, _state, ae_true);
ae_vector_init(&rbuf2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&ibuf2, 0, DT_INT, _state, ae_true);
ae_vector_set_length(&ibuf, n, _state);
ae_vector_set_length(&rbuf, n, _state);
for(i=0; i<=n-1; i++)
{
ibuf.ptr.p_int[i] = i;
}
tagsortfasti(x, &ibuf, &rbuf2, &ibuf2, n, _state);
for(i=0; i<=n-1; i++)
{
rbuf.ptr.p_double[i] = y->ptr.p_double[ibuf.ptr.p_int[i]];
}
ae_v_move(&y->ptr.p_double[0], 1, &rbuf.ptr.p_double[0], 1, ae_v_len(0,n-1));
for(i=0; i<=n-1; i++)
{
rbuf.ptr.p_double[i] = d->ptr.p_double[ibuf.ptr.p_int[i]];
}
ae_v_move(&d->ptr.p_double[0], 1, &rbuf.ptr.p_double[0], 1, ae_v_len(0,n-1));
ae_frame_leave(_state);
}
/*************************************************************************
This procedure search roots of an quadratic equation inside [0;1] and it number of roots.
INPUT PARAMETERS:
P0 - value of a function at 0
M0 - value of a derivative at 0
P1 - value of a function at 1
M1 - value of a derivative at 1
OUTPUT PARAMETERS:
X0 - first root of an equation
X1 - second root of an equation
NR - number of roots
RESTRICTIONS OF PARAMETERS:
Parameters for this procedure has't to be zero simultaneously. Is expected,
that input polinom is't degenerate or constant identicaly ZERO.
REMARK:
The procedure always fill value for X1 and X2, even if it is't belongs to [0;1].
But first true root(even if existing one) is in X1.
Number of roots is NR.
-- ALGLIB PROJECT --
Copyright 26.09.2011 by Bochkanov Sergey
*************************************************************************/
void solvepolinom2(double p0,
double m0,
double p1,
double m1,
double* x0,
double* x1,
ae_int_t* nr,
ae_state *_state)
{
double a;
double b;
double c;
double dd;
double tmp;
double exf;
double extr;
*x0 = 0;
*x1 = 0;
*nr = 0;
/*
*calculate parameters for equation: A, B and C
*/
a = 6*p0+3*m0-6*p1+3*m1;
b = -6*p0-4*m0+6*p1-2*m1;
c = m0;
/*
*check case, when A=0
*we are considering the linear equation
*/
if( ae_fp_eq(a,0) )
{
/*
*B<>0 and root inside [0;1]
*one root
*/
if( (ae_fp_neq(b,0)&&ae_sign(c, _state)*ae_sign(b, _state)<=0)&&ae_fp_greater_eq(ae_fabs(b, _state),ae_fabs(c, _state)) )
{
*x0 = -c/b;
*nr = 1;
return;
}
else
{
*nr = 0;
return;
}
}
/*
*consider case, when extremumu outside (0;1)
*exist one root only
*/
if( ae_fp_less_eq(ae_fabs(2*a, _state),ae_fabs(b, _state))||ae_sign(b, _state)*ae_sign(a, _state)>=0 )
{
if( ae_sign(m0, _state)*ae_sign(m1, _state)>0 )
{
*nr = 0;
return;
}
/*
*consider case, when the one exist
*same sign of derivative
*/
if( ae_sign(m0, _state)*ae_sign(m1, _state)<0 )
{
*nr = 1;
extr = -b/(2*a);
dd = b*b-4*a*c;
if( ae_fp_less(dd,0) )
{
return;
}
*x0 = (-b-ae_sqrt(dd, _state))/(2*a);
*x1 = (-b+ae_sqrt(dd, _state))/(2*a);
if( (ae_fp_greater_eq(extr,1)&&ae_fp_less_eq(*x1,extr))||(ae_fp_less_eq(extr,0)&&ae_fp_greater_eq(*x1,extr)) )
{
*x0 = *x1;
}
return;
}
/*
*consider case, when the one is 0
*/
if( ae_fp_eq(m0,0) )
{
*x0 = 0;
*nr = 1;
return;
}
if( ae_fp_eq(m1,0) )
{
*x0 = 1;
*nr = 1;
return;
}
}
else
{
/*
*consider case, when both of derivatives is 0
*/
if( ae_fp_eq(m0,0)&&ae_fp_eq(m1,0) )
{
*x0 = 0;
*x1 = 1;
*nr = 2;
return;
}
/*
*consider case, when derivative at 0 is 0, and derivative at 1 is't 0
*/
if( ae_fp_eq(m0,0)&&ae_fp_neq(m1,0) )
{
dd = b*b-4*a*c;
if( ae_fp_less(dd,0) )
{
*x0 = 0;
*nr = 1;
return;
}
*x0 = (-b-ae_sqrt(dd, _state))/(2*a);
*x1 = (-b+ae_sqrt(dd, _state))/(2*a);
extr = -b/(2*a);
exf = a*extr*extr+b*extr+c;
if( ae_sign(exf, _state)*ae_sign(m1, _state)>0 )
{
*x0 = 0;
*nr = 1;
return;
}
else
{
if( ae_fp_greater(extr,*x0) )
{
*x0 = 0;
}
else
{
*x1 = 0;
}
*nr = 2;
/*
*roots must placed ascending
*/
if( ae_fp_greater(*x0,*x1) )
{
tmp = *x0;
*x0 = *x1;
*x1 = tmp;
}
return;
}
}
if( ae_fp_eq(m1,0)&&ae_fp_neq(m0,0) )
{
dd = b*b-4*a*c;
if( ae_fp_less(dd,0) )
{
*x0 = 1;
*nr = 1;
return;
}
*x0 = (-b-ae_sqrt(dd, _state))/(2*a);
*x1 = (-b+ae_sqrt(dd, _state))/(2*a);
extr = -b/(2*a);
exf = a*extr*extr+b*extr+c;
if( ae_sign(exf, _state)*ae_sign(m0, _state)>0 )
{
*x0 = 1;
*nr = 1;
return;
}
else
{
if( ae_fp_less(extr,*x0) )
{
*x0 = 1;
}
else
{
*x1 = 1;
}
*nr = 2;
/*
*roots must placed ascending
*/
if( ae_fp_greater(*x0,*x1) )
{
tmp = *x0;
*x0 = *x1;
*x1 = tmp;
}
return;
}
}
else
{
extr = -b/(2*a);
exf = a*extr*extr+b*extr+c;
if( ae_sign(exf, _state)*ae_sign(m0, _state)>0&&ae_sign(exf, _state)*ae_sign(m1, _state)>0 )
{
*nr = 0;
return;
}
dd = b*b-4*a*c;
if( ae_fp_less(dd,0) )
{
*nr = 0;
return;
}
*x0 = (-b-ae_sqrt(dd, _state))/(2*a);
*x1 = (-b+ae_sqrt(dd, _state))/(2*a);
/*
*if EXF and m0, EXF and m1 has different signs, then equation has two roots
*/
if( ae_sign(exf, _state)*ae_sign(m0, _state)<0&&ae_sign(exf, _state)*ae_sign(m1, _state)<0 )
{
*nr = 2;
/*
*roots must placed ascending
*/
if( ae_fp_greater(*x0,*x1) )
{
tmp = *x0;
*x0 = *x1;
*x1 = tmp;
}
return;
}
else
{
*nr = 1;
if( ae_sign(exf, _state)*ae_sign(m0, _state)<0 )
{
if( ae_fp_less(*x1,extr) )
{
*x0 = *x1;
}
return;
}
if( ae_sign(exf, _state)*ae_sign(m1, _state)<0 )
{
if( ae_fp_greater(*x1,extr) )
{
*x0 = *x1;
}
return;
}
}
}
}
}
/*************************************************************************
This procedure search roots of an cubic equation inside [A;B], it number of roots
and number of extremums.
INPUT PARAMETERS:
pA - value of a function at A
mA - value of a derivative at A
pB - value of a function at B
mB - value of a derivative at B
A0 - left border [A0;B0]
B0 - right border [A0;B0]
OUTPUT PARAMETERS:
X0 - first root of an equation
X1 - second root of an equation
X2 - third root of an equation
EX0 - first extremum of a function
EX0 - second extremum of a function
NR - number of roots
NR - number of extrmums
RESTRICTIONS OF PARAMETERS:
Length of [A;B] must be positive and is't zero, i.e. A<>B and A<B.
REMARK:
If 'NR' is -1 it's mean, than polinom has infiniti roots.
If 'NE' is -1 it's mean, than polinom has infiniti extremums.
-- ALGLIB PROJECT --
Copyright 26.09.2011 by Bochkanov Sergey
*************************************************************************/
void solvecubicpolinom(double pa,
double ma,
double pb,
double mb,
double a,
double b,
double* x0,
double* x1,
double* x2,
double* ex0,
double* ex1,
ae_int_t* nr,
ae_int_t* ne,
/* Real */ ae_vector* tempdata,
ae_state *_state)
{
ae_int_t i;
double tmpma;
double tmpmb;
double tex0;
double tex1;
*x0 = 0;
*x1 = 0;
*x2 = 0;
*ex0 = 0;
*ex1 = 0;
*nr = 0;
*ne = 0;
rvectorsetlengthatleast(tempdata, 3, _state);
/*
*case, when A>B
*/
ae_assert(ae_fp_less(a,b), "\nSolveCubicPolinom: incorrect borders for [A;B]!\n", _state);
/*
*case 1
*function can be identicaly to ZERO
*/
if( ((ae_fp_eq(ma,0)&&ae_fp_eq(mb,0))&&ae_fp_eq(pa,pb))&&ae_fp_eq(pa,0) )
{
*nr = -1;
*ne = -1;
return;
}
if( (ae_fp_eq(ma,0)&&ae_fp_eq(mb,0))&&ae_fp_eq(pa,pb) )
{
*nr = 0;
*ne = -1;
return;
}
tmpma = ma*(b-a);
tmpmb = mb*(b-a);
solvepolinom2(pa, tmpma, pb, tmpmb, ex0, ex1, ne, _state);
*ex0 = spline1d_rescaleval(0, 1, a, b, *ex0, _state);
*ex1 = spline1d_rescaleval(0, 1, a, b, *ex1, _state);
/*
*case 3.1
*no extremums at [A;B]
*/
if( *ne==0 )
{
*nr = bisectmethod(pa, tmpma, pb, tmpmb, 0, 1, x0, _state);
if( *nr==1 )
{
*x0 = spline1d_rescaleval(0, 1, a, b, *x0, _state);
}
return;
}
/*
*case 3.2
*one extremum
*/
if( *ne==1 )
{
if( ae_fp_eq(*ex0,a)||ae_fp_eq(*ex0,b) )
{
*nr = bisectmethod(pa, tmpma, pb, tmpmb, 0, 1, x0, _state);
if( *nr==1 )
{
*x0 = spline1d_rescaleval(0, 1, a, b, *x0, _state);
}
return;
}
else
{
*nr = 0;
i = 0;
tex0 = spline1d_rescaleval(a, b, 0, 1, *ex0, _state);
*nr = bisectmethod(pa, tmpma, pb, tmpmb, 0, tex0, x0, _state)+(*nr);
if( *nr>i )
{
tempdata->ptr.p_double[i] = spline1d_rescaleval(0, tex0, a, *ex0, *x0, _state);
i = i+1;
}
*nr = bisectmethod(pa, tmpma, pb, tmpmb, tex0, 1, x0, _state)+(*nr);
if( *nr>i )
{
*x0 = spline1d_rescaleval(tex0, 1, *ex0, b, *x0, _state);
if( i>0 )
{
if( ae_fp_neq(*x0,tempdata->ptr.p_double[i-1]) )
{
tempdata->ptr.p_double[i] = *x0;
i = i+1;
}
else
{
*nr = *nr-1;
}
}
else
{
tempdata->ptr.p_double[i] = *x0;
i = i+1;
}
}
if( *nr>0 )
{
*x0 = tempdata->ptr.p_double[0];
if( *nr>1 )
{
*x1 = tempdata->ptr.p_double[1];
}
return;
}
}
return;
}
else
{
/*
*case 3.3
*two extremums(or more, but it's impossible)
*
*
*case 3.3.0
*both extremums at the border
*/
if( ae_fp_eq(*ex0,a)&&ae_fp_eq(*ex1,b) )
{
*nr = bisectmethod(pa, tmpma, pb, tmpmb, 0, 1, x0, _state);
if( *nr==1 )
{
*x0 = spline1d_rescaleval(0, 1, a, b, *x0, _state);
}
return;
}
if( ae_fp_eq(*ex0,a)&&ae_fp_neq(*ex1,b) )
{
*nr = 0;
i = 0;
tex1 = spline1d_rescaleval(a, b, 0, 1, *ex1, _state);
*nr = bisectmethod(pa, tmpma, pb, tmpmb, 0, tex1, x0, _state)+(*nr);
if( *nr>i )
{
tempdata->ptr.p_double[i] = spline1d_rescaleval(0, tex1, a, *ex1, *x0, _state);
i = i+1;
}
*nr = bisectmethod(pa, tmpma, pb, tmpmb, tex1, 1, x0, _state)+(*nr);
if( *nr>i )
{
*x0 = spline1d_rescaleval(tex1, 1, *ex1, b, *x0, _state);
if( ae_fp_neq(*x0,tempdata->ptr.p_double[i-1]) )
{
tempdata->ptr.p_double[i] = *x0;
i = i+1;
}
else
{
*nr = *nr-1;
}
}
if( *nr>0 )
{
*x0 = tempdata->ptr.p_double[0];
if( *nr>1 )
{
*x1 = tempdata->ptr.p_double[1];
}
return;
}
}
if( ae_fp_eq(*ex1,b)&&ae_fp_neq(*ex0,a) )
{
*nr = 0;
i = 0;
tex0 = spline1d_rescaleval(a, b, 0, 1, *ex0, _state);
*nr = bisectmethod(pa, tmpma, pb, tmpmb, 0, tex0, x0, _state)+(*nr);
if( *nr>i )
{
tempdata->ptr.p_double[i] = spline1d_rescaleval(0, tex0, a, *ex0, *x0, _state);
i = i+1;
}
*nr = bisectmethod(pa, tmpma, pb, tmpmb, tex0, 1, x0, _state)+(*nr);
if( *nr>i )
{
*x0 = spline1d_rescaleval(tex0, 1, *ex0, b, *x0, _state);
if( i>0 )
{
if( ae_fp_neq(*x0,tempdata->ptr.p_double[i-1]) )
{
tempdata->ptr.p_double[i] = *x0;
i = i+1;
}
else
{
*nr = *nr-1;
}
}
else
{
tempdata->ptr.p_double[i] = *x0;
i = i+1;
}
}
if( *nr>0 )
{
*x0 = tempdata->ptr.p_double[0];
if( *nr>1 )
{
*x1 = tempdata->ptr.p_double[1];
}
return;
}
}
else
{
/*
*case 3.3.2
*both extremums inside (0;1)
*/
*nr = 0;
i = 0;
tex0 = spline1d_rescaleval(a, b, 0, 1, *ex0, _state);
tex1 = spline1d_rescaleval(a, b, 0, 1, *ex1, _state);
*nr = bisectmethod(pa, tmpma, pb, tmpmb, 0, tex0, x0, _state)+(*nr);
if( *nr>i )
{
tempdata->ptr.p_double[i] = spline1d_rescaleval(0, tex0, a, *ex0, *x0, _state);
i = i+1;
}
*nr = bisectmethod(pa, tmpma, pb, tmpmb, tex0, tex1, x0, _state)+(*nr);
if( *nr>i )
{
*x0 = spline1d_rescaleval(tex0, tex1, *ex0, *ex1, *x0, _state);
if( i>0 )
{
if( ae_fp_neq(*x0,tempdata->ptr.p_double[i-1]) )
{
tempdata->ptr.p_double[i] = *x0;
i = i+1;
}
else
{
*nr = *nr-1;
}
}
else
{
tempdata->ptr.p_double[i] = *x0;
i = i+1;
}
}
*nr = bisectmethod(pa, tmpma, pb, tmpmb, tex1, 1, x0, _state)+(*nr);
if( *nr>i )
{
*x0 = spline1d_rescaleval(tex1, 1, *ex1, b, *x0, _state);
if( i>0 )
{
if( ae_fp_neq(*x0,tempdata->ptr.p_double[i-1]) )
{
tempdata->ptr.p_double[i] = *x0;
i = i+1;
}
else
{
*nr = *nr-1;
}
}
else
{
tempdata->ptr.p_double[i] = *x0;
i = i+1;
}
}
/*
*write are found roots
*/
if( *nr>0 )
{
*x0 = tempdata->ptr.p_double[0];
if( *nr>1 )
{
*x1 = tempdata->ptr.p_double[1];
}
if( *nr>2 )
{
*x2 = tempdata->ptr.p_double[2];
}
return;
}
}
}
}
/*************************************************************************
Function for searching a root at [A;B] by bisection method and return number of roots
(0 or 1)
INPUT PARAMETERS:
pA - value of a function at A
mA - value of a derivative at A
pB - value of a function at B
mB - value of a derivative at B
A0 - left border [A0;B0]
B0 - right border [A0;B0]
RESTRICTIONS OF PARAMETERS:
We assume, that B0>A0.
REMARK:
Assume, that exist one root only at [A;B], else
function may be work incorrectly.
The function dont check value A0,B0!
-- ALGLIB PROJECT --
Copyright 26.09.2011 by Bochkanov Sergey
*************************************************************************/
ae_int_t bisectmethod(double pa,
double ma,
double pb,
double mb,
double a,
double b,
double* x,
ae_state *_state)
{
double vacuum;
double eps;
double a0;
double b0;
double m;
double lf;
double rf;
double mf;
ae_int_t result;
*x = 0;
/*
*accuracy
*/
eps = 1000*(b-a)*ae_machineepsilon;
/*
*initialization left and right borders
*/
a0 = a;
b0 = b;
/*
*initialize function value at 'A' and 'B'
*/
spline1d_hermitecalc(pa, ma, pb, mb, a, &lf, &vacuum, _state);
spline1d_hermitecalc(pa, ma, pb, mb, b, &rf, &vacuum, _state);
/*
*check, that 'A' and 'B' are't roots,
*and that root exist
*/
if( ae_sign(lf, _state)*ae_sign(rf, _state)>0 )
{
result = 0;
return result;
}
else
{
if( ae_fp_eq(lf,0) )
{
*x = a;
result = 1;
return result;
}
else
{
if( ae_fp_eq(rf,0) )
{
*x = b;
result = 1;
return result;
}
}
}
/*
*searching a root
*/
do
{
m = (b0+a0)/2;
spline1d_hermitecalc(pa, ma, pb, mb, a0, &lf, &vacuum, _state);
spline1d_hermitecalc(pa, ma, pb, mb, b0, &rf, &vacuum, _state);
spline1d_hermitecalc(pa, ma, pb, mb, m, &mf, &vacuum, _state);
if( ae_sign(mf, _state)*ae_sign(lf, _state)<0 )
{
b0 = m;
}
else
{
if( ae_sign(mf, _state)*ae_sign(rf, _state)<0 )
{
a0 = m;
}
else
{
if( ae_fp_eq(lf,0) )
{
*x = a0;
result = 1;
return result;
}
if( ae_fp_eq(rf,0) )
{
*x = b0;
result = 1;
return result;
}
if( ae_fp_eq(mf,0) )
{
*x = m;
result = 1;
return result;
}
}
}
}
while(ae_fp_greater_eq(ae_fabs(b0-a0, _state),eps));
*x = m;
result = 1;
return result;
}
/*************************************************************************
This function builds monotone cubic Hermite interpolant. This interpolant
is monotonic in [x(0),x(n-1)] and is constant outside of this interval.
In case y[] form non-monotonic sequence, interpolant is piecewise
monotonic. Say, for x=(0,1,2,3,4) and y=(0,1,2,1,0) interpolant will
monotonically grow at [0..2] and monotonically decrease at [2..4].
INPUT PARAMETERS:
X - spline nodes, array[0..N-1]. Subroutine automatically
sorts points, so caller may pass unsorted array.
Y - function values, array[0..N-1]
N - the number of points(N>=2).
OUTPUT PARAMETERS:
C - spline interpolant.
-- ALGLIB PROJECT --
Copyright 21.06.2012 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildmonotone(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
spline1dinterpolant* c,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _x;
ae_vector _y;
ae_vector d;
ae_vector ex;
ae_vector ey;
ae_vector p;
double delta;
double alpha;
double beta;
ae_int_t tmpn;
ae_int_t sn;
double ca;
double cb;
double epsilon;
ae_int_t i;
ae_int_t j;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_x, x, _state, ae_true);
x = &_x;
ae_vector_init_copy(&_y, y, _state, ae_true);
y = &_y;
_spline1dinterpolant_clear(c);
ae_vector_init(&d, 0, DT_REAL, _state, ae_true);
ae_vector_init(&ex, 0, DT_REAL, _state, ae_true);
ae_vector_init(&ey, 0, DT_REAL, _state, ae_true);
ae_vector_init(&p, 0, DT_INT, _state, ae_true);
/*
* Check lengths of arguments
*/
ae_assert(n>=2, "Spline1DBuildMonotone: N<2", _state);
ae_assert(x->cnt>=n, "Spline1DBuildMonotone: Length(X)<N", _state);
ae_assert(y->cnt>=n, "Spline1DBuildMonotone: Length(Y)<N", _state);
/*
* Check and sort points
*/
ae_assert(isfinitevector(x, n, _state), "Spline1DBuildMonotone: X contains infinite or NAN values", _state);
ae_assert(isfinitevector(y, n, _state), "Spline1DBuildMonotone: Y contains infinite or NAN values", _state);
spline1d_heapsortppoints(x, y, &p, n, _state);
ae_assert(aredistinct(x, n, _state), "Spline1DBuildMonotone: at least two consequent points are too close", _state);
epsilon = ae_machineepsilon;
n = n+2;
ae_vector_set_length(&d, n, _state);
ae_vector_set_length(&ex, n, _state);
ae_vector_set_length(&ey, n, _state);
ex.ptr.p_double[0] = x->ptr.p_double[0]-ae_fabs(x->ptr.p_double[1]-x->ptr.p_double[0], _state);
ex.ptr.p_double[n-1] = x->ptr.p_double[n-3]+ae_fabs(x->ptr.p_double[n-3]-x->ptr.p_double[n-4], _state);
ey.ptr.p_double[0] = y->ptr.p_double[0];
ey.ptr.p_double[n-1] = y->ptr.p_double[n-3];
for(i=1; i<=n-2; i++)
{
ex.ptr.p_double[i] = x->ptr.p_double[i-1];
ey.ptr.p_double[i] = y->ptr.p_double[i-1];
}
/*
* Init sign of the function for first segment
*/
i = 0;
ca = 0;
do
{
ca = ey.ptr.p_double[i+1]-ey.ptr.p_double[i];
i = i+1;
}
while(!(ae_fp_neq(ca,0)||i>n-2));
if( ae_fp_neq(ca,0) )
{
ca = ca/ae_fabs(ca, _state);
}
i = 0;
while(i<n-1)
{
/*
* Partition of the segment [X0;Xn]
*/
tmpn = 1;
for(j=i; j<=n-2; j++)
{
cb = ey.ptr.p_double[j+1]-ey.ptr.p_double[j];
if( ae_fp_greater_eq(ca*cb,0) )
{
tmpn = tmpn+1;
}
else
{
ca = cb/ae_fabs(cb, _state);
break;
}
}
sn = i+tmpn;
ae_assert(tmpn>=2, "Spline1DBuildMonotone: internal error", _state);
/*
* Calculate derivatives for current segment
*/
d.ptr.p_double[i] = 0;
d.ptr.p_double[sn-1] = 0;
for(j=i+1; j<=sn-2; j++)
{
d.ptr.p_double[j] = ((ey.ptr.p_double[j]-ey.ptr.p_double[j-1])/(ex.ptr.p_double[j]-ex.ptr.p_double[j-1])+(ey.ptr.p_double[j+1]-ey.ptr.p_double[j])/(ex.ptr.p_double[j+1]-ex.ptr.p_double[j]))/2;
}
for(j=i; j<=sn-2; j++)
{
delta = (ey.ptr.p_double[j+1]-ey.ptr.p_double[j])/(ex.ptr.p_double[j+1]-ex.ptr.p_double[j]);
if( ae_fp_less_eq(ae_fabs(delta, _state),epsilon) )
{
d.ptr.p_double[j] = 0;
d.ptr.p_double[j+1] = 0;
}
else
{
alpha = d.ptr.p_double[j]/delta;
beta = d.ptr.p_double[j+1]/delta;
if( ae_fp_neq(alpha,0) )
{
cb = alpha*ae_sqrt(1+ae_sqr(beta/alpha, _state), _state);
}
else
{
if( ae_fp_neq(beta,0) )
{
cb = beta;
}
else
{
continue;
}
}
if( ae_fp_greater(cb,3) )
{
d.ptr.p_double[j] = 3*alpha*delta/cb;
d.ptr.p_double[j+1] = 3*beta*delta/cb;
}
}
}
/*
* Transition to next segment
*/
i = sn-1;
}
spline1dbuildhermite(&ex, &ey, &d, n, c, _state);
c->continuity = 2;
ae_frame_leave(_state);
}
/*************************************************************************
Internal version of Spline1DGridDiffCubic.
Accepts pre-ordered X/Y, temporary arrays (which may be preallocated, if
you want to save time, or not) and output array (which may be preallocated
too).
Y is passed as var-parameter because we may need to force last element to
be equal to the first one (if periodic boundary conditions are specified).
-- ALGLIB PROJECT --
Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
static void spline1d_spline1dgriddiffcubicinternal(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_int_t boundltype,
double boundl,
ae_int_t boundrtype,
double boundr,
/* Real */ ae_vector* d,
/* Real */ ae_vector* a1,
/* Real */ ae_vector* a2,
/* Real */ ae_vector* a3,
/* Real */ ae_vector* b,
/* Real */ ae_vector* dt,
ae_state *_state)
{
ae_int_t i;
/*
* allocate arrays
*/
if( d->cnt<n )
{
ae_vector_set_length(d, n, _state);
}
if( a1->cnt<n )
{
ae_vector_set_length(a1, n, _state);
}
if( a2->cnt<n )
{
ae_vector_set_length(a2, n, _state);
}
if( a3->cnt<n )
{
ae_vector_set_length(a3, n, _state);
}
if( b->cnt<n )
{
ae_vector_set_length(b, n, _state);
}
if( dt->cnt<n )
{
ae_vector_set_length(dt, n, _state);
}
/*
* Special cases:
* * N=2, parabolic terminated boundary condition on both ends
* * N=2, periodic boundary condition
*/
if( (n==2&&boundltype==0)&&boundrtype==0 )
{
d->ptr.p_double[0] = (y->ptr.p_double[1]-y->ptr.p_double[0])/(x->ptr.p_double[1]-x->ptr.p_double[0]);
d->ptr.p_double[1] = d->ptr.p_double[0];
return;
}
if( (n==2&&boundltype==-1)&&boundrtype==-1 )
{
d->ptr.p_double[0] = 0;
d->ptr.p_double[1] = 0;
return;
}
/*
* Periodic and non-periodic boundary conditions are
* two separate classes
*/
if( boundrtype==-1&&boundltype==-1 )
{
/*
* Periodic boundary conditions
*/
y->ptr.p_double[n-1] = y->ptr.p_double[0];
/*
* Boundary conditions at N-1 points
* (one point less because last point is the same as first point).
*/
a1->ptr.p_double[0] = x->ptr.p_double[1]-x->ptr.p_double[0];
a2->ptr.p_double[0] = 2*(x->ptr.p_double[1]-x->ptr.p_double[0]+x->ptr.p_double[n-1]-x->ptr.p_double[n-2]);
a3->ptr.p_double[0] = x->ptr.p_double[n-1]-x->ptr.p_double[n-2];
b->ptr.p_double[0] = 3*(y->ptr.p_double[n-1]-y->ptr.p_double[n-2])/(x->ptr.p_double[n-1]-x->ptr.p_double[n-2])*(x->ptr.p_double[1]-x->ptr.p_double[0])+3*(y->ptr.p_double[1]-y->ptr.p_double[0])/(x->ptr.p_double[1]-x->ptr.p_double[0])*(x->ptr.p_double[n-1]-x->ptr.p_double[n-2]);
for(i=1; i<=n-2; i++)
{
/*
* Altough last point is [N-2], we use X[N-1] and Y[N-1]
* (because of periodicity)
*/
a1->ptr.p_double[i] = x->ptr.p_double[i+1]-x->ptr.p_double[i];
a2->ptr.p_double[i] = 2*(x->ptr.p_double[i+1]-x->ptr.p_double[i-1]);
a3->ptr.p_double[i] = x->ptr.p_double[i]-x->ptr.p_double[i-1];
b->ptr.p_double[i] = 3*(y->ptr.p_double[i]-y->ptr.p_double[i-1])/(x->ptr.p_double[i]-x->ptr.p_double[i-1])*(x->ptr.p_double[i+1]-x->ptr.p_double[i])+3*(y->ptr.p_double[i+1]-y->ptr.p_double[i])/(x->ptr.p_double[i+1]-x->ptr.p_double[i])*(x->ptr.p_double[i]-x->ptr.p_double[i-1]);
}
/*
* Solve, add last point (with index N-1)
*/
spline1d_solvecyclictridiagonal(a1, a2, a3, b, n-1, dt, _state);
ae_v_move(&d->ptr.p_double[0], 1, &dt->ptr.p_double[0], 1, ae_v_len(0,n-2));
d->ptr.p_double[n-1] = d->ptr.p_double[0];
}
else
{
/*
* Non-periodic boundary condition.
* Left boundary conditions.
*/
if( boundltype==0 )
{
a1->ptr.p_double[0] = 0;
a2->ptr.p_double[0] = 1;
a3->ptr.p_double[0] = 1;
b->ptr.p_double[0] = 2*(y->ptr.p_double[1]-y->ptr.p_double[0])/(x->ptr.p_double[1]-x->ptr.p_double[0]);
}
if( boundltype==1 )
{
a1->ptr.p_double[0] = 0;
a2->ptr.p_double[0] = 1;
a3->ptr.p_double[0] = 0;
b->ptr.p_double[0] = boundl;
}
if( boundltype==2 )
{
a1->ptr.p_double[0] = 0;
a2->ptr.p_double[0] = 2;
a3->ptr.p_double[0] = 1;
b->ptr.p_double[0] = 3*(y->ptr.p_double[1]-y->ptr.p_double[0])/(x->ptr.p_double[1]-x->ptr.p_double[0])-0.5*boundl*(x->ptr.p_double[1]-x->ptr.p_double[0]);
}
/*
* Central conditions
*/
for(i=1; i<=n-2; i++)
{
a1->ptr.p_double[i] = x->ptr.p_double[i+1]-x->ptr.p_double[i];
a2->ptr.p_double[i] = 2*(x->ptr.p_double[i+1]-x->ptr.p_double[i-1]);
a3->ptr.p_double[i] = x->ptr.p_double[i]-x->ptr.p_double[i-1];
b->ptr.p_double[i] = 3*(y->ptr.p_double[i]-y->ptr.p_double[i-1])/(x->ptr.p_double[i]-x->ptr.p_double[i-1])*(x->ptr.p_double[i+1]-x->ptr.p_double[i])+3*(y->ptr.p_double[i+1]-y->ptr.p_double[i])/(x->ptr.p_double[i+1]-x->ptr.p_double[i])*(x->ptr.p_double[i]-x->ptr.p_double[i-1]);
}
/*
* Right boundary conditions
*/
if( boundrtype==0 )
{
a1->ptr.p_double[n-1] = 1;
a2->ptr.p_double[n-1] = 1;
a3->ptr.p_double[n-1] = 0;
b->ptr.p_double[n-1] = 2*(y->ptr.p_double[n-1]-y->ptr.p_double[n-2])/(x->ptr.p_double[n-1]-x->ptr.p_double[n-2]);
}
if( boundrtype==1 )
{
a1->ptr.p_double[n-1] = 0;
a2->ptr.p_double[n-1] = 1;
a3->ptr.p_double[n-1] = 0;
b->ptr.p_double[n-1] = boundr;
}
if( boundrtype==2 )
{
a1->ptr.p_double[n-1] = 1;
a2->ptr.p_double[n-1] = 2;
a3->ptr.p_double[n-1] = 0;
b->ptr.p_double[n-1] = 3*(y->ptr.p_double[n-1]-y->ptr.p_double[n-2])/(x->ptr.p_double[n-1]-x->ptr.p_double[n-2])+0.5*boundr*(x->ptr.p_double[n-1]-x->ptr.p_double[n-2]);
}
/*
* Solve
*/
spline1d_solvetridiagonal(a1, a2, a3, b, n, d, _state);
}
}
/*************************************************************************
Internal subroutine. Heap sort.
*************************************************************************/
static void spline1d_heapsortpoints(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector bufx;
ae_vector bufy;
ae_frame_make(_state, &_frame_block);
ae_vector_init(&bufx, 0, DT_REAL, _state, ae_true);
ae_vector_init(&bufy, 0, DT_REAL, _state, ae_true);
tagsortfastr(x, y, &bufx, &bufy, n, _state);
ae_frame_leave(_state);
}
/*************************************************************************
Internal subroutine. Heap sort.
Accepts:
X, Y - points
P - empty or preallocated array
Returns:
X, Y - sorted by X
P - array of permutations; I-th position of output
arrays X/Y contains (X[P[I]],Y[P[I]])
*************************************************************************/
static void spline1d_heapsortppoints(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Integer */ ae_vector* p,
ae_int_t n,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector rbuf;
ae_vector ibuf;
ae_int_t i;
ae_frame_make(_state, &_frame_block);
ae_vector_init(&rbuf, 0, DT_REAL, _state, ae_true);
ae_vector_init(&ibuf, 0, DT_INT, _state, ae_true);
if( p->cnt<n )
{
ae_vector_set_length(p, n, _state);
}
ae_vector_set_length(&rbuf, n, _state);
for(i=0; i<=n-1; i++)
{
p->ptr.p_int[i] = i;
}
tagsortfasti(x, p, &rbuf, &ibuf, n, _state);
for(i=0; i<=n-1; i++)
{
rbuf.ptr.p_double[i] = y->ptr.p_double[p->ptr.p_int[i]];
}
ae_v_move(&y->ptr.p_double[0], 1, &rbuf.ptr.p_double[0], 1, ae_v_len(0,n-1));
ae_frame_leave(_state);
}
/*************************************************************************
Internal subroutine. Tridiagonal solver. Solves
( B[0] C[0]
( A[1] B[1] C[1] )
( A[2] B[2] C[2] )
( .......... ) * X = D
( .......... )
( A[N-2] B[N-2] C[N-2] )
( A[N-1] B[N-1] )
*************************************************************************/
static void spline1d_solvetridiagonal(/* Real */ ae_vector* a,
/* Real */ ae_vector* b,
/* Real */ ae_vector* c,
/* Real */ ae_vector* d,
ae_int_t n,
/* Real */ ae_vector* x,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _b;
ae_vector _d;
ae_int_t k;
double t;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_b, b, _state, ae_true);
b = &_b;
ae_vector_init_copy(&_d, d, _state, ae_true);
d = &_d;
if( x->cnt<n )
{
ae_vector_set_length(x, n, _state);
}
for(k=1; k<=n-1; k++)
{
t = a->ptr.p_double[k]/b->ptr.p_double[k-1];
b->ptr.p_double[k] = b->ptr.p_double[k]-t*c->ptr.p_double[k-1];
d->ptr.p_double[k] = d->ptr.p_double[k]-t*d->ptr.p_double[k-1];
}
x->ptr.p_double[n-1] = d->ptr.p_double[n-1]/b->ptr.p_double[n-1];
for(k=n-2; k>=0; k--)
{
x->ptr.p_double[k] = (d->ptr.p_double[k]-c->ptr.p_double[k]*x->ptr.p_double[k+1])/b->ptr.p_double[k];
}
ae_frame_leave(_state);
}
/*************************************************************************
Internal subroutine. Cyclic tridiagonal solver. Solves
( B[0] C[0] A[0] )
( A[1] B[1] C[1] )
( A[2] B[2] C[2] )
( .......... ) * X = D
( .......... )
( A[N-2] B[N-2] C[N-2] )
( C[N-1] A[N-1] B[N-1] )
*************************************************************************/
static void spline1d_solvecyclictridiagonal(/* Real */ ae_vector* a,
/* Real */ ae_vector* b,
/* Real */ ae_vector* c,
/* Real */ ae_vector* d,
ae_int_t n,
/* Real */ ae_vector* x,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _b;
ae_int_t k;
double alpha;
double beta;
double gamma;
ae_vector y;
ae_vector z;
ae_vector u;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_b, b, _state, ae_true);
b = &_b;
ae_vector_init(&y, 0, DT_REAL, _state, ae_true);
ae_vector_init(&z, 0, DT_REAL, _state, ae_true);
ae_vector_init(&u, 0, DT_REAL, _state, ae_true);
if( x->cnt<n )
{
ae_vector_set_length(x, n, _state);
}
beta = a->ptr.p_double[0];
alpha = c->ptr.p_double[n-1];
gamma = -b->ptr.p_double[0];
b->ptr.p_double[0] = 2*b->ptr.p_double[0];
b->ptr.p_double[n-1] = b->ptr.p_double[n-1]-alpha*beta/gamma;
ae_vector_set_length(&u, n, _state);
for(k=0; k<=n-1; k++)
{
u.ptr.p_double[k] = 0;
}
u.ptr.p_double[0] = gamma;
u.ptr.p_double[n-1] = alpha;
spline1d_solvetridiagonal(a, b, c, d, n, &y, _state);
spline1d_solvetridiagonal(a, b, c, &u, n, &z, _state);
for(k=0; k<=n-1; k++)
{
x->ptr.p_double[k] = y.ptr.p_double[k]-(y.ptr.p_double[0]+beta/gamma*y.ptr.p_double[n-1])/(1+z.ptr.p_double[0]+beta/gamma*z.ptr.p_double[n-1])*z.ptr.p_double[k];
}
ae_frame_leave(_state);
}
/*************************************************************************
Internal subroutine. Three-point differentiation
*************************************************************************/
static double spline1d_diffthreepoint(double t,
double x0,
double f0,
double x1,
double f1,
double x2,
double f2,
ae_state *_state)
{
double a;
double b;
double result;
t = t-x0;
x1 = x1-x0;
x2 = x2-x0;
a = (f2-f0-x2/x1*(f1-f0))/(ae_sqr(x2, _state)-x1*x2);
b = (f1-f0-a*ae_sqr(x1, _state))/x1;
result = 2*a*t+b;
return result;
}
/*************************************************************************
Procedure for calculating value of a function is providet in the form of
Hermite polinom
INPUT PARAMETERS:
P0 - value of a function at 0
M0 - value of a derivative at 0
P1 - value of a function at 1
M1 - value of a derivative at 1
T - point inside [0;1]
OUTPUT PARAMETERS:
S - value of a function at T
B0 - value of a derivative function at T
-- ALGLIB PROJECT --
Copyright 26.09.2011 by Bochkanov Sergey
*************************************************************************/
static void spline1d_hermitecalc(double p0,
double m0,
double p1,
double m1,
double t,
double* s,
double* ds,
ae_state *_state)
{
*s = 0;
*ds = 0;
*s = p0*(1+2*t)*(1-t)*(1-t)+m0*t*(1-t)*(1-t)+p1*(3-2*t)*t*t+m1*t*t*(t-1);
*ds = -p0*6*t*(1-t)+m0*(1-t)*(1-3*t)+p1*6*t*(1-t)+m1*t*(3*t-2);
}
/*************************************************************************
Function for mapping from [A0;B0] to [A1;B1]
INPUT PARAMETERS:
A0 - left border [A0;B0]
B0 - right border [A0;B0]
A1 - left border [A1;B1]
B1 - right border [A1;B1]
T - value inside [A0;B0]
RESTRICTIONS OF PARAMETERS:
We assume, that B0>A0 and B1>A1. But we chech, that T is inside [A0;B0],
and if T<A0 then T become A1, if T>B0 then T - B1.
INPUT PARAMETERS:
A0 - left border for segment [A0;B0] from 'T' is converted to [A1;B1]
B0 - right border for segment [A0;B0] from 'T' is converted to [A1;B1]
A1 - left border for segment [A1;B1] to 'T' is converted from [A0;B0]
B1 - right border for segment [A1;B1] to 'T' is converted from [A0;B0]
T - the parameter is mapped from [A0;B0] to [A1;B1]
Result:
is converted value for 'T' from [A0;B0] to [A1;B1]
REMARK:
The function dont check value A0,B0 and A1,B1!
-- ALGLIB PROJECT --
Copyright 26.09.2011 by Bochkanov Sergey
*************************************************************************/
static double spline1d_rescaleval(double a0,
double b0,
double a1,
double b1,
double t,
ae_state *_state)
{
double result;
/*
*return left border
*/
if( ae_fp_less_eq(t,a0) )
{
result = a1;
return result;
}
/*
*return right border
*/
if( ae_fp_greater_eq(t,b0) )
{
result = b1;
return result;
}
/*
*return value between left and right borders
*/
result = (b1-a1)*(t-a0)/(b0-a0)+a1;
return result;
}
ae_bool _spline1dinterpolant_init(void* _p, ae_state *_state, ae_bool make_automatic)
{
spline1dinterpolant *p = (spline1dinterpolant*)_p;
ae_touch_ptr((void*)p);
if( !ae_vector_init(&p->x, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->c, 0, DT_REAL, _state, make_automatic) )
return ae_false;
return ae_true;
}
ae_bool _spline1dinterpolant_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic)
{
spline1dinterpolant *dst = (spline1dinterpolant*)_dst;
spline1dinterpolant *src = (spline1dinterpolant*)_src;
dst->periodic = src->periodic;
dst->n = src->n;
dst->k = src->k;
dst->continuity = src->continuity;
if( !ae_vector_init_copy(&dst->x, &src->x, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->c, &src->c, _state, make_automatic) )
return ae_false;
return ae_true;
}
void _spline1dinterpolant_clear(void* _p)
{
spline1dinterpolant *p = (spline1dinterpolant*)_p;
ae_touch_ptr((void*)p);
ae_vector_clear(&p->x);
ae_vector_clear(&p->c);
}
void _spline1dinterpolant_destroy(void* _p)
{
spline1dinterpolant *p = (spline1dinterpolant*)_p;
ae_touch_ptr((void*)p);
ae_vector_destroy(&p->x);
ae_vector_destroy(&p->c);
}
/*************************************************************************
Fitting by polynomials in barycentric form. This function provides simple
unterface for unconstrained unweighted fitting. See PolynomialFitWC() if
you need constrained fitting.
Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver
SEE ALSO:
PolynomialFitWC()
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
N - number of points, N>0
* if given, only leading N elements of X/Y are used
* if not given, automatically determined from sizes of X/Y
M - number of basis functions (= polynomial_degree + 1), M>=1
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearW() subroutine:
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
P - interpolant in barycentric form.
Rep - report, same format as in LSFitLinearW() subroutine.
Following fields are set:
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
NOTES:
you can convert P from barycentric form to the power or Chebyshev
basis with PolynomialBar2Pow() or PolynomialBar2Cheb() functions from
POLINT subpackage.
-- ALGLIB PROJECT --
Copyright 10.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialfit(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_int_t m,
ae_int_t* info,
barycentricinterpolant* p,
polynomialfitreport* rep,
ae_state *_state)
{
ae_frame _frame_block;
ae_int_t i;
ae_vector w;
ae_vector xc;
ae_vector yc;
ae_vector dc;
ae_frame_make(_state, &_frame_block);
*info = 0;
_barycentricinterpolant_clear(p);
_polynomialfitreport_clear(rep);
ae_vector_init(&w, 0, DT_REAL, _state, ae_true);
ae_vector_init(&xc, 0, DT_REAL, _state, ae_true);
ae_vector_init(&yc, 0, DT_REAL, _state, ae_true);
ae_vector_init(&dc, 0, DT_INT, _state, ae_true);
ae_assert(n>0, "PolynomialFit: N<=0!", _state);
ae_assert(m>0, "PolynomialFit: M<=0!", _state);
ae_assert(x->cnt>=n, "PolynomialFit: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "PolynomialFit: Length(Y)<N!", _state);
ae_assert(isfinitevector(x, n, _state), "PolynomialFit: X contains infinite or NaN values!", _state);
ae_assert(isfinitevector(y, n, _state), "PolynomialFit: Y contains infinite or NaN values!", _state);
ae_vector_set_length(&w, n, _state);
for(i=0; i<=n-1; i++)
{
w.ptr.p_double[i] = 1;
}
polynomialfitwc(x, y, &w, n, &xc, &yc, &dc, 0, m, info, p, rep, _state);
ae_frame_leave(_state);
}
/*************************************************************************
Weighted fitting by polynomials in barycentric form, with constraints on
function values or first derivatives.
Small regularizing term is used when solving constrained tasks (to improve
stability).
Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver
SEE ALSO:
PolynomialFit()
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
W - weights, array[0..N-1]
Each summand in square sum of approximation deviations from
given values is multiplied by the square of corresponding
weight. Fill it by 1's if you don't want to solve weighted
task.
N - number of points, N>0.
* if given, only leading N elements of X/Y/W are used
* if not given, automatically determined from sizes of X/Y/W
XC - points where polynomial values/derivatives are constrained,
array[0..K-1].
YC - values of constraints, array[0..K-1]
DC - array[0..K-1], types of constraints:
* DC[i]=0 means that P(XC[i])=YC[i]
* DC[i]=1 means that P'(XC[i])=YC[i]
SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
K - number of constraints, 0<=K<M.
K=0 means no constraints (XC/YC/DC are not used in such cases)
M - number of basis functions (= polynomial_degree + 1), M>=1
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearW() subroutine:
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
-3 means inconsistent constraints
P - interpolant in barycentric form.
Rep - report, same format as in LSFitLinearW() subroutine.
Following fields are set:
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
NOTES:
you can convert P from barycentric form to the power or Chebyshev
basis with PolynomialBar2Pow() or PolynomialBar2Cheb() functions from
POLINT subpackage.
SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:
Setting constraints can lead to undesired results, like ill-conditioned
behavior, or inconsistency being detected. From the other side, it allows
us to improve quality of the fit. Here we summarize our experience with
constrained regression splines:
* even simple constraints can be inconsistent, see Wikipedia article on
this subject: http://en.wikipedia.org/wiki/Birkhoff_interpolation
* the greater is M (given fixed constraints), the more chances that
constraints will be consistent
* in the general case, consistency of constraints is NOT GUARANTEED.
* in the one special cases, however, we can guarantee consistency. This
case is: M>1 and constraints on the function values (NOT DERIVATIVES)
Our final recommendation is to use constraints WHEN AND ONLY when you
can't solve your task without them. Anything beyond special cases given
above is not guaranteed and may result in inconsistency.
-- ALGLIB PROJECT --
Copyright 10.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialfitwc(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
ae_int_t n,
/* Real */ ae_vector* xc,
/* Real */ ae_vector* yc,
/* Integer */ ae_vector* dc,
ae_int_t k,
ae_int_t m,
ae_int_t* info,
barycentricinterpolant* p,
polynomialfitreport* rep,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _x;
ae_vector _y;
ae_vector _w;
ae_vector _xc;
ae_vector _yc;
double xa;
double xb;
double sa;
double sb;
ae_vector xoriginal;
ae_vector yoriginal;
ae_vector y2;
ae_vector w2;
ae_vector tmp;
ae_vector tmp2;
ae_vector bx;
ae_vector by;
ae_vector bw;
ae_int_t i;
ae_int_t j;
double u;
double v;
double s;
ae_int_t relcnt;
lsfitreport lrep;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_x, x, _state, ae_true);
x = &_x;
ae_vector_init_copy(&_y, y, _state, ae_true);
y = &_y;
ae_vector_init_copy(&_w, w, _state, ae_true);
w = &_w;
ae_vector_init_copy(&_xc, xc, _state, ae_true);
xc = &_xc;
ae_vector_init_copy(&_yc, yc, _state, ae_true);
yc = &_yc;
*info = 0;
_barycentricinterpolant_clear(p);
_polynomialfitreport_clear(rep);
ae_vector_init(&xoriginal, 0, DT_REAL, _state, ae_true);
ae_vector_init(&yoriginal, 0, DT_REAL, _state, ae_true);
ae_vector_init(&y2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&w2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmp, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmp2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&bx, 0, DT_REAL, _state, ae_true);
ae_vector_init(&by, 0, DT_REAL, _state, ae_true);
ae_vector_init(&bw, 0, DT_REAL, _state, ae_true);
_lsfitreport_init(&lrep, _state, ae_true);
ae_assert(n>0, "PolynomialFitWC: N<=0!", _state);
ae_assert(m>0, "PolynomialFitWC: M<=0!", _state);
ae_assert(k>=0, "PolynomialFitWC: K<0!", _state);
ae_assert(k<m, "PolynomialFitWC: K>=M!", _state);
ae_assert(x->cnt>=n, "PolynomialFitWC: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "PolynomialFitWC: Length(Y)<N!", _state);
ae_assert(w->cnt>=n, "PolynomialFitWC: Length(W)<N!", _state);
ae_assert(xc->cnt>=k, "PolynomialFitWC: Length(XC)<K!", _state);
ae_assert(yc->cnt>=k, "PolynomialFitWC: Length(YC)<K!", _state);
ae_assert(dc->cnt>=k, "PolynomialFitWC: Length(DC)<K!", _state);
ae_assert(isfinitevector(x, n, _state), "PolynomialFitWC: X contains infinite or NaN values!", _state);
ae_assert(isfinitevector(y, n, _state), "PolynomialFitWC: Y contains infinite or NaN values!", _state);
ae_assert(isfinitevector(w, n, _state), "PolynomialFitWC: X contains infinite or NaN values!", _state);
ae_assert(isfinitevector(xc, k, _state), "PolynomialFitWC: XC contains infinite or NaN values!", _state);
ae_assert(isfinitevector(yc, k, _state), "PolynomialFitWC: YC contains infinite or NaN values!", _state);
for(i=0; i<=k-1; i++)
{
ae_assert(dc->ptr.p_int[i]==0||dc->ptr.p_int[i]==1, "PolynomialFitWC: one of DC[] is not 0 or 1!", _state);
}
/*
* Scale X, Y, XC, YC.
* Solve scaled problem using internal Chebyshev fitting function.
*/
lsfitscalexy(x, y, w, n, xc, yc, dc, k, &xa, &xb, &sa, &sb, &xoriginal, &yoriginal, _state);
lsfit_internalchebyshevfit(x, y, w, n, xc, yc, dc, k, m, info, &tmp, &lrep, _state);
if( *info<0 )
{
ae_frame_leave(_state);
return;
}
/*
* Generate barycentric model and scale it
* * BX, BY store barycentric model nodes
* * FMatrix is reused (remember - it is at least MxM, what we need)
*
* Model intialization is done in O(M^2). In principle, it can be
* done in O(M*log(M)), but before it we solved task with O(N*M^2)
* complexity, so it is only a small amount of total time spent.
*/
ae_vector_set_length(&bx, m, _state);
ae_vector_set_length(&by, m, _state);
ae_vector_set_length(&bw, m, _state);
ae_vector_set_length(&tmp2, m, _state);
s = 1;
for(i=0; i<=m-1; i++)
{
if( m!=1 )
{
u = ae_cos(ae_pi*i/(m-1), _state);
}
else
{
u = 0;
}
v = 0;
for(j=0; j<=m-1; j++)
{
if( j==0 )
{
tmp2.ptr.p_double[j] = 1;
}
else
{
if( j==1 )
{
tmp2.ptr.p_double[j] = u;
}
else
{
tmp2.ptr.p_double[j] = 2*u*tmp2.ptr.p_double[j-1]-tmp2.ptr.p_double[j-2];
}
}
v = v+tmp.ptr.p_double[j]*tmp2.ptr.p_double[j];
}
bx.ptr.p_double[i] = u;
by.ptr.p_double[i] = v;
bw.ptr.p_double[i] = s;
if( i==0||i==m-1 )
{
bw.ptr.p_double[i] = 0.5*bw.ptr.p_double[i];
}
s = -s;
}
barycentricbuildxyw(&bx, &by, &bw, m, p, _state);
barycentriclintransx(p, 2/(xb-xa), -(xa+xb)/(xb-xa), _state);
barycentriclintransy(p, sb-sa, sa, _state);
/*
* Scale absolute errors obtained from LSFitLinearW.
* Relative error should be calculated separately
* (because of shifting/scaling of the task)
*/
rep->taskrcond = lrep.taskrcond;
rep->rmserror = lrep.rmserror*(sb-sa);
rep->avgerror = lrep.avgerror*(sb-sa);
rep->maxerror = lrep.maxerror*(sb-sa);
rep->avgrelerror = 0;
relcnt = 0;
for(i=0; i<=n-1; i++)
{
if( ae_fp_neq(yoriginal.ptr.p_double[i],0) )
{
rep->avgrelerror = rep->avgrelerror+ae_fabs(barycentriccalc(p, xoriginal.ptr.p_double[i], _state)-yoriginal.ptr.p_double[i], _state)/ae_fabs(yoriginal.ptr.p_double[i], _state);
relcnt = relcnt+1;
}
}
if( relcnt!=0 )
{
rep->avgrelerror = rep->avgrelerror/relcnt;
}
ae_frame_leave(_state);
}
/*************************************************************************
Weghted rational least squares fitting using Floater-Hormann rational
functions with optimal D chosen from [0,9], with constraints and
individual weights.
Equidistant grid with M node on [min(x),max(x)] is used to build basis
functions. Different values of D are tried, optimal D (least WEIGHTED root
mean square error) is chosen. Task is linear, so linear least squares
solver is used. Complexity of this computational scheme is O(N*M^2)
(mostly dominated by the least squares solver).
SEE ALSO
* BarycentricFitFloaterHormann(), "lightweight" fitting without invididual
weights and constraints.
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
W - weights, array[0..N-1]
Each summand in square sum of approximation deviations from
given values is multiplied by the square of corresponding
weight. Fill it by 1's if you don't want to solve weighted
task.
N - number of points, N>0.
XC - points where function values/derivatives are constrained,
array[0..K-1].
YC - values of constraints, array[0..K-1]
DC - array[0..K-1], types of constraints:
* DC[i]=0 means that S(XC[i])=YC[i]
* DC[i]=1 means that S'(XC[i])=YC[i]
SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
K - number of constraints, 0<=K<M.
K=0 means no constraints (XC/YC/DC are not used in such cases)
M - number of basis functions ( = number_of_nodes), M>=2.
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearWC() subroutine.
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
-3 means inconsistent constraints
-1 means another errors in parameters passed
(N<=0, for example)
B - barycentric interpolant.
Rep - report, same format as in LSFitLinearWC() subroutine.
Following fields are set:
* DBest best value of the D parameter
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroutine doesn't calculate task's condition number for K<>0.
SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:
Setting constraints can lead to undesired results, like ill-conditioned
behavior, or inconsistency being detected. From the other side, it allows
us to improve quality of the fit. Here we summarize our experience with
constrained barycentric interpolants:
* excessive constraints can be inconsistent. Floater-Hormann basis
functions aren't as flexible as splines (although they are very smooth).
* the more evenly constraints are spread across [min(x),max(x)], the more
chances that they will be consistent
* the greater is M (given fixed constraints), the more chances that
constraints will be consistent
* in the general case, consistency of constraints IS NOT GUARANTEED.
* in the several special cases, however, we CAN guarantee consistency.
* one of this cases is constraints on the function VALUES at the interval
boundaries. Note that consustency of the constraints on the function
DERIVATIVES is NOT guaranteed (you can use in such cases cubic splines
which are more flexible).
* another special case is ONE constraint on the function value (OR, but
not AND, derivative) anywhere in the interval
Our final recommendation is to use constraints WHEN AND ONLY WHEN you
can't solve your task without them. Anything beyond special cases given
above is not guaranteed and may result in inconsistency.
-- ALGLIB PROJECT --
Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentricfitfloaterhormannwc(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
ae_int_t n,
/* Real */ ae_vector* xc,
/* Real */ ae_vector* yc,
/* Integer */ ae_vector* dc,
ae_int_t k,
ae_int_t m,
ae_int_t* info,
barycentricinterpolant* b,
barycentricfitreport* rep,
ae_state *_state)
{
ae_frame _frame_block;
ae_int_t d;
ae_int_t i;
double wrmscur;
double wrmsbest;
barycentricinterpolant locb;
barycentricfitreport locrep;
ae_int_t locinfo;
ae_frame_make(_state, &_frame_block);
*info = 0;
_barycentricinterpolant_clear(b);
_barycentricfitreport_clear(rep);
_barycentricinterpolant_init(&locb, _state, ae_true);
_barycentricfitreport_init(&locrep, _state, ae_true);
ae_assert(n>0, "BarycentricFitFloaterHormannWC: N<=0!", _state);
ae_assert(m>0, "BarycentricFitFloaterHormannWC: M<=0!", _state);
ae_assert(k>=0, "BarycentricFitFloaterHormannWC: K<0!", _state);
ae_assert(k<m, "BarycentricFitFloaterHormannWC: K>=M!", _state);
ae_assert(x->cnt>=n, "BarycentricFitFloaterHormannWC: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "BarycentricFitFloaterHormannWC: Length(Y)<N!", _state);
ae_assert(w->cnt>=n, "BarycentricFitFloaterHormannWC: Length(W)<N!", _state);
ae_assert(xc->cnt>=k, "BarycentricFitFloaterHormannWC: Length(XC)<K!", _state);
ae_assert(yc->cnt>=k, "BarycentricFitFloaterHormannWC: Length(YC)<K!", _state);
ae_assert(dc->cnt>=k, "BarycentricFitFloaterHormannWC: Length(DC)<K!", _state);
ae_assert(isfinitevector(x, n, _state), "BarycentricFitFloaterHormannWC: X contains infinite or NaN values!", _state);
ae_assert(isfinitevector(y, n, _state), "BarycentricFitFloaterHormannWC: Y contains infinite or NaN values!", _state);
ae_assert(isfinitevector(w, n, _state), "BarycentricFitFloaterHormannWC: X contains infinite or NaN values!", _state);
ae_assert(isfinitevector(xc, k, _state), "BarycentricFitFloaterHormannWC: XC contains infinite or NaN values!", _state);
ae_assert(isfinitevector(yc, k, _state), "BarycentricFitFloaterHormannWC: YC contains infinite or NaN values!", _state);
for(i=0; i<=k-1; i++)
{
ae_assert(dc->ptr.p_int[i]==0||dc->ptr.p_int[i]==1, "BarycentricFitFloaterHormannWC: one of DC[] is not 0 or 1!", _state);
}
/*
* Find optimal D
*
* Info is -3 by default (degenerate constraints).
* If LocInfo will always be equal to -3, Info will remain equal to -3.
* If at least once LocInfo will be -4, Info will be -4.
*/
wrmsbest = ae_maxrealnumber;
rep->dbest = -1;
*info = -3;
for(d=0; d<=ae_minint(9, n-1, _state); d++)
{
lsfit_barycentricfitwcfixedd(x, y, w, n, xc, yc, dc, k, m, d, &locinfo, &locb, &locrep, _state);
ae_assert((locinfo==-4||locinfo==-3)||locinfo>0, "BarycentricFitFloaterHormannWC: unexpected result from BarycentricFitWCFixedD!", _state);
if( locinfo>0 )
{
/*
* Calculate weghted RMS
*/
wrmscur = 0;
for(i=0; i<=n-1; i++)
{
wrmscur = wrmscur+ae_sqr(w->ptr.p_double[i]*(y->ptr.p_double[i]-barycentriccalc(&locb, x->ptr.p_double[i], _state)), _state);
}
wrmscur = ae_sqrt(wrmscur/n, _state);
if( ae_fp_less(wrmscur,wrmsbest)||rep->dbest<0 )
{
barycentriccopy(&locb, b, _state);
rep->dbest = d;
*info = 1;
rep->rmserror = locrep.rmserror;
rep->avgerror = locrep.avgerror;
rep->avgrelerror = locrep.avgrelerror;
rep->maxerror = locrep.maxerror;
rep->taskrcond = locrep.taskrcond;
wrmsbest = wrmscur;
}
}
else
{
if( locinfo!=-3&&*info<0 )
{
*info = locinfo;
}
}
}
ae_frame_leave(_state);
}
/*************************************************************************
Rational least squares fitting using Floater-Hormann rational functions
with optimal D chosen from [0,9].
Equidistant grid with M node on [min(x),max(x)] is used to build basis
functions. Different values of D are tried, optimal D (least root mean
square error) is chosen. Task is linear, so linear least squares solver
is used. Complexity of this computational scheme is O(N*M^2) (mostly
dominated by the least squares solver).
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
N - number of points, N>0.
M - number of basis functions ( = number_of_nodes), M>=2.
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearWC() subroutine.
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
-3 means inconsistent constraints
B - barycentric interpolant.
Rep - report, same format as in LSFitLinearWC() subroutine.
Following fields are set:
* DBest best value of the D parameter
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
-- ALGLIB PROJECT --
Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentricfitfloaterhormann(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_int_t m,
ae_int_t* info,
barycentricinterpolant* b,
barycentricfitreport* rep,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector w;
ae_vector xc;
ae_vector yc;
ae_vector dc;
ae_int_t i;
ae_frame_make(_state, &_frame_block);
*info = 0;
_barycentricinterpolant_clear(b);
_barycentricfitreport_clear(rep);
ae_vector_init(&w, 0, DT_REAL, _state, ae_true);
ae_vector_init(&xc, 0, DT_REAL, _state, ae_true);
ae_vector_init(&yc, 0, DT_REAL, _state, ae_true);
ae_vector_init(&dc, 0, DT_INT, _state, ae_true);
ae_assert(n>0, "BarycentricFitFloaterHormann: N<=0!", _state);
ae_assert(m>0, "BarycentricFitFloaterHormann: M<=0!", _state);
ae_assert(x->cnt>=n, "BarycentricFitFloaterHormann: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "BarycentricFitFloaterHormann: Length(Y)<N!", _state);
ae_assert(isfinitevector(x, n, _state), "BarycentricFitFloaterHormann: X contains infinite or NaN values!", _state);
ae_assert(isfinitevector(y, n, _state), "BarycentricFitFloaterHormann: Y contains infinite or NaN values!", _state);
ae_vector_set_length(&w, n, _state);
for(i=0; i<=n-1; i++)
{
w.ptr.p_double[i] = 1;
}
barycentricfitfloaterhormannwc(x, y, &w, n, &xc, &yc, &dc, 0, m, info, b, rep, _state);
ae_frame_leave(_state);
}
/*************************************************************************
Rational least squares fitting using Floater-Hormann rational functions
with optimal D chosen from [0,9].
Equidistant grid with M node on [min(x),max(x)] is used to build basis
functions. Different values of D are tried, optimal D (least root mean
square error) is chosen. Task is linear, so linear least squares solver
is used. Complexity of this computational scheme is O(N*M^2) (mostly
dominated by the least squares solver).
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
N - number of points, N>0.
M - number of basis functions ( = number_of_nodes), M>=2.
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearWC() subroutine.
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
-3 means inconsistent constraints
B - barycentric interpolant.
Rep - report, same format as in LSFitLinearWC() subroutine.
Following fields are set:
* DBest best value of the D parameter
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
-- ALGLIB PROJECT --
Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfitpenalized(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_int_t m,
double rho,
ae_int_t* info,
spline1dinterpolant* s,
spline1dfitreport* rep,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _x;
ae_vector _y;
ae_vector w;
ae_int_t i;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_x, x, _state, ae_true);
x = &_x;
ae_vector_init_copy(&_y, y, _state, ae_true);
y = &_y;
*info = 0;
_spline1dinterpolant_clear(s);
_spline1dfitreport_clear(rep);
ae_vector_init(&w, 0, DT_REAL, _state, ae_true);
ae_assert(n>=1, "Spline1DFitPenalized: N<1!", _state);
ae_assert(m>=4, "Spline1DFitPenalized: M<4!", _state);
ae_assert(x->cnt>=n, "Spline1DFitPenalized: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "Spline1DFitPenalized: Length(Y)<N!", _state);
ae_assert(isfinitevector(x, n, _state), "Spline1DFitPenalized: X contains infinite or NAN values!", _state);
ae_assert(isfinitevector(y, n, _state), "Spline1DFitPenalized: Y contains infinite or NAN values!", _state);
ae_assert(ae_isfinite(rho, _state), "Spline1DFitPenalized: Rho is infinite!", _state);
ae_vector_set_length(&w, n, _state);
for(i=0; i<=n-1; i++)
{
w.ptr.p_double[i] = 1;
}
spline1dfitpenalizedw(x, y, &w, n, m, rho, info, s, rep, _state);
ae_frame_leave(_state);
}
/*************************************************************************
Weighted fitting by penalized cubic spline.
Equidistant grid with M nodes on [min(x,xc),max(x,xc)] is used to build
basis functions. Basis functions are cubic splines with natural boundary
conditions. Problem is regularized by adding non-linearity penalty to the
usual least squares penalty function:
S(x) = arg min { LS + P }, where
LS = SUM { w[i]^2*(y[i] - S(x[i]))^2 } - least squares penalty
P = C*10^rho*integral{ S''(x)^2*dx } - non-linearity penalty
rho - tunable constant given by user
C - automatically determined scale parameter,
makes penalty invariant with respect to scaling of X, Y, W.
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
W - weights, array[0..N-1]
Each summand in square sum of approximation deviations from
given values is multiplied by the square of corresponding
weight. Fill it by 1's if you don't want to solve weighted
problem.
N - number of points (optional):
* N>0
* if given, only first N elements of X/Y/W are processed
* if not given, automatically determined from X/Y/W sizes
M - number of basis functions ( = number_of_nodes), M>=4.
Rho - regularization constant passed by user. It penalizes
nonlinearity in the regression spline. It is logarithmically
scaled, i.e. actual value of regularization constant is
calculated as 10^Rho. It is automatically scaled so that:
* Rho=2.0 corresponds to moderate amount of nonlinearity
* generally, it should be somewhere in the [-8.0,+8.0]
If you do not want to penalize nonlineary,
pass small Rho. Values as low as -15 should work.
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearWC() subroutine.
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD or
Cholesky decomposition; problem may be
too ill-conditioned (very rare)
S - spline interpolant.
Rep - Following fields are set:
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
NOTE 1: additional nodes are added to the spline outside of the fitting
interval to force linearity when x<min(x,xc) or x>max(x,xc). It is done
for consistency - we penalize non-linearity at [min(x,xc),max(x,xc)], so
it is natural to force linearity outside of this interval.
NOTE 2: function automatically sorts points, so caller may pass unsorted
array.
-- ALGLIB PROJECT --
Copyright 19.10.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dfitpenalizedw(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
ae_int_t n,
ae_int_t m,
double rho,
ae_int_t* info,
spline1dinterpolant* s,
spline1dfitreport* rep,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _x;
ae_vector _y;
ae_vector _w;
ae_int_t i;
ae_int_t j;
ae_int_t b;
double v;
double relcnt;
double xa;
double xb;
double sa;
double sb;
ae_vector xoriginal;
ae_vector yoriginal;
double pdecay;
double tdecay;
ae_matrix fmatrix;
ae_vector fcolumn;
ae_vector y2;
ae_vector w2;
ae_vector xc;
ae_vector yc;
ae_vector dc;
double fdmax;
double admax;
ae_matrix amatrix;
ae_matrix d2matrix;
double fa;
double ga;
double fb;
double gb;
double lambdav;
ae_vector bx;
ae_vector by;
ae_vector bd1;
ae_vector bd2;
ae_vector tx;
ae_vector ty;
ae_vector td;
spline1dinterpolant bs;
ae_matrix nmatrix;
ae_vector rightpart;
fblslincgstate cgstate;
ae_vector c;
ae_vector tmp0;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_x, x, _state, ae_true);
x = &_x;
ae_vector_init_copy(&_y, y, _state, ae_true);
y = &_y;
ae_vector_init_copy(&_w, w, _state, ae_true);
w = &_w;
*info = 0;
_spline1dinterpolant_clear(s);
_spline1dfitreport_clear(rep);
ae_vector_init(&xoriginal, 0, DT_REAL, _state, ae_true);
ae_vector_init(&yoriginal, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&fmatrix, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&fcolumn, 0, DT_REAL, _state, ae_true);
ae_vector_init(&y2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&w2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&xc, 0, DT_REAL, _state, ae_true);
ae_vector_init(&yc, 0, DT_REAL, _state, ae_true);
ae_vector_init(&dc, 0, DT_INT, _state, ae_true);
ae_matrix_init(&amatrix, 0, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&d2matrix, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&bx, 0, DT_REAL, _state, ae_true);
ae_vector_init(&by, 0, DT_REAL, _state, ae_true);
ae_vector_init(&bd1, 0, DT_REAL, _state, ae_true);
ae_vector_init(&bd2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tx, 0, DT_REAL, _state, ae_true);
ae_vector_init(&ty, 0, DT_REAL, _state, ae_true);
ae_vector_init(&td, 0, DT_REAL, _state, ae_true);
_spline1dinterpolant_init(&bs, _state, ae_true);
ae_matrix_init(&nmatrix, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&rightpart, 0, DT_REAL, _state, ae_true);
_fblslincgstate_init(&cgstate, _state, ae_true);
ae_vector_init(&c, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmp0, 0, DT_REAL, _state, ae_true);
ae_assert(n>=1, "Spline1DFitPenalizedW: N<1!", _state);
ae_assert(m>=4, "Spline1DFitPenalizedW: M<4!", _state);
ae_assert(x->cnt>=n, "Spline1DFitPenalizedW: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "Spline1DFitPenalizedW: Length(Y)<N!", _state);
ae_assert(w->cnt>=n, "Spline1DFitPenalizedW: Length(W)<N!", _state);
ae_assert(isfinitevector(x, n, _state), "Spline1DFitPenalizedW: X contains infinite or NAN values!", _state);
ae_assert(isfinitevector(y, n, _state), "Spline1DFitPenalizedW: Y contains infinite or NAN values!", _state);
ae_assert(isfinitevector(w, n, _state), "Spline1DFitPenalizedW: Y contains infinite or NAN values!", _state);
ae_assert(ae_isfinite(rho, _state), "Spline1DFitPenalizedW: Rho is infinite!", _state);
/*
* Prepare LambdaV
*/
v = -ae_log(ae_machineepsilon, _state)/ae_log(10, _state);
if( ae_fp_less(rho,-v) )
{
rho = -v;
}
if( ae_fp_greater(rho,v) )
{
rho = v;
}
lambdav = ae_pow(10, rho, _state);
/*
* Sort X, Y, W
*/
heapsortdpoints(x, y, w, n, _state);
/*
* Scale X, Y, XC, YC
*/
lsfitscalexy(x, y, w, n, &xc, &yc, &dc, 0, &xa, &xb, &sa, &sb, &xoriginal, &yoriginal, _state);
/*
* Allocate space
*/
ae_matrix_set_length(&fmatrix, n, m, _state);
ae_matrix_set_length(&amatrix, m, m, _state);
ae_matrix_set_length(&d2matrix, m, m, _state);
ae_vector_set_length(&bx, m, _state);
ae_vector_set_length(&by, m, _state);
ae_vector_set_length(&fcolumn, n, _state);
ae_matrix_set_length(&nmatrix, m, m, _state);
ae_vector_set_length(&rightpart, m, _state);
ae_vector_set_length(&tmp0, ae_maxint(m, n, _state), _state);
ae_vector_set_length(&c, m, _state);
/*
* Fill:
* * FMatrix by values of basis functions
* * TmpAMatrix by second derivatives of I-th function at J-th point
* * CMatrix by constraints
*/
fdmax = 0;
for(b=0; b<=m-1; b++)
{
/*
* Prepare I-th basis function
*/
for(j=0; j<=m-1; j++)
{
bx.ptr.p_double[j] = (double)(2*j)/(double)(m-1)-1;
by.ptr.p_double[j] = 0;
}
by.ptr.p_double[b] = 1;
spline1dgriddiff2cubic(&bx, &by, m, 2, 0.0, 2, 0.0, &bd1, &bd2, _state);
spline1dbuildcubic(&bx, &by, m, 2, 0.0, 2, 0.0, &bs, _state);
/*
* Calculate B-th column of FMatrix
* Update FDMax (maximum column norm)
*/
spline1dconvcubic(&bx, &by, m, 2, 0.0, 2, 0.0, x, n, &fcolumn, _state);
ae_v_move(&fmatrix.ptr.pp_double[0][b], fmatrix.stride, &fcolumn.ptr.p_double[0], 1, ae_v_len(0,n-1));
v = 0;
for(i=0; i<=n-1; i++)
{
v = v+ae_sqr(w->ptr.p_double[i]*fcolumn.ptr.p_double[i], _state);
}
fdmax = ae_maxreal(fdmax, v, _state);
/*
* Fill temporary with second derivatives of basis function
*/
ae_v_move(&d2matrix.ptr.pp_double[b][0], 1, &bd2.ptr.p_double[0], 1, ae_v_len(0,m-1));
}
/*
* * calculate penalty matrix A
* * calculate max of diagonal elements of A
* * calculate PDecay - coefficient before penalty matrix
*/
for(i=0; i<=m-1; i++)
{
for(j=i; j<=m-1; j++)
{
/*
* calculate integral(B_i''*B_j'') where B_i and B_j are
* i-th and j-th basis splines.
* B_i and B_j are piecewise linear functions.
*/
v = 0;
for(b=0; b<=m-2; b++)
{
fa = d2matrix.ptr.pp_double[i][b];
fb = d2matrix.ptr.pp_double[i][b+1];
ga = d2matrix.ptr.pp_double[j][b];
gb = d2matrix.ptr.pp_double[j][b+1];
v = v+(bx.ptr.p_double[b+1]-bx.ptr.p_double[b])*(fa*ga+(fa*(gb-ga)+ga*(fb-fa))/2+(fb-fa)*(gb-ga)/3);
}
amatrix.ptr.pp_double[i][j] = v;
amatrix.ptr.pp_double[j][i] = v;
}
}
admax = 0;
for(i=0; i<=m-1; i++)
{
admax = ae_maxreal(admax, ae_fabs(amatrix.ptr.pp_double[i][i], _state), _state);
}
pdecay = lambdav*fdmax/admax;
/*
* Calculate TDecay for Tikhonov regularization
*/
tdecay = fdmax*(1+pdecay)*10*ae_machineepsilon;
/*
* Prepare system
*
* NOTE: FMatrix is spoiled during this process
*/
for(i=0; i<=n-1; i++)
{
v = w->ptr.p_double[i];
ae_v_muld(&fmatrix.ptr.pp_double[i][0], 1, ae_v_len(0,m-1), v);
}
rmatrixgemm(m, m, n, 1.0, &fmatrix, 0, 0, 1, &fmatrix, 0, 0, 0, 0.0, &nmatrix, 0, 0, _state);
for(i=0; i<=m-1; i++)
{
for(j=0; j<=m-1; j++)
{
nmatrix.ptr.pp_double[i][j] = nmatrix.ptr.pp_double[i][j]+pdecay*amatrix.ptr.pp_double[i][j];
}
}
for(i=0; i<=m-1; i++)
{
nmatrix.ptr.pp_double[i][i] = nmatrix.ptr.pp_double[i][i]+tdecay;
}
for(i=0; i<=m-1; i++)
{
rightpart.ptr.p_double[i] = 0;
}
for(i=0; i<=n-1; i++)
{
v = y->ptr.p_double[i]*w->ptr.p_double[i];
ae_v_addd(&rightpart.ptr.p_double[0], 1, &fmatrix.ptr.pp_double[i][0], 1, ae_v_len(0,m-1), v);
}
/*
* Solve system
*/
if( !spdmatrixcholesky(&nmatrix, m, ae_true, _state) )
{
*info = -4;
ae_frame_leave(_state);
return;
}
fblscholeskysolve(&nmatrix, 1.0, m, ae_true, &rightpart, &tmp0, _state);
ae_v_move(&c.ptr.p_double[0], 1, &rightpart.ptr.p_double[0], 1, ae_v_len(0,m-1));
/*
* add nodes to force linearity outside of the fitting interval
*/
spline1dgriddiffcubic(&bx, &c, m, 2, 0.0, 2, 0.0, &bd1, _state);
ae_vector_set_length(&tx, m+2, _state);
ae_vector_set_length(&ty, m+2, _state);
ae_vector_set_length(&td, m+2, _state);
ae_v_move(&tx.ptr.p_double[1], 1, &bx.ptr.p_double[0], 1, ae_v_len(1,m));
ae_v_move(&ty.ptr.p_double[1], 1, &rightpart.ptr.p_double[0], 1, ae_v_len(1,m));
ae_v_move(&td.ptr.p_double[1], 1, &bd1.ptr.p_double[0], 1, ae_v_len(1,m));
tx.ptr.p_double[0] = tx.ptr.p_double[1]-(tx.ptr.p_double[2]-tx.ptr.p_double[1]);
ty.ptr.p_double[0] = ty.ptr.p_double[1]-td.ptr.p_double[1]*(tx.ptr.p_double[2]-tx.ptr.p_double[1]);
td.ptr.p_double[0] = td.ptr.p_double[1];
tx.ptr.p_double[m+1] = tx.ptr.p_double[m]+(tx.ptr.p_double[m]-tx.ptr.p_double[m-1]);
ty.ptr.p_double[m+1] = ty.ptr.p_double[m]+td.ptr.p_double[m]*(tx.ptr.p_double[m]-tx.ptr.p_double[m-1]);
td.ptr.p_double[m+1] = td.ptr.p_double[m];
spline1dbuildhermite(&tx, &ty, &td, m+2, s, _state);
spline1dlintransx(s, 2/(xb-xa), -(xa+xb)/(xb-xa), _state);
spline1dlintransy(s, sb-sa, sa, _state);
*info = 1;
/*
* Fill report
*/
rep->rmserror = 0;
rep->avgerror = 0;
rep->avgrelerror = 0;
rep->maxerror = 0;
relcnt = 0;
spline1dconvcubic(&bx, &rightpart, m, 2, 0.0, 2, 0.0, x, n, &fcolumn, _state);
for(i=0; i<=n-1; i++)
{
v = (sb-sa)*fcolumn.ptr.p_double[i]+sa;
rep->rmserror = rep->rmserror+ae_sqr(v-yoriginal.ptr.p_double[i], _state);
rep->avgerror = rep->avgerror+ae_fabs(v-yoriginal.ptr.p_double[i], _state);
if( ae_fp_neq(yoriginal.ptr.p_double[i],0) )
{
rep->avgrelerror = rep->avgrelerror+ae_fabs(v-yoriginal.ptr.p_double[i], _state)/ae_fabs(yoriginal.ptr.p_double[i], _state);
relcnt = relcnt+1;
}
rep->maxerror = ae_maxreal(rep->maxerror, ae_fabs(v-yoriginal.ptr.p_double[i], _state), _state);
}
rep->rmserror = ae_sqrt(rep->rmserror/n, _state);
rep->avgerror = rep->avgerror/n;
if( ae_fp_neq(relcnt,0) )
{
rep->avgrelerror = rep->avgrelerror/relcnt;
}
ae_frame_leave(_state);
}
/*************************************************************************
Weighted fitting by cubic spline, with constraints on function values or
derivatives.
Equidistant grid with M-2 nodes on [min(x,xc),max(x,xc)] is used to build
basis functions. Basis functions are cubic splines with continuous second
derivatives and non-fixed first derivatives at interval ends. Small
regularizing term is used when solving constrained tasks (to improve
stability).
Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver
SEE ALSO
Spline1DFitHermiteWC() - fitting by Hermite splines (more flexible,
less smooth)
Spline1DFitCubic() - "lightweight" fitting by cubic splines,
without invididual weights and constraints
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
W - weights, array[0..N-1]
Each summand in square sum of approximation deviations from
given values is multiplied by the square of corresponding
weight. Fill it by 1's if you don't want to solve weighted
task.
N - number of points (optional):
* N>0
* if given, only first N elements of X/Y/W are processed
* if not given, automatically determined from X/Y/W sizes
XC - points where spline values/derivatives are constrained,
array[0..K-1].
YC - values of constraints, array[0..K-1]
DC - array[0..K-1], types of constraints:
* DC[i]=0 means that S(XC[i])=YC[i]
* DC[i]=1 means that S'(XC[i])=YC[i]
SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
K - number of constraints (optional):
* 0<=K<M.
* K=0 means no constraints (XC/YC/DC are not used)
* if given, only first K elements of XC/YC/DC are used
* if not given, automatically determined from XC/YC/DC
M - number of basis functions ( = number_of_nodes+2), M>=4.
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearWC() subroutine.
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
-3 means inconsistent constraints
S - spline interpolant.
Rep - report, same format as in LSFitLinearWC() subroutine.
Following fields are set:
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:
Setting constraints can lead to undesired results, like ill-conditioned
behavior, or inconsistency being detected. From the other side, it allows
us to improve quality of the fit. Here we summarize our experience with
constrained regression splines:
* excessive constraints can be inconsistent. Splines are piecewise cubic
functions, and it is easy to create an example, where large number of
constraints concentrated in small area will result in inconsistency.
Just because spline is not flexible enough to satisfy all of them. And
same constraints spread across the [min(x),max(x)] will be perfectly
consistent.
* the more evenly constraints are spread across [min(x),max(x)], the more
chances that they will be consistent
* the greater is M (given fixed constraints), the more chances that
constraints will be consistent
* in the general case, consistency of constraints IS NOT GUARANTEED.
* in the several special cases, however, we CAN guarantee consistency.
* one of this cases is constraints on the function values AND/OR its
derivatives at the interval boundaries.
* another special case is ONE constraint on the function value (OR, but
not AND, derivative) anywhere in the interval
Our final recommendation is to use constraints WHEN AND ONLY WHEN you
can't solve your task without them. Anything beyond special cases given
above is not guaranteed and may result in inconsistency.
-- ALGLIB PROJECT --
Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfitcubicwc(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
ae_int_t n,
/* Real */ ae_vector* xc,
/* Real */ ae_vector* yc,
/* Integer */ ae_vector* dc,
ae_int_t k,
ae_int_t m,
ae_int_t* info,
spline1dinterpolant* s,
spline1dfitreport* rep,
ae_state *_state)
{
ae_int_t i;
*info = 0;
_spline1dinterpolant_clear(s);
_spline1dfitreport_clear(rep);
ae_assert(n>=1, "Spline1DFitCubicWC: N<1!", _state);
ae_assert(m>=4, "Spline1DFitCubicWC: M<4!", _state);
ae_assert(k>=0, "Spline1DFitCubicWC: K<0!", _state);
ae_assert(k<m, "Spline1DFitCubicWC: K>=M!", _state);
ae_assert(x->cnt>=n, "Spline1DFitCubicWC: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "Spline1DFitCubicWC: Length(Y)<N!", _state);
ae_assert(w->cnt>=n, "Spline1DFitCubicWC: Length(W)<N!", _state);
ae_assert(xc->cnt>=k, "Spline1DFitCubicWC: Length(XC)<K!", _state);
ae_assert(yc->cnt>=k, "Spline1DFitCubicWC: Length(YC)<K!", _state);
ae_assert(dc->cnt>=k, "Spline1DFitCubicWC: Length(DC)<K!", _state);
ae_assert(isfinitevector(x, n, _state), "Spline1DFitCubicWC: X contains infinite or NAN values!", _state);
ae_assert(isfinitevector(y, n, _state), "Spline1DFitCubicWC: Y contains infinite or NAN values!", _state);
ae_assert(isfinitevector(w, n, _state), "Spline1DFitCubicWC: Y contains infinite or NAN values!", _state);
ae_assert(isfinitevector(xc, k, _state), "Spline1DFitCubicWC: X contains infinite or NAN values!", _state);
ae_assert(isfinitevector(yc, k, _state), "Spline1DFitCubicWC: Y contains infinite or NAN values!", _state);
for(i=0; i<=k-1; i++)
{
ae_assert(dc->ptr.p_int[i]==0||dc->ptr.p_int[i]==1, "Spline1DFitCubicWC: DC[i] is neither 0 or 1!", _state);
}
lsfit_spline1dfitinternal(0, x, y, w, n, xc, yc, dc, k, m, info, s, rep, _state);
}
/*************************************************************************
Weighted fitting by Hermite spline, with constraints on function values
or first derivatives.
Equidistant grid with M nodes on [min(x,xc),max(x,xc)] is used to build
basis functions. Basis functions are Hermite splines. Small regularizing
term is used when solving constrained tasks (to improve stability).
Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver
SEE ALSO
Spline1DFitCubicWC() - fitting by Cubic splines (less flexible,
more smooth)
Spline1DFitHermite() - "lightweight" Hermite fitting, without
invididual weights and constraints
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
W - weights, array[0..N-1]
Each summand in square sum of approximation deviations from
given values is multiplied by the square of corresponding
weight. Fill it by 1's if you don't want to solve weighted
task.
N - number of points (optional):
* N>0
* if given, only first N elements of X/Y/W are processed
* if not given, automatically determined from X/Y/W sizes
XC - points where spline values/derivatives are constrained,
array[0..K-1].
YC - values of constraints, array[0..K-1]
DC - array[0..K-1], types of constraints:
* DC[i]=0 means that S(XC[i])=YC[i]
* DC[i]=1 means that S'(XC[i])=YC[i]
SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
K - number of constraints (optional):
* 0<=K<M.
* K=0 means no constraints (XC/YC/DC are not used)
* if given, only first K elements of XC/YC/DC are used
* if not given, automatically determined from XC/YC/DC
M - number of basis functions (= 2 * number of nodes),
M>=4,
M IS EVEN!
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearW() subroutine:
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
-3 means inconsistent constraints
-2 means odd M was passed (which is not supported)
-1 means another errors in parameters passed
(N<=0, for example)
S - spline interpolant.
Rep - report, same format as in LSFitLinearW() subroutine.
Following fields are set:
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
IMPORTANT:
this subroitine supports only even M's
ORDER OF POINTS
Subroutine automatically sorts points, so caller may pass unsorted array.
SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:
Setting constraints can lead to undesired results, like ill-conditioned
behavior, or inconsistency being detected. From the other side, it allows
us to improve quality of the fit. Here we summarize our experience with
constrained regression splines:
* excessive constraints can be inconsistent. Splines are piecewise cubic
functions, and it is easy to create an example, where large number of
constraints concentrated in small area will result in inconsistency.
Just because spline is not flexible enough to satisfy all of them. And
same constraints spread across the [min(x),max(x)] will be perfectly
consistent.
* the more evenly constraints are spread across [min(x),max(x)], the more
chances that they will be consistent
* the greater is M (given fixed constraints), the more chances that
constraints will be consistent
* in the general case, consistency of constraints is NOT GUARANTEED.
* in the several special cases, however, we can guarantee consistency.
* one of this cases is M>=4 and constraints on the function value
(AND/OR its derivative) at the interval boundaries.
* another special case is M>=4 and ONE constraint on the function value
(OR, BUT NOT AND, derivative) anywhere in [min(x),max(x)]
Our final recommendation is to use constraints WHEN AND ONLY when you
can't solve your task without them. Anything beyond special cases given
above is not guaranteed and may result in inconsistency.
-- ALGLIB PROJECT --
Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfithermitewc(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
ae_int_t n,
/* Real */ ae_vector* xc,
/* Real */ ae_vector* yc,
/* Integer */ ae_vector* dc,
ae_int_t k,
ae_int_t m,
ae_int_t* info,
spline1dinterpolant* s,
spline1dfitreport* rep,
ae_state *_state)
{
ae_int_t i;
*info = 0;
_spline1dinterpolant_clear(s);
_spline1dfitreport_clear(rep);
ae_assert(n>=1, "Spline1DFitHermiteWC: N<1!", _state);
ae_assert(m>=4, "Spline1DFitHermiteWC: M<4!", _state);
ae_assert(m%2==0, "Spline1DFitHermiteWC: M is odd!", _state);
ae_assert(k>=0, "Spline1DFitHermiteWC: K<0!", _state);
ae_assert(k<m, "Spline1DFitHermiteWC: K>=M!", _state);
ae_assert(x->cnt>=n, "Spline1DFitHermiteWC: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "Spline1DFitHermiteWC: Length(Y)<N!", _state);
ae_assert(w->cnt>=n, "Spline1DFitHermiteWC: Length(W)<N!", _state);
ae_assert(xc->cnt>=k, "Spline1DFitHermiteWC: Length(XC)<K!", _state);
ae_assert(yc->cnt>=k, "Spline1DFitHermiteWC: Length(YC)<K!", _state);
ae_assert(dc->cnt>=k, "Spline1DFitHermiteWC: Length(DC)<K!", _state);
ae_assert(isfinitevector(x, n, _state), "Spline1DFitHermiteWC: X contains infinite or NAN values!", _state);
ae_assert(isfinitevector(y, n, _state), "Spline1DFitHermiteWC: Y contains infinite or NAN values!", _state);
ae_assert(isfinitevector(w, n, _state), "Spline1DFitHermiteWC: Y contains infinite or NAN values!", _state);
ae_assert(isfinitevector(xc, k, _state), "Spline1DFitHermiteWC: X contains infinite or NAN values!", _state);
ae_assert(isfinitevector(yc, k, _state), "Spline1DFitHermiteWC: Y contains infinite or NAN values!", _state);
for(i=0; i<=k-1; i++)
{
ae_assert(dc->ptr.p_int[i]==0||dc->ptr.p_int[i]==1, "Spline1DFitHermiteWC: DC[i] is neither 0 or 1!", _state);
}
lsfit_spline1dfitinternal(1, x, y, w, n, xc, yc, dc, k, m, info, s, rep, _state);
}
/*************************************************************************
Least squares fitting by cubic spline.
This subroutine is "lightweight" alternative for more complex and feature-
rich Spline1DFitCubicWC(). See Spline1DFitCubicWC() for more information
about subroutine parameters (we don't duplicate it here because of length)
-- ALGLIB PROJECT --
Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfitcubic(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_int_t m,
ae_int_t* info,
spline1dinterpolant* s,
spline1dfitreport* rep,
ae_state *_state)
{
ae_frame _frame_block;
ae_int_t i;
ae_vector w;
ae_vector xc;
ae_vector yc;
ae_vector dc;
ae_frame_make(_state, &_frame_block);
*info = 0;
_spline1dinterpolant_clear(s);
_spline1dfitreport_clear(rep);
ae_vector_init(&w, 0, DT_REAL, _state, ae_true);
ae_vector_init(&xc, 0, DT_REAL, _state, ae_true);
ae_vector_init(&yc, 0, DT_REAL, _state, ae_true);
ae_vector_init(&dc, 0, DT_INT, _state, ae_true);
ae_assert(n>=1, "Spline1DFitCubic: N<1!", _state);
ae_assert(m>=4, "Spline1DFitCubic: M<4!", _state);
ae_assert(x->cnt>=n, "Spline1DFitCubic: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "Spline1DFitCubic: Length(Y)<N!", _state);
ae_assert(isfinitevector(x, n, _state), "Spline1DFitCubic: X contains infinite or NAN values!", _state);
ae_assert(isfinitevector(y, n, _state), "Spline1DFitCubic: Y contains infinite or NAN values!", _state);
ae_vector_set_length(&w, n, _state);
for(i=0; i<=n-1; i++)
{
w.ptr.p_double[i] = 1;
}
spline1dfitcubicwc(x, y, &w, n, &xc, &yc, &dc, 0, m, info, s, rep, _state);
ae_frame_leave(_state);
}
/*************************************************************************
Least squares fitting by Hermite spline.
This subroutine is "lightweight" alternative for more complex and feature-
rich Spline1DFitHermiteWC(). See Spline1DFitHermiteWC() description for
more information about subroutine parameters (we don't duplicate it here
because of length).
-- ALGLIB PROJECT --
Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfithermite(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_int_t m,
ae_int_t* info,
spline1dinterpolant* s,
spline1dfitreport* rep,
ae_state *_state)
{
ae_frame _frame_block;
ae_int_t i;
ae_vector w;
ae_vector xc;
ae_vector yc;
ae_vector dc;
ae_frame_make(_state, &_frame_block);
*info = 0;
_spline1dinterpolant_clear(s);
_spline1dfitreport_clear(rep);
ae_vector_init(&w, 0, DT_REAL, _state, ae_true);
ae_vector_init(&xc, 0, DT_REAL, _state, ae_true);
ae_vector_init(&yc, 0, DT_REAL, _state, ae_true);
ae_vector_init(&dc, 0, DT_INT, _state, ae_true);
ae_assert(n>=1, "Spline1DFitHermite: N<1!", _state);
ae_assert(m>=4, "Spline1DFitHermite: M<4!", _state);
ae_assert(m%2==0, "Spline1DFitHermite: M is odd!", _state);
ae_assert(x->cnt>=n, "Spline1DFitHermite: Length(X)<N!", _state);
ae_assert(y->cnt>=n, "Spline1DFitHermite: Length(Y)<N!", _state);
ae_assert(isfinitevector(x, n, _state), "Spline1DFitHermite: X contains infinite or NAN values!", _state);
ae_assert(isfinitevector(y, n, _state), "Spline1DFitHermite: Y contains infinite or NAN values!", _state);
ae_vector_set_length(&w, n, _state);
for(i=0; i<=n-1; i++)
{
w.ptr.p_double[i] = 1;
}
spline1dfithermitewc(x, y, &w, n, &xc, &yc, &dc, 0, m, info, s, rep, _state);
ae_frame_leave(_state);
}
/*************************************************************************
Weighted linear least squares fitting.
QR decomposition is used to reduce task to MxM, then triangular solver or
SVD-based solver is used depending on condition number of the system. It
allows to maximize speed and retain decent accuracy.
IMPORTANT: if you want to perform polynomial fitting, it may be more
convenient to use PolynomialFit() function. This function gives
best results on polynomial problems and solves numerical
stability issues which arise when you fit high-degree
polynomials to your data.
INPUT PARAMETERS:
Y - array[0..N-1] Function values in N points.
W - array[0..N-1] Weights corresponding to function values.
Each summand in square sum of approximation deviations
from given values is multiplied by the square of
corresponding weight.
FMatrix - a table of basis functions values, array[0..N-1, 0..M-1].
FMatrix[I, J] - value of J-th basis function in I-th point.
N - number of points used. N>=1.
M - number of basis functions, M>=1.
OUTPUT PARAMETERS:
Info - error code:
* -4 internal SVD decomposition subroutine failed (very
rare and for degenerate systems only)
* -1 incorrect N/M were specified
* 1 task is solved
C - decomposition coefficients, array[0..M-1]
Rep - fitting report. Following fields are set:
* Rep.TaskRCond reciprocal of condition number
* R2 non-adjusted coefficient of determination
(non-weighted)
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
ERRORS IN PARAMETERS
This solver also calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar covariance matrix for parameters, array[K,K].
* Rep.ErrPar errors in parameters, array[K],
errpar = sqrt(diag(CovPar))
* Rep.ErrCurve vector of fit errors - standard deviations of empirical
best-fit curve from "ideal" best-fit curve built with
infinite number of samples, array[N].
errcurve = sqrt(diag(F*CovPar*F')),
where F is functions matrix.
* Rep.Noise vector of per-point estimates of noise, array[N]
NOTE: noise in the data is estimated as follows:
* for fitting without user-supplied weights all points are
assumed to have same level of noise, which is estimated from
the data
* for fitting with user-supplied weights we assume that noise
level in I-th point is inversely proportional to Ith weight.
Coefficient of proportionality is estimated from the data.
NOTE: we apply small amount of regularization when we invert squared
Jacobian and calculate covariance matrix. It guarantees that
algorithm won't divide by zero during inversion, but skews
error estimates a bit (fractional error is about 10^-9).
However, we believe that this difference is insignificant for
all practical purposes except for the situation when you want
to compare ALGLIB results with "reference" implementation up
to the last significant digit.
NOTE: covariance matrix is estimated using correction for degrees
of freedom (covariances are divided by N-M instead of dividing
by N).
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitlinearw(/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
/* Real */ ae_matrix* fmatrix,
ae_int_t n,
ae_int_t m,
ae_int_t* info,
/* Real */ ae_vector* c,
lsfitreport* rep,
ae_state *_state)
{
*info = 0;
ae_vector_clear(c);
_lsfitreport_clear(rep);
ae_assert(n>=1, "LSFitLinearW: N<1!", _state);
ae_assert(m>=1, "LSFitLinearW: M<1!", _state);
ae_assert(y->cnt>=n, "LSFitLinearW: length(Y)<N!", _state);
ae_assert(isfinitevector(y, n, _state), "LSFitLinearW: Y contains infinite or NaN values!", _state);
ae_assert(w->cnt>=n, "LSFitLinearW: length(W)<N!", _state);
ae_assert(isfinitevector(w, n, _state), "LSFitLinearW: W contains infinite or NaN values!", _state);
ae_assert(fmatrix->rows>=n, "LSFitLinearW: rows(FMatrix)<N!", _state);
ae_assert(fmatrix->cols>=m, "LSFitLinearW: cols(FMatrix)<M!", _state);
ae_assert(apservisfinitematrix(fmatrix, n, m, _state), "LSFitLinearW: FMatrix contains infinite or NaN values!", _state);
lsfit_lsfitlinearinternal(y, w, fmatrix, n, m, info, c, rep, _state);
}
/*************************************************************************
Weighted constained linear least squares fitting.
This is variation of LSFitLinearW(), which searchs for min|A*x=b| given
that K additional constaints C*x=bc are satisfied. It reduces original
task to modified one: min|B*y-d| WITHOUT constraints, then LSFitLinearW()
is called.
IMPORTANT: if you want to perform polynomial fitting, it may be more
convenient to use PolynomialFit() function. This function gives
best results on polynomial problems and solves numerical
stability issues which arise when you fit high-degree
polynomials to your data.
INPUT PARAMETERS:
Y - array[0..N-1] Function values in N points.
W - array[0..N-1] Weights corresponding to function values.
Each summand in square sum of approximation deviations
from given values is multiplied by the square of
corresponding weight.
FMatrix - a table of basis functions values, array[0..N-1, 0..M-1].
FMatrix[I,J] - value of J-th basis function in I-th point.
CMatrix - a table of constaints, array[0..K-1,0..M].
I-th row of CMatrix corresponds to I-th linear constraint:
CMatrix[I,0]*C[0] + ... + CMatrix[I,M-1]*C[M-1] = CMatrix[I,M]
N - number of points used. N>=1.
M - number of basis functions, M>=1.
K - number of constraints, 0 <= K < M
K=0 corresponds to absence of constraints.
OUTPUT PARAMETERS:
Info - error code:
* -4 internal SVD decomposition subroutine failed (very
rare and for degenerate systems only)
* -3 either too many constraints (M or more),
degenerate constraints (some constraints are
repetead twice) or inconsistent constraints were
specified.
* 1 task is solved
C - decomposition coefficients, array[0..M-1]
Rep - fitting report. Following fields are set:
* R2 non-adjusted coefficient of determination
(non-weighted)
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
ERRORS IN PARAMETERS
This solver also calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar covariance matrix for parameters, array[K,K].
* Rep.ErrPar errors in parameters, array[K],
errpar = sqrt(diag(CovPar))
* Rep.ErrCurve vector of fit errors - standard deviations of empirical
best-fit curve from "ideal" best-fit curve built with
infinite number of samples, array[N].
errcurve = sqrt(diag(F*CovPar*F')),
where F is functions matrix.
* Rep.Noise vector of per-point estimates of noise, array[N]
IMPORTANT: errors in parameters are calculated without taking into
account boundary/linear constraints! Presence of constraints
changes distribution of errors, but there is no easy way to
account for constraints when you calculate covariance matrix.
NOTE: noise in the data is estimated as follows:
* for fitting without user-supplied weights all points are
assumed to have same level of noise, which is estimated from
the data
* for fitting with user-supplied weights we assume that noise
level in I-th point is inversely proportional to Ith weight.
Coefficient of proportionality is estimated from the data.
NOTE: we apply small amount of regularization when we invert squared
Jacobian and calculate covariance matrix. It guarantees that
algorithm won't divide by zero during inversion, but skews
error estimates a bit (fractional error is about 10^-9).
However, we believe that this difference is insignificant for
all practical purposes except for the situation when you want
to compare ALGLIB results with "reference" implementation up
to the last significant digit.
NOTE: covariance matrix is estimated using correction for degrees
of freedom (covariances are divided by N-M instead of dividing
by N).
-- ALGLIB --
Copyright 07.09.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitlinearwc(/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
/* Real */ ae_matrix* fmatrix,
/* Real */ ae_matrix* cmatrix,
ae_int_t n,
ae_int_t m,
ae_int_t k,
ae_int_t* info,
/* Real */ ae_vector* c,
lsfitreport* rep,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _y;
ae_matrix _cmatrix;
ae_int_t i;
ae_int_t j;
ae_vector tau;
ae_matrix q;
ae_matrix f2;
ae_vector tmp;
ae_vector c0;
double v;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_y, y, _state, ae_true);
y = &_y;
ae_matrix_init_copy(&_cmatrix, cmatrix, _state, ae_true);
cmatrix = &_cmatrix;
*info = 0;
ae_vector_clear(c);
_lsfitreport_clear(rep);
ae_vector_init(&tau, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&q, 0, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&f2, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmp, 0, DT_REAL, _state, ae_true);
ae_vector_init(&c0, 0, DT_REAL, _state, ae_true);
ae_assert(n>=1, "LSFitLinearWC: N<1!", _state);
ae_assert(m>=1, "LSFitLinearWC: M<1!", _state);
ae_assert(k>=0, "LSFitLinearWC: K<0!", _state);
ae_assert(y->cnt>=n, "LSFitLinearWC: length(Y)<N!", _state);
ae_assert(isfinitevector(y, n, _state), "LSFitLinearWC: Y contains infinite or NaN values!", _state);
ae_assert(w->cnt>=n, "LSFitLinearWC: length(W)<N!", _state);
ae_assert(isfinitevector(w, n, _state), "LSFitLinearWC: W contains infinite or NaN values!", _state);
ae_assert(fmatrix->rows>=n, "LSFitLinearWC: rows(FMatrix)<N!", _state);
ae_assert(fmatrix->cols>=m, "LSFitLinearWC: cols(FMatrix)<M!", _state);
ae_assert(apservisfinitematrix(fmatrix, n, m, _state), "LSFitLinearWC: FMatrix contains infinite or NaN values!", _state);
ae_assert(cmatrix->rows>=k, "LSFitLinearWC: rows(CMatrix)<K!", _state);
ae_assert(cmatrix->cols>=m+1||k==0, "LSFitLinearWC: cols(CMatrix)<M+1!", _state);
ae_assert(apservisfinitematrix(cmatrix, k, m+1, _state), "LSFitLinearWC: CMatrix contains infinite or NaN values!", _state);
if( k>=m )
{
*info = -3;
ae_frame_leave(_state);
return;
}
/*
* Solve
*/
if( k==0 )
{
/*
* no constraints
*/
lsfit_lsfitlinearinternal(y, w, fmatrix, n, m, info, c, rep, _state);
}
else
{
/*
* First, find general form solution of constraints system:
* * factorize C = L*Q
* * unpack Q
* * fill upper part of C with zeros (for RCond)
*
* We got C=C0+Q2'*y where Q2 is lower M-K rows of Q.
*/
rmatrixlq(cmatrix, k, m, &tau, _state);
rmatrixlqunpackq(cmatrix, k, m, &tau, m, &q, _state);
for(i=0; i<=k-1; i++)
{
for(j=i+1; j<=m-1; j++)
{
cmatrix->ptr.pp_double[i][j] = 0.0;
}
}
if( ae_fp_less(rmatrixlurcondinf(cmatrix, k, _state),1000*ae_machineepsilon) )
{
*info = -3;
ae_frame_leave(_state);
return;
}
ae_vector_set_length(&tmp, k, _state);
for(i=0; i<=k-1; i++)
{
if( i>0 )
{
v = ae_v_dotproduct(&cmatrix->ptr.pp_double[i][0], 1, &tmp.ptr.p_double[0], 1, ae_v_len(0,i-1));
}
else
{
v = 0;
}
tmp.ptr.p_double[i] = (cmatrix->ptr.pp_double[i][m]-v)/cmatrix->ptr.pp_double[i][i];
}
ae_vector_set_length(&c0, m, _state);
for(i=0; i<=m-1; i++)
{
c0.ptr.p_double[i] = 0;
}
for(i=0; i<=k-1; i++)
{
v = tmp.ptr.p_double[i];
ae_v_addd(&c0.ptr.p_double[0], 1, &q.ptr.pp_double[i][0], 1, ae_v_len(0,m-1), v);
}
/*
* Second, prepare modified matrix F2 = F*Q2' and solve modified task
*/
ae_vector_set_length(&tmp, ae_maxint(n, m, _state)+1, _state);
ae_matrix_set_length(&f2, n, m-k, _state);
matrixvectormultiply(fmatrix, 0, n-1, 0, m-1, ae_false, &c0, 0, m-1, -1.0, y, 0, n-1, 1.0, _state);
matrixmatrixmultiply(fmatrix, 0, n-1, 0, m-1, ae_false, &q, k, m-1, 0, m-1, ae_true, 1.0, &f2, 0, n-1, 0, m-k-1, 0.0, &tmp, _state);
lsfit_lsfitlinearinternal(y, w, &f2, n, m-k, info, &tmp, rep, _state);
rep->taskrcond = -1;
if( *info<=0 )
{
ae_frame_leave(_state);
return;
}
/*
* then, convert back to original answer: C = C0 + Q2'*Y0
*/
ae_vector_set_length(c, m, _state);
ae_v_move(&c->ptr.p_double[0], 1, &c0.ptr.p_double[0], 1, ae_v_len(0,m-1));
matrixvectormultiply(&q, k, m-1, 0, m-1, ae_true, &tmp, 0, m-k-1, 1.0, c, 0, m-1, 1.0, _state);
}
ae_frame_leave(_state);
}
/*************************************************************************
Linear least squares fitting.
QR decomposition is used to reduce task to MxM, then triangular solver or
SVD-based solver is used depending on condition number of the system. It
allows to maximize speed and retain decent accuracy.
IMPORTANT: if you want to perform polynomial fitting, it may be more
convenient to use PolynomialFit() function. This function gives
best results on polynomial problems and solves numerical
stability issues which arise when you fit high-degree
polynomials to your data.
INPUT PARAMETERS:
Y - array[0..N-1] Function values in N points.
FMatrix - a table of basis functions values, array[0..N-1, 0..M-1].
FMatrix[I, J] - value of J-th basis function in I-th point.
N - number of points used. N>=1.
M - number of basis functions, M>=1.
OUTPUT PARAMETERS:
Info - error code:
* -4 internal SVD decomposition subroutine failed (very
rare and for degenerate systems only)
* 1 task is solved
C - decomposition coefficients, array[0..M-1]
Rep - fitting report. Following fields are set:
* Rep.TaskRCond reciprocal of condition number
* R2 non-adjusted coefficient of determination
(non-weighted)
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
ERRORS IN PARAMETERS
This solver also calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar covariance matrix for parameters, array[K,K].
* Rep.ErrPar errors in parameters, array[K],
errpar = sqrt(diag(CovPar))
* Rep.ErrCurve vector of fit errors - standard deviations of empirical
best-fit curve from "ideal" best-fit curve built with
infinite number of samples, array[N].
errcurve = sqrt(diag(F*CovPar*F')),
where F is functions matrix.
* Rep.Noise vector of per-point estimates of noise, array[N]
NOTE: noise in the data is estimated as follows:
* for fitting without user-supplied weights all points are
assumed to have same level of noise, which is estimated from
the data
* for fitting with user-supplied weights we assume that noise
level in I-th point is inversely proportional to Ith weight.
Coefficient of proportionality is estimated from the data.
NOTE: we apply small amount of regularization when we invert squared
Jacobian and calculate covariance matrix. It guarantees that
algorithm won't divide by zero during inversion, but skews
error estimates a bit (fractional error is about 10^-9).
However, we believe that this difference is insignificant for
all practical purposes except for the situation when you want
to compare ALGLIB results with "reference" implementation up
to the last significant digit.
NOTE: covariance matrix is estimated using correction for degrees
of freedom (covariances are divided by N-M instead of dividing
by N).
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitlinear(/* Real */ ae_vector* y,
/* Real */ ae_matrix* fmatrix,
ae_int_t n,
ae_int_t m,
ae_int_t* info,
/* Real */ ae_vector* c,
lsfitreport* rep,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector w;
ae_int_t i;
ae_frame_make(_state, &_frame_block);
*info = 0;
ae_vector_clear(c);
_lsfitreport_clear(rep);
ae_vector_init(&w, 0, DT_REAL, _state, ae_true);
ae_assert(n>=1, "LSFitLinear: N<1!", _state);
ae_assert(m>=1, "LSFitLinear: M<1!", _state);
ae_assert(y->cnt>=n, "LSFitLinear: length(Y)<N!", _state);
ae_assert(isfinitevector(y, n, _state), "LSFitLinear: Y contains infinite or NaN values!", _state);
ae_assert(fmatrix->rows>=n, "LSFitLinear: rows(FMatrix)<N!", _state);
ae_assert(fmatrix->cols>=m, "LSFitLinear: cols(FMatrix)<M!", _state);
ae_assert(apservisfinitematrix(fmatrix, n, m, _state), "LSFitLinear: FMatrix contains infinite or NaN values!", _state);
ae_vector_set_length(&w, n, _state);
for(i=0; i<=n-1; i++)
{
w.ptr.p_double[i] = 1;
}
lsfit_lsfitlinearinternal(y, &w, fmatrix, n, m, info, c, rep, _state);
ae_frame_leave(_state);
}
/*************************************************************************
Constained linear least squares fitting.
This is variation of LSFitLinear(), which searchs for min|A*x=b| given
that K additional constaints C*x=bc are satisfied. It reduces original
task to modified one: min|B*y-d| WITHOUT constraints, then LSFitLinear()
is called.
IMPORTANT: if you want to perform polynomial fitting, it may be more
convenient to use PolynomialFit() function. This function gives
best results on polynomial problems and solves numerical
stability issues which arise when you fit high-degree
polynomials to your data.
INPUT PARAMETERS:
Y - array[0..N-1] Function values in N points.
FMatrix - a table of basis functions values, array[0..N-1, 0..M-1].
FMatrix[I,J] - value of J-th basis function in I-th point.
CMatrix - a table of constaints, array[0..K-1,0..M].
I-th row of CMatrix corresponds to I-th linear constraint:
CMatrix[I,0]*C[0] + ... + CMatrix[I,M-1]*C[M-1] = CMatrix[I,M]
N - number of points used. N>=1.
M - number of basis functions, M>=1.
K - number of constraints, 0 <= K < M
K=0 corresponds to absence of constraints.
OUTPUT PARAMETERS:
Info - error code:
* -4 internal SVD decomposition subroutine failed (very
rare and for degenerate systems only)
* -3 either too many constraints (M or more),
degenerate constraints (some constraints are
repetead twice) or inconsistent constraints were
specified.
* 1 task is solved
C - decomposition coefficients, array[0..M-1]
Rep - fitting report. Following fields are set:
* R2 non-adjusted coefficient of determination
(non-weighted)
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
ERRORS IN PARAMETERS
This solver also calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar covariance matrix for parameters, array[K,K].
* Rep.ErrPar errors in parameters, array[K],
errpar = sqrt(diag(CovPar))
* Rep.ErrCurve vector of fit errors - standard deviations of empirical
best-fit curve from "ideal" best-fit curve built with
infinite number of samples, array[N].
errcurve = sqrt(diag(F*CovPar*F')),
where F is functions matrix.
* Rep.Noise vector of per-point estimates of noise, array[N]
IMPORTANT: errors in parameters are calculated without taking into
account boundary/linear constraints! Presence of constraints
changes distribution of errors, but there is no easy way to
account for constraints when you calculate covariance matrix.
NOTE: noise in the data is estimated as follows:
* for fitting without user-supplied weights all points are
assumed to have same level of noise, which is estimated from
the data
* for fitting with user-supplied weights we assume that noise
level in I-th point is inversely proportional to Ith weight.
Coefficient of proportionality is estimated from the data.
NOTE: we apply small amount of regularization when we invert squared
Jacobian and calculate covariance matrix. It guarantees that
algorithm won't divide by zero during inversion, but skews
error estimates a bit (fractional error is about 10^-9).
However, we believe that this difference is insignificant for
all practical purposes except for the situation when you want
to compare ALGLIB results with "reference" implementation up
to the last significant digit.
NOTE: covariance matrix is estimated using correction for degrees
of freedom (covariances are divided by N-M instead of dividing
by N).
-- ALGLIB --
Copyright 07.09.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitlinearc(/* Real */ ae_vector* y,
/* Real */ ae_matrix* fmatrix,
/* Real */ ae_matrix* cmatrix,
ae_int_t n,
ae_int_t m,
ae_int_t k,
ae_int_t* info,
/* Real */ ae_vector* c,
lsfitreport* rep,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _y;
ae_vector w;
ae_int_t i;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_y, y, _state, ae_true);
y = &_y;
*info = 0;
ae_vector_clear(c);
_lsfitreport_clear(rep);
ae_vector_init(&w, 0, DT_REAL, _state, ae_true);
ae_assert(n>=1, "LSFitLinearC: N<1!", _state);
ae_assert(m>=1, "LSFitLinearC: M<1!", _state);
ae_assert(k>=0, "LSFitLinearC: K<0!", _state);
ae_assert(y->cnt>=n, "LSFitLinearC: length(Y)<N!", _state);
ae_assert(isfinitevector(y, n, _state), "LSFitLinearC: Y contains infinite or NaN values!", _state);
ae_assert(fmatrix->rows>=n, "LSFitLinearC: rows(FMatrix)<N!", _state);
ae_assert(fmatrix->cols>=m, "LSFitLinearC: cols(FMatrix)<M!", _state);
ae_assert(apservisfinitematrix(fmatrix, n, m, _state), "LSFitLinearC: FMatrix contains infinite or NaN values!", _state);
ae_assert(cmatrix->rows>=k, "LSFitLinearC: rows(CMatrix)<K!", _state);
ae_assert(cmatrix->cols>=m+1||k==0, "LSFitLinearC: cols(CMatrix)<M+1!", _state);
ae_assert(apservisfinitematrix(cmatrix, k, m+1, _state), "LSFitLinearC: CMatrix contains infinite or NaN values!", _state);
ae_vector_set_length(&w, n, _state);
for(i=0; i<=n-1; i++)
{
w.ptr.p_double[i] = 1;
}
lsfitlinearwc(y, &w, fmatrix, cmatrix, n, m, k, info, c, rep, _state);
ae_frame_leave(_state);
}
/*************************************************************************
Weighted nonlinear least squares fitting using function values only.
Combination of numerical differentiation and secant updates is used to
obtain function Jacobian.
Nonlinear task min(F(c)) is solved, where
F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2,
* N is a number of points,
* M is a dimension of a space points belong to,
* K is a dimension of a space of parameters being fitted,
* w is an N-dimensional vector of weight coefficients,
* x is a set of N points, each of them is an M-dimensional vector,
* c is a K-dimensional vector of parameters being fitted
This subroutine uses only f(c,x[i]).
INPUT PARAMETERS:
X - array[0..N-1,0..M-1], points (one row = one point)
Y - array[0..N-1], function values.
W - weights, array[0..N-1]
C - array[0..K-1], initial approximation to the solution,
N - number of points, N>1
M - dimension of space
K - number of parameters being fitted
DiffStep- numerical differentiation step;
should not be very small or large;
large = loss of accuracy
small = growth of round-off errors
OUTPUT PARAMETERS:
State - structure which stores algorithm state
-- ALGLIB --
Copyright 18.10.2008 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatewf(/* Real */ ae_matrix* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
/* Real */ ae_vector* c,
ae_int_t n,
ae_int_t m,
ae_int_t k,
double diffstep,
lsfitstate* state,
ae_state *_state)
{
ae_int_t i;
_lsfitstate_clear(state);
ae_assert(n>=1, "LSFitCreateWF: N<1!", _state);
ae_assert(m>=1, "LSFitCreateWF: M<1!", _state);
ae_assert(k>=1, "LSFitCreateWF: K<1!", _state);
ae_assert(c->cnt>=k, "LSFitCreateWF: length(C)<K!", _state);
ae_assert(isfinitevector(c, k, _state), "LSFitCreateWF: C contains infinite or NaN values!", _state);
ae_assert(y->cnt>=n, "LSFitCreateWF: length(Y)<N!", _state);
ae_assert(isfinitevector(y, n, _state), "LSFitCreateWF: Y contains infinite or NaN values!", _state);
ae_assert(w->cnt>=n, "LSFitCreateWF: length(W)<N!", _state);
ae_assert(isfinitevector(w, n, _state), "LSFitCreateWF: W contains infinite or NaN values!", _state);
ae_assert(x->rows>=n, "LSFitCreateWF: rows(X)<N!", _state);
ae_assert(x->cols>=m, "LSFitCreateWF: cols(X)<M!", _state);
ae_assert(apservisfinitematrix(x, n, m, _state), "LSFitCreateWF: X contains infinite or NaN values!", _state);
ae_assert(ae_isfinite(diffstep, _state), "LSFitCreateWF: DiffStep is not finite!", _state);
ae_assert(ae_fp_greater(diffstep,0), "LSFitCreateWF: DiffStep<=0!", _state);
state->teststep = 0;
state->diffstep = diffstep;
state->npoints = n;
state->nweights = n;
state->wkind = 1;
state->m = m;
state->k = k;
lsfitsetcond(state, 0.0, 0.0, 0, _state);
lsfitsetstpmax(state, 0.0, _state);
lsfitsetxrep(state, ae_false, _state);
ae_matrix_set_length(&state->taskx, n, m, _state);
ae_vector_set_length(&state->tasky, n, _state);
ae_vector_set_length(&state->taskw, n, _state);
ae_vector_set_length(&state->c, k, _state);
ae_vector_set_length(&state->x, m, _state);
ae_v_move(&state->c.ptr.p_double[0], 1, &c->ptr.p_double[0], 1, ae_v_len(0,k-1));
ae_v_move(&state->taskw.ptr.p_double[0], 1, &w->ptr.p_double[0], 1, ae_v_len(0,n-1));
for(i=0; i<=n-1; i++)
{
ae_v_move(&state->taskx.ptr.pp_double[i][0], 1, &x->ptr.pp_double[i][0], 1, ae_v_len(0,m-1));
state->tasky.ptr.p_double[i] = y->ptr.p_double[i];
}
ae_vector_set_length(&state->s, k, _state);
ae_vector_set_length(&state->bndl, k, _state);
ae_vector_set_length(&state->bndu, k, _state);
for(i=0; i<=k-1; i++)
{
state->s.ptr.p_double[i] = 1.0;
state->bndl.ptr.p_double[i] = _state->v_neginf;
state->bndu.ptr.p_double[i] = _state->v_posinf;
}
state->optalgo = 0;
state->prevnpt = -1;
state->prevalgo = -1;
minlmcreatev(k, n, &state->c, diffstep, &state->optstate, _state);
lsfit_lsfitclearrequestfields(state, _state);
ae_vector_set_length(&state->rstate.ia, 6+1, _state);
ae_vector_set_length(&state->rstate.ra, 8+1, _state);
state->rstate.stage = -1;
}
/*************************************************************************
Nonlinear least squares fitting using function values only.
Combination of numerical differentiation and secant updates is used to
obtain function Jacobian.
Nonlinear task min(F(c)) is solved, where
F(c) = (f(c,x[0])-y[0])^2 + ... + (f(c,x[n-1])-y[n-1])^2,
* N is a number of points,
* M is a dimension of a space points belong to,
* K is a dimension of a space of parameters being fitted,
* w is an N-dimensional vector of weight coefficients,
* x is a set of N points, each of them is an M-dimensional vector,
* c is a K-dimensional vector of parameters being fitted
This subroutine uses only f(c,x[i]).
INPUT PARAMETERS:
X - array[0..N-1,0..M-1], points (one row = one point)
Y - array[0..N-1], function values.
C - array[0..K-1], initial approximation to the solution,
N - number of points, N>1
M - dimension of space
K - number of parameters being fitted
DiffStep- numerical differentiation step;
should not be very small or large;
large = loss of accuracy
small = growth of round-off errors
OUTPUT PARAMETERS:
State - structure which stores algorithm state
-- ALGLIB --
Copyright 18.10.2008 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatef(/* Real */ ae_matrix* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* c,
ae_int_t n,
ae_int_t m,
ae_int_t k,
double diffstep,
lsfitstate* state,
ae_state *_state)
{
ae_int_t i;
_lsfitstate_clear(state);
ae_assert(n>=1, "LSFitCreateF: N<1!", _state);
ae_assert(m>=1, "LSFitCreateF: M<1!", _state);
ae_assert(k>=1, "LSFitCreateF: K<1!", _state);
ae_assert(c->cnt>=k, "LSFitCreateF: length(C)<K!", _state);
ae_assert(isfinitevector(c, k, _state), "LSFitCreateF: C contains infinite or NaN values!", _state);
ae_assert(y->cnt>=n, "LSFitCreateF: length(Y)<N!", _state);
ae_assert(isfinitevector(y, n, _state), "LSFitCreateF: Y contains infinite or NaN values!", _state);
ae_assert(x->rows>=n, "LSFitCreateF: rows(X)<N!", _state);
ae_assert(x->cols>=m, "LSFitCreateF: cols(X)<M!", _state);
ae_assert(apservisfinitematrix(x, n, m, _state), "LSFitCreateF: X contains infinite or NaN values!", _state);
ae_assert(x->rows>=n, "LSFitCreateF: rows(X)<N!", _state);
ae_assert(x->cols>=m, "LSFitCreateF: cols(X)<M!", _state);
ae_assert(apservisfinitematrix(x, n, m, _state), "LSFitCreateF: X contains infinite or NaN values!", _state);
ae_assert(ae_isfinite(diffstep, _state), "LSFitCreateF: DiffStep is not finite!", _state);
ae_assert(ae_fp_greater(diffstep,0), "LSFitCreateF: DiffStep<=0!", _state);
state->teststep = 0;
state->diffstep = diffstep;
state->npoints = n;
state->wkind = 0;
state->m = m;
state->k = k;
lsfitsetcond(state, 0.0, 0.0, 0, _state);
lsfitsetstpmax(state, 0.0, _state);
lsfitsetxrep(state, ae_false, _state);
ae_matrix_set_length(&state->taskx, n, m, _state);
ae_vector_set_length(&state->tasky, n, _state);
ae_vector_set_length(&state->c, k, _state);
ae_vector_set_length(&state->x, m, _state);
ae_v_move(&state->c.ptr.p_double[0], 1, &c->ptr.p_double[0], 1, ae_v_len(0,k-1));
for(i=0; i<=n-1; i++)
{
ae_v_move(&state->taskx.ptr.pp_double[i][0], 1, &x->ptr.pp_double[i][0], 1, ae_v_len(0,m-1));
state->tasky.ptr.p_double[i] = y->ptr.p_double[i];
}
ae_vector_set_length(&state->s, k, _state);
ae_vector_set_length(&state->bndl, k, _state);
ae_vector_set_length(&state->bndu, k, _state);
for(i=0; i<=k-1; i++)
{
state->s.ptr.p_double[i] = 1.0;
state->bndl.ptr.p_double[i] = _state->v_neginf;
state->bndu.ptr.p_double[i] = _state->v_posinf;
}
state->optalgo = 0;
state->prevnpt = -1;
state->prevalgo = -1;
minlmcreatev(k, n, &state->c, diffstep, &state->optstate, _state);
lsfit_lsfitclearrequestfields(state, _state);
ae_vector_set_length(&state->rstate.ia, 6+1, _state);
ae_vector_set_length(&state->rstate.ra, 8+1, _state);
state->rstate.stage = -1;
}
/*************************************************************************
Weighted nonlinear least squares fitting using gradient only.
Nonlinear task min(F(c)) is solved, where
F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2,
* N is a number of points,
* M is a dimension of a space points belong to,
* K is a dimension of a space of parameters being fitted,
* w is an N-dimensional vector of weight coefficients,
* x is a set of N points, each of them is an M-dimensional vector,
* c is a K-dimensional vector of parameters being fitted
This subroutine uses only f(c,x[i]) and its gradient.
INPUT PARAMETERS:
X - array[0..N-1,0..M-1], points (one row = one point)
Y - array[0..N-1], function values.
W - weights, array[0..N-1]
C - array[0..K-1], initial approximation to the solution,
N - number of points, N>1
M - dimension of space
K - number of parameters being fitted
CheapFG - boolean flag, which is:
* True if both function and gradient calculation complexity
are less than O(M^2). An improved algorithm can
be used which corresponds to FGJ scheme from
MINLM unit.
* False otherwise.
Standard Jacibian-bases Levenberg-Marquardt algo
will be used (FJ scheme).
OUTPUT PARAMETERS:
State - structure which stores algorithm state
See also:
LSFitResults
LSFitCreateFG (fitting without weights)
LSFitCreateWFGH (fitting using Hessian)
LSFitCreateFGH (fitting using Hessian, without weights)
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatewfg(/* Real */ ae_matrix* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
/* Real */ ae_vector* c,
ae_int_t n,
ae_int_t m,
ae_int_t k,
ae_bool cheapfg,
lsfitstate* state,
ae_state *_state)
{
ae_int_t i;
_lsfitstate_clear(state);
ae_assert(n>=1, "LSFitCreateWFG: N<1!", _state);
ae_assert(m>=1, "LSFitCreateWFG: M<1!", _state);
ae_assert(k>=1, "LSFitCreateWFG: K<1!", _state);
ae_assert(c->cnt>=k, "LSFitCreateWFG: length(C)<K!", _state);
ae_assert(isfinitevector(c, k, _state), "LSFitCreateWFG: C contains infinite or NaN values!", _state);
ae_assert(y->cnt>=n, "LSFitCreateWFG: length(Y)<N!", _state);
ae_assert(isfinitevector(y, n, _state), "LSFitCreateWFG: Y contains infinite or NaN values!", _state);
ae_assert(w->cnt>=n, "LSFitCreateWFG: length(W)<N!", _state);
ae_assert(isfinitevector(w, n, _state), "LSFitCreateWFG: W contains infinite or NaN values!", _state);
ae_assert(x->rows>=n, "LSFitCreateWFG: rows(X)<N!", _state);
ae_assert(x->cols>=m, "LSFitCreateWFG: cols(X)<M!", _state);
ae_assert(apservisfinitematrix(x, n, m, _state), "LSFitCreateWFG: X contains infinite or NaN values!", _state);
state->teststep = 0;
state->diffstep = 0;
state->npoints = n;
state->nweights = n;
state->wkind = 1;
state->m = m;
state->k = k;
lsfitsetcond(state, 0.0, 0.0, 0, _state);
lsfitsetstpmax(state, 0.0, _state);
lsfitsetxrep(state, ae_false, _state);
ae_matrix_set_length(&state->taskx, n, m, _state);
ae_vector_set_length(&state->tasky, n, _state);
ae_vector_set_length(&state->taskw, n, _state);
ae_vector_set_length(&state->c, k, _state);
ae_vector_set_length(&state->x, m, _state);
ae_vector_set_length(&state->g, k, _state);
ae_v_move(&state->c.ptr.p_double[0], 1, &c->ptr.p_double[0], 1, ae_v_len(0,k-1));
ae_v_move(&state->taskw.ptr.p_double[0], 1, &w->ptr.p_double[0], 1, ae_v_len(0,n-1));
for(i=0; i<=n-1; i++)
{
ae_v_move(&state->taskx.ptr.pp_double[i][0], 1, &x->ptr.pp_double[i][0], 1, ae_v_len(0,m-1));
state->tasky.ptr.p_double[i] = y->ptr.p_double[i];
}
ae_vector_set_length(&state->s, k, _state);
ae_vector_set_length(&state->bndl, k, _state);
ae_vector_set_length(&state->bndu, k, _state);
for(i=0; i<=k-1; i++)
{
state->s.ptr.p_double[i] = 1.0;
state->bndl.ptr.p_double[i] = _state->v_neginf;
state->bndu.ptr.p_double[i] = _state->v_posinf;
}
state->optalgo = 1;
state->prevnpt = -1;
state->prevalgo = -1;
if( cheapfg )
{
minlmcreatevgj(k, n, &state->c, &state->optstate, _state);
}
else
{
minlmcreatevj(k, n, &state->c, &state->optstate, _state);
}
lsfit_lsfitclearrequestfields(state, _state);
ae_vector_set_length(&state->rstate.ia, 6+1, _state);
ae_vector_set_length(&state->rstate.ra, 8+1, _state);
state->rstate.stage = -1;
}
/*************************************************************************
Nonlinear least squares fitting using gradient only, without individual
weights.
Nonlinear task min(F(c)) is solved, where
F(c) = ((f(c,x[0])-y[0]))^2 + ... + ((f(c,x[n-1])-y[n-1]))^2,
* N is a number of points,
* M is a dimension of a space points belong to,
* K is a dimension of a space of parameters being fitted,
* x is a set of N points, each of them is an M-dimensional vector,
* c is a K-dimensional vector of parameters being fitted
This subroutine uses only f(c,x[i]) and its gradient.
INPUT PARAMETERS:
X - array[0..N-1,0..M-1], points (one row = one point)
Y - array[0..N-1], function values.
C - array[0..K-1], initial approximation to the solution,
N - number of points, N>1
M - dimension of space
K - number of parameters being fitted
CheapFG - boolean flag, which is:
* True if both function and gradient calculation complexity
are less than O(M^2). An improved algorithm can
be used which corresponds to FGJ scheme from
MINLM unit.
* False otherwise.
Standard Jacibian-bases Levenberg-Marquardt algo
will be used (FJ scheme).
OUTPUT PARAMETERS:
State - structure which stores algorithm state
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatefg(/* Real */ ae_matrix* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* c,
ae_int_t n,
ae_int_t m,
ae_int_t k,
ae_bool cheapfg,
lsfitstate* state,
ae_state *_state)
{
ae_int_t i;
_lsfitstate_clear(state);
ae_assert(n>=1, "LSFitCreateFG: N<1!", _state);
ae_assert(m>=1, "LSFitCreateFG: M<1!", _state);
ae_assert(k>=1, "LSFitCreateFG: K<1!", _state);
ae_assert(c->cnt>=k, "LSFitCreateFG: length(C)<K!", _state);
ae_assert(isfinitevector(c, k, _state), "LSFitCreateFG: C contains infinite or NaN values!", _state);
ae_assert(y->cnt>=n, "LSFitCreateFG: length(Y)<N!", _state);
ae_assert(isfinitevector(y, n, _state), "LSFitCreateFG: Y contains infinite or NaN values!", _state);
ae_assert(x->rows>=n, "LSFitCreateFG: rows(X)<N!", _state);
ae_assert(x->cols>=m, "LSFitCreateFG: cols(X)<M!", _state);
ae_assert(apservisfinitematrix(x, n, m, _state), "LSFitCreateFG: X contains infinite or NaN values!", _state);
ae_assert(x->rows>=n, "LSFitCreateFG: rows(X)<N!", _state);
ae_assert(x->cols>=m, "LSFitCreateFG: cols(X)<M!", _state);
ae_assert(apservisfinitematrix(x, n, m, _state), "LSFitCreateFG: X contains infinite or NaN values!", _state);
state->teststep = 0;
state->diffstep = 0;
state->npoints = n;
state->wkind = 0;
state->m = m;
state->k = k;
lsfitsetcond(state, 0.0, 0.0, 0, _state);
lsfitsetstpmax(state, 0.0, _state);
lsfitsetxrep(state, ae_false, _state);
ae_matrix_set_length(&state->taskx, n, m, _state);
ae_vector_set_length(&state->tasky, n, _state);
ae_vector_set_length(&state->c, k, _state);
ae_vector_set_length(&state->x, m, _state);
ae_vector_set_length(&state->g, k, _state);
ae_v_move(&state->c.ptr.p_double[0], 1, &c->ptr.p_double[0], 1, ae_v_len(0,k-1));
for(i=0; i<=n-1; i++)
{
ae_v_move(&state->taskx.ptr.pp_double[i][0], 1, &x->ptr.pp_double[i][0], 1, ae_v_len(0,m-1));
state->tasky.ptr.p_double[i] = y->ptr.p_double[i];
}
ae_vector_set_length(&state->s, k, _state);
ae_vector_set_length(&state->bndl, k, _state);
ae_vector_set_length(&state->bndu, k, _state);
for(i=0; i<=k-1; i++)
{
state->s.ptr.p_double[i] = 1.0;
state->bndl.ptr.p_double[i] = _state->v_neginf;
state->bndu.ptr.p_double[i] = _state->v_posinf;
}
state->optalgo = 1;
state->prevnpt = -1;
state->prevalgo = -1;
if( cheapfg )
{
minlmcreatevgj(k, n, &state->c, &state->optstate, _state);
}
else
{
minlmcreatevj(k, n, &state->c, &state->optstate, _state);
}
lsfit_lsfitclearrequestfields(state, _state);
ae_vector_set_length(&state->rstate.ia, 6+1, _state);
ae_vector_set_length(&state->rstate.ra, 8+1, _state);
state->rstate.stage = -1;
}
/*************************************************************************
Weighted nonlinear least squares fitting using gradient/Hessian.
Nonlinear task min(F(c)) is solved, where
F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2,
* N is a number of points,
* M is a dimension of a space points belong to,
* K is a dimension of a space of parameters being fitted,
* w is an N-dimensional vector of weight coefficients,
* x is a set of N points, each of them is an M-dimensional vector,
* c is a K-dimensional vector of parameters being fitted
This subroutine uses f(c,x[i]), its gradient and its Hessian.
INPUT PARAMETERS:
X - array[0..N-1,0..M-1], points (one row = one point)
Y - array[0..N-1], function values.
W - weights, array[0..N-1]
C - array[0..K-1], initial approximation to the solution,
N - number of points, N>1
M - dimension of space
K - number of parameters being fitted
OUTPUT PARAMETERS:
State - structure which stores algorithm state
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatewfgh(/* Real */ ae_matrix* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
/* Real */ ae_vector* c,
ae_int_t n,
ae_int_t m,
ae_int_t k,
lsfitstate* state,
ae_state *_state)
{
ae_int_t i;
_lsfitstate_clear(state);
ae_assert(n>=1, "LSFitCreateWFGH: N<1!", _state);
ae_assert(m>=1, "LSFitCreateWFGH: M<1!", _state);
ae_assert(k>=1, "LSFitCreateWFGH: K<1!", _state);
ae_assert(c->cnt>=k, "LSFitCreateWFGH: length(C)<K!", _state);
ae_assert(isfinitevector(c, k, _state), "LSFitCreateWFGH: C contains infinite or NaN values!", _state);
ae_assert(y->cnt>=n, "LSFitCreateWFGH: length(Y)<N!", _state);
ae_assert(isfinitevector(y, n, _state), "LSFitCreateWFGH: Y contains infinite or NaN values!", _state);
ae_assert(w->cnt>=n, "LSFitCreateWFGH: length(W)<N!", _state);
ae_assert(isfinitevector(w, n, _state), "LSFitCreateWFGH: W contains infinite or NaN values!", _state);
ae_assert(x->rows>=n, "LSFitCreateWFGH: rows(X)<N!", _state);
ae_assert(x->cols>=m, "LSFitCreateWFGH: cols(X)<M!", _state);
ae_assert(apservisfinitematrix(x, n, m, _state), "LSFitCreateWFGH: X contains infinite or NaN values!", _state);
state->teststep = 0;
state->diffstep = 0;
state->npoints = n;
state->nweights = n;
state->wkind = 1;
state->m = m;
state->k = k;
lsfitsetcond(state, 0.0, 0.0, 0, _state);
lsfitsetstpmax(state, 0.0, _state);
lsfitsetxrep(state, ae_false, _state);
ae_matrix_set_length(&state->taskx, n, m, _state);
ae_vector_set_length(&state->tasky, n, _state);
ae_vector_set_length(&state->taskw, n, _state);
ae_vector_set_length(&state->c, k, _state);
ae_matrix_set_length(&state->h, k, k, _state);
ae_vector_set_length(&state->x, m, _state);
ae_vector_set_length(&state->g, k, _state);
ae_v_move(&state->c.ptr.p_double[0], 1, &c->ptr.p_double[0], 1, ae_v_len(0,k-1));
ae_v_move(&state->taskw.ptr.p_double[0], 1, &w->ptr.p_double[0], 1, ae_v_len(0,n-1));
for(i=0; i<=n-1; i++)
{
ae_v_move(&state->taskx.ptr.pp_double[i][0], 1, &x->ptr.pp_double[i][0], 1, ae_v_len(0,m-1));
state->tasky.ptr.p_double[i] = y->ptr.p_double[i];
}
ae_vector_set_length(&state->s, k, _state);
ae_vector_set_length(&state->bndl, k, _state);
ae_vector_set_length(&state->bndu, k, _state);
for(i=0; i<=k-1; i++)
{
state->s.ptr.p_double[i] = 1.0;
state->bndl.ptr.p_double[i] = _state->v_neginf;
state->bndu.ptr.p_double[i] = _state->v_posinf;
}
state->optalgo = 2;
state->prevnpt = -1;
state->prevalgo = -1;
minlmcreatefgh(k, &state->c, &state->optstate, _state);
lsfit_lsfitclearrequestfields(state, _state);
ae_vector_set_length(&state->rstate.ia, 6+1, _state);
ae_vector_set_length(&state->rstate.ra, 8+1, _state);
state->rstate.stage = -1;
}
/*************************************************************************
Nonlinear least squares fitting using gradient/Hessian, without individial
weights.
Nonlinear task min(F(c)) is solved, where
F(c) = ((f(c,x[0])-y[0]))^2 + ... + ((f(c,x[n-1])-y[n-1]))^2,
* N is a number of points,
* M is a dimension of a space points belong to,
* K is a dimension of a space of parameters being fitted,
* x is a set of N points, each of them is an M-dimensional vector,
* c is a K-dimensional vector of parameters being fitted
This subroutine uses f(c,x[i]), its gradient and its Hessian.
INPUT PARAMETERS:
X - array[0..N-1,0..M-1], points (one row = one point)
Y - array[0..N-1], function values.
C - array[0..K-1], initial approximation to the solution,
N - number of points, N>1
M - dimension of space
K - number of parameters being fitted
OUTPUT PARAMETERS:
State - structure which stores algorithm state
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatefgh(/* Real */ ae_matrix* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* c,
ae_int_t n,
ae_int_t m,
ae_int_t k,
lsfitstate* state,
ae_state *_state)
{
ae_int_t i;
_lsfitstate_clear(state);
ae_assert(n>=1, "LSFitCreateFGH: N<1!", _state);
ae_assert(m>=1, "LSFitCreateFGH: M<1!", _state);
ae_assert(k>=1, "LSFitCreateFGH: K<1!", _state);
ae_assert(c->cnt>=k, "LSFitCreateFGH: length(C)<K!", _state);
ae_assert(isfinitevector(c, k, _state), "LSFitCreateFGH: C contains infinite or NaN values!", _state);
ae_assert(y->cnt>=n, "LSFitCreateFGH: length(Y)<N!", _state);
ae_assert(isfinitevector(y, n, _state), "LSFitCreateFGH: Y contains infinite or NaN values!", _state);
ae_assert(x->rows>=n, "LSFitCreateFGH: rows(X)<N!", _state);
ae_assert(x->cols>=m, "LSFitCreateFGH: cols(X)<M!", _state);
ae_assert(apservisfinitematrix(x, n, m, _state), "LSFitCreateFGH: X contains infinite or NaN values!", _state);
state->teststep = 0;
state->diffstep = 0;
state->npoints = n;
state->wkind = 0;
state->m = m;
state->k = k;
lsfitsetcond(state, 0.0, 0.0, 0, _state);
lsfitsetstpmax(state, 0.0, _state);
lsfitsetxrep(state, ae_false, _state);
ae_matrix_set_length(&state->taskx, n, m, _state);
ae_vector_set_length(&state->tasky, n, _state);
ae_vector_set_length(&state->c, k, _state);
ae_matrix_set_length(&state->h, k, k, _state);
ae_vector_set_length(&state->x, m, _state);
ae_vector_set_length(&state->g, k, _state);
ae_v_move(&state->c.ptr.p_double[0], 1, &c->ptr.p_double[0], 1, ae_v_len(0,k-1));
for(i=0; i<=n-1; i++)
{
ae_v_move(&state->taskx.ptr.pp_double[i][0], 1, &x->ptr.pp_double[i][0], 1, ae_v_len(0,m-1));
state->tasky.ptr.p_double[i] = y->ptr.p_double[i];
}
ae_vector_set_length(&state->s, k, _state);
ae_vector_set_length(&state->bndl, k, _state);
ae_vector_set_length(&state->bndu, k, _state);
for(i=0; i<=k-1; i++)
{
state->s.ptr.p_double[i] = 1.0;
state->bndl.ptr.p_double[i] = _state->v_neginf;
state->bndu.ptr.p_double[i] = _state->v_posinf;
}
state->optalgo = 2;
state->prevnpt = -1;
state->prevalgo = -1;
minlmcreatefgh(k, &state->c, &state->optstate, _state);
lsfit_lsfitclearrequestfields(state, _state);
ae_vector_set_length(&state->rstate.ia, 6+1, _state);
ae_vector_set_length(&state->rstate.ra, 8+1, _state);
state->rstate.stage = -1;
}
/*************************************************************************
Stopping conditions for nonlinear least squares fitting.
INPUT PARAMETERS:
State - structure which stores algorithm state
EpsF - stopping criterion. Algorithm stops if
|F(k+1)-F(k)| <= EpsF*max{|F(k)|, |F(k+1)|, 1}
EpsX - >=0
The subroutine finishes its work if on k+1-th iteration
the condition |v|<=EpsX is fulfilled, where:
* |.| means Euclidian norm
* v - scaled step vector, v[i]=dx[i]/s[i]
* dx - ste pvector, dx=X(k+1)-X(k)
* s - scaling coefficients set by LSFitSetScale()
MaxIts - maximum number of iterations. If MaxIts=0, the number of
iterations is unlimited. Only Levenberg-Marquardt
iterations are counted (L-BFGS/CG iterations are NOT
counted because their cost is very low compared to that of
LM).
NOTE
Passing EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic
stopping criterion selection (according to the scheme used by MINLM unit).
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitsetcond(lsfitstate* state,
double epsf,
double epsx,
ae_int_t maxits,
ae_state *_state)
{
ae_assert(ae_isfinite(epsf, _state), "LSFitSetCond: EpsF is not finite!", _state);
ae_assert(ae_fp_greater_eq(epsf,0), "LSFitSetCond: negative EpsF!", _state);
ae_assert(ae_isfinite(epsx, _state), "LSFitSetCond: EpsX is not finite!", _state);
ae_assert(ae_fp_greater_eq(epsx,0), "LSFitSetCond: negative EpsX!", _state);
ae_assert(maxits>=0, "LSFitSetCond: negative MaxIts!", _state);
state->epsf = epsf;
state->epsx = epsx;
state->maxits = maxits;
}
/*************************************************************************
This function sets maximum step length
INPUT PARAMETERS:
State - structure which stores algorithm state
StpMax - maximum step length, >=0. Set StpMax to 0.0, if you don't
want to limit step length.
Use this subroutine when you optimize target function which contains exp()
or other fast growing functions, and optimization algorithm makes too
large steps which leads to overflow. This function allows us to reject
steps that are too large (and therefore expose us to the possible
overflow) without actually calculating function value at the x+stp*d.
NOTE: non-zero StpMax leads to moderate performance degradation because
intermediate step of preconditioned L-BFGS optimization is incompatible
with limits on step size.
-- ALGLIB --
Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void lsfitsetstpmax(lsfitstate* state, double stpmax, ae_state *_state)
{
ae_assert(ae_fp_greater_eq(stpmax,0), "LSFitSetStpMax: StpMax<0!", _state);
state->stpmax = stpmax;
}
/*************************************************************************
This function turns on/off reporting.
INPUT PARAMETERS:
State - structure which stores algorithm state
NeedXRep- whether iteration reports are needed or not
When reports are needed, State.C (current parameters) and State.F (current
value of fitting function) are reported.
-- ALGLIB --
Copyright 15.08.2010 by Bochkanov Sergey
*************************************************************************/
void lsfitsetxrep(lsfitstate* state, ae_bool needxrep, ae_state *_state)
{
state->xrep = needxrep;
}
/*************************************************************************
This function sets scaling coefficients for underlying optimizer.
ALGLIB optimizers use scaling matrices to test stopping conditions (step
size and gradient are scaled before comparison with tolerances). Scale of
the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function
Generally, scale is NOT considered to be a form of preconditioner. But LM
optimizer is unique in that it uses scaling matrix both in the stopping
condition tests and as Marquardt damping factor.
Proper scaling is very important for the algorithm performance. It is less
important for the quality of results, but still has some influence (it is
easier to converge when variables are properly scaled, so premature
stopping is possible when very badly scalled variables are combined with
relaxed stopping conditions).
INPUT PARAMETERS:
State - structure stores algorithm state
S - array[N], non-zero scaling coefficients
S[i] may be negative, sign doesn't matter.
-- ALGLIB --
Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void lsfitsetscale(lsfitstate* state,
/* Real */ ae_vector* s,
ae_state *_state)
{
ae_int_t i;
ae_assert(s->cnt>=state->k, "LSFitSetScale: Length(S)<K", _state);
for(i=0; i<=state->k-1; i++)
{
ae_assert(ae_isfinite(s->ptr.p_double[i], _state), "LSFitSetScale: S contains infinite or NAN elements", _state);
ae_assert(ae_fp_neq(s->ptr.p_double[i],0), "LSFitSetScale: S contains infinite or NAN elements", _state);
state->s.ptr.p_double[i] = ae_fabs(s->ptr.p_double[i], _state);
}
}
/*************************************************************************
This function sets boundary constraints for underlying optimizer
Boundary constraints are inactive by default (after initial creation).
They are preserved until explicitly turned off with another SetBC() call.
INPUT PARAMETERS:
State - structure stores algorithm state
BndL - lower bounds, array[K].
If some (all) variables are unbounded, you may specify
very small number or -INF (latter is recommended because
it will allow solver to use better algorithm).
BndU - upper bounds, array[K].
If some (all) variables are unbounded, you may specify
very large number or +INF (latter is recommended because
it will allow solver to use better algorithm).
NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th
variable will be "frozen" at X[i]=BndL[i]=BndU[i].
NOTE 2: unlike other constrained optimization algorithms, this solver has
following useful properties:
* bound constraints are always satisfied exactly
* function is evaluated only INSIDE area specified by bound constraints
-- ALGLIB --
Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void lsfitsetbc(lsfitstate* state,
/* Real */ ae_vector* bndl,
/* Real */ ae_vector* bndu,
ae_state *_state)
{
ae_int_t i;
ae_int_t k;
k = state->k;
ae_assert(bndl->cnt>=k, "LSFitSetBC: Length(BndL)<K", _state);
ae_assert(bndu->cnt>=k, "LSFitSetBC: Length(BndU)<K", _state);
for(i=0; i<=k-1; i++)
{
ae_assert(ae_isfinite(bndl->ptr.p_double[i], _state)||ae_isneginf(bndl->ptr.p_double[i], _state), "LSFitSetBC: BndL contains NAN or +INF", _state);
ae_assert(ae_isfinite(bndu->ptr.p_double[i], _state)||ae_isposinf(bndu->ptr.p_double[i], _state), "LSFitSetBC: BndU contains NAN or -INF", _state);
if( ae_isfinite(bndl->ptr.p_double[i], _state)&&ae_isfinite(bndu->ptr.p_double[i], _state) )
{
ae_assert(ae_fp_less_eq(bndl->ptr.p_double[i],bndu->ptr.p_double[i]), "LSFitSetBC: BndL[i]>BndU[i]", _state);
}
state->bndl.ptr.p_double[i] = bndl->ptr.p_double[i];
state->bndu.ptr.p_double[i] = bndu->ptr.p_double[i];
}
}
/*************************************************************************
NOTES:
1. this algorithm is somewhat unusual because it works with parameterized
function f(C,X), where X is a function argument (we have many points
which are characterized by different argument values), and C is a
parameter to fit.
For example, if we want to do linear fit by f(c0,c1,x) = c0*x+c1, then
x will be argument, and {c0,c1} will be parameters.
It is important to understand that this algorithm finds minimum in the
space of function PARAMETERS (not arguments), so it needs derivatives
of f() with respect to C, not X.
In the example above it will need f=c0*x+c1 and {df/dc0,df/dc1} = {x,1}
instead of {df/dx} = {c0}.
2. Callback functions accept C as the first parameter, and X as the second
3. If state was created with LSFitCreateFG(), algorithm needs just
function and its gradient, but if state was created with
LSFitCreateFGH(), algorithm will need function, gradient and Hessian.
According to the said above, there ase several versions of this
function, which accept different sets of callbacks.
This flexibility opens way to subtle errors - you may create state with
LSFitCreateFGH() (optimization using Hessian), but call function which
does not accept Hessian. So when algorithm will request Hessian, there
will be no callback to call. In this case exception will be thrown.
Be careful to avoid such errors because there is no way to find them at
compile time - you can see them at runtime only.
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
ae_bool lsfititeration(lsfitstate* state, ae_state *_state)
{
double lx;
double lf;
double ld;
double rx;
double rf;
double rd;
ae_int_t n;
ae_int_t m;
ae_int_t k;
double v;
double vv;
double relcnt;
ae_int_t i;
ae_int_t j;
ae_int_t j1;
ae_int_t info;
ae_bool result;
/*
* Reverse communication preparations
* I know it looks ugly, but it works the same way
* anywhere from C++ to Python.
*
* This code initializes locals by:
* * random values determined during code
* generation - on first subroutine call
* * values from previous call - on subsequent calls
*/
if( state->rstate.stage>=0 )
{
n = state->rstate.ia.ptr.p_int[0];
m = state->rstate.ia.ptr.p_int[1];
k = state->rstate.ia.ptr.p_int[2];
i = state->rstate.ia.ptr.p_int[3];
j = state->rstate.ia.ptr.p_int[4];
j1 = state->rstate.ia.ptr.p_int[5];
info = state->rstate.ia.ptr.p_int[6];
lx = state->rstate.ra.ptr.p_double[0];
lf = state->rstate.ra.ptr.p_double[1];
ld = state->rstate.ra.ptr.p_double[2];
rx = state->rstate.ra.ptr.p_double[3];
rf = state->rstate.ra.ptr.p_double[4];
rd = state->rstate.ra.ptr.p_double[5];
v = state->rstate.ra.ptr.p_double[6];
vv = state->rstate.ra.ptr.p_double[7];
relcnt = state->rstate.ra.ptr.p_double[8];
}
else
{
n = -983;
m = -989;
k = -834;
i = 900;
j = -287;
j1 = 364;
info = 214;
lx = -338;
lf = -686;
ld = 912;
rx = 585;
rf = 497;
rd = -271;
v = -581;
vv = 745;
relcnt = -533;
}
if( state->rstate.stage==0 )
{
goto lbl_0;
}
if( state->rstate.stage==1 )
{
goto lbl_1;
}
if( state->rstate.stage==2 )
{
goto lbl_2;
}
if( state->rstate.stage==3 )
{
goto lbl_3;
}
if( state->rstate.stage==4 )
{
goto lbl_4;
}
if( state->rstate.stage==5 )
{
goto lbl_5;
}
if( state->rstate.stage==6 )
{
goto lbl_6;
}
if( state->rstate.stage==7 )
{
goto lbl_7;
}
if( state->rstate.stage==8 )
{
goto lbl_8;
}
if( state->rstate.stage==9 )
{
goto lbl_9;
}
if( state->rstate.stage==10 )
{
goto lbl_10;
}
if( state->rstate.stage==11 )
{
goto lbl_11;
}
if( state->rstate.stage==12 )
{
goto lbl_12;
}
if( state->rstate.stage==13 )
{
goto lbl_13;
}
/*
* Routine body
*/
/*
* Init
*/
if( state->wkind==1 )
{
ae_assert(state->npoints==state->nweights, "LSFitFit: number of points is not equal to the number of weights", _state);
}
state->repvaridx = -1;
n = state->npoints;
m = state->m;
k = state->k;
minlmsetcond(&state->optstate, 0.0, state->epsf, state->epsx, state->maxits, _state);
minlmsetstpmax(&state->optstate, state->stpmax, _state);
minlmsetxrep(&state->optstate, state->xrep, _state);
minlmsetscale(&state->optstate, &state->s, _state);
minlmsetbc(&state->optstate, &state->bndl, &state->bndu, _state);
/*
* Check that user-supplied gradient is correct
*/
lsfit_lsfitclearrequestfields(state, _state);
if( !(ae_fp_greater(state->teststep,0)&&state->optalgo==1) )
{
goto lbl_14;
}
for(i=0; i<=k-1; i++)
{
if( ae_isfinite(state->bndl.ptr.p_double[i], _state) )
{
state->c.ptr.p_double[i] = ae_maxreal(state->c.ptr.p_double[i], state->bndl.ptr.p_double[i], _state);
}
if( ae_isfinite(state->bndu.ptr.p_double[i], _state) )
{
state->c.ptr.p_double[i] = ae_minreal(state->c.ptr.p_double[i], state->bndu.ptr.p_double[i], _state);
}
}
state->needfg = ae_true;
i = 0;
lbl_16:
if( i>k-1 )
{
goto lbl_18;
}
ae_assert(ae_fp_less_eq(state->bndl.ptr.p_double[i],state->c.ptr.p_double[i])&&ae_fp_less_eq(state->c.ptr.p_double[i],state->bndu.ptr.p_double[i]), "LSFitIteration: internal error(State.C is out of bounds)", _state);
v = state->c.ptr.p_double[i];
j = 0;
lbl_19:
if( j>n-1 )
{
goto lbl_21;
}
ae_v_move(&state->x.ptr.p_double[0], 1, &state->taskx.ptr.pp_double[j][0], 1, ae_v_len(0,m-1));
state->c.ptr.p_double[i] = v-state->teststep*state->s.ptr.p_double[i];
if( ae_isfinite(state->bndl.ptr.p_double[i], _state) )
{
state->c.ptr.p_double[i] = ae_maxreal(state->c.ptr.p_double[i], state->bndl.ptr.p_double[i], _state);
}
lx = state->c.ptr.p_double[i];
state->rstate.stage = 0;
goto lbl_rcomm;
lbl_0:
lf = state->f;
ld = state->g.ptr.p_double[i];
state->c.ptr.p_double[i] = v+state->teststep*state->s.ptr.p_double[i];
if( ae_isfinite(state->bndu.ptr.p_double[i], _state) )
{
state->c.ptr.p_double[i] = ae_minreal(state->c.ptr.p_double[i], state->bndu.ptr.p_double[i], _state);
}
rx = state->c.ptr.p_double[i];
state->rstate.stage = 1;
goto lbl_rcomm;
lbl_1:
rf = state->f;
rd = state->g.ptr.p_double[i];
state->c.ptr.p_double[i] = (lx+rx)/2;
if( ae_isfinite(state->bndl.ptr.p_double[i], _state) )
{
state->c.ptr.p_double[i] = ae_maxreal(state->c.ptr.p_double[i], state->bndl.ptr.p_double[i], _state);
}
if( ae_isfinite(state->bndu.ptr.p_double[i], _state) )
{
state->c.ptr.p_double[i] = ae_minreal(state->c.ptr.p_double[i], state->bndu.ptr.p_double[i], _state);
}
state->rstate.stage = 2;
goto lbl_rcomm;
lbl_2:
state->c.ptr.p_double[i] = v;
if( !derivativecheck(lf, ld, rf, rd, state->f, state->g.ptr.p_double[i], rx-lx, _state) )
{
state->repvaridx = i;
state->repterminationtype = -7;
result = ae_false;
return result;
}
j = j+1;
goto lbl_19;
lbl_21:
i = i+1;
goto lbl_16;
lbl_18:
state->needfg = ae_false;
lbl_14:
/*
* Fill WCur by weights:
* * for WKind=0 unit weights are chosen
* * for WKind=1 we use user-supplied weights stored in State.TaskW
*/
rvectorsetlengthatleast(&state->wcur, n, _state);
for(i=0; i<=n-1; i++)
{
state->wcur.ptr.p_double[i] = 1.0;
if( state->wkind==1 )
{
state->wcur.ptr.p_double[i] = state->taskw.ptr.p_double[i];
}
}
/*
* Optimize
*/
lbl_22:
if( !minlmiteration(&state->optstate, _state) )
{
goto lbl_23;
}
if( !state->optstate.needfi )
{
goto lbl_24;
}
/*
* calculate f[] = wi*(f(xi,c)-yi)
*/
i = 0;
lbl_26:
if( i>n-1 )
{
goto lbl_28;
}
ae_v_move(&state->c.ptr.p_double[0], 1, &state->optstate.x.ptr.p_double[0], 1, ae_v_len(0,k-1));
ae_v_move(&state->x.ptr.p_double[0], 1, &state->taskx.ptr.pp_double[i][0], 1, ae_v_len(0,m-1));
state->pointindex = i;
lsfit_lsfitclearrequestfields(state, _state);
state->needf = ae_true;
state->rstate.stage = 3;
goto lbl_rcomm;
lbl_3:
state->needf = ae_false;
vv = state->wcur.ptr.p_double[i];
state->optstate.fi.ptr.p_double[i] = vv*(state->f-state->tasky.ptr.p_double[i]);
i = i+1;
goto lbl_26;
lbl_28:
goto lbl_22;
lbl_24:
if( !state->optstate.needf )
{
goto lbl_29;
}
/*
* calculate F = sum (wi*(f(xi,c)-yi))^2
*/
state->optstate.f = 0;
i = 0;
lbl_31:
if( i>n-1 )
{
goto lbl_33;
}
ae_v_move(&state->c.ptr.p_double[0], 1, &state->optstate.x.ptr.p_double[0], 1, ae_v_len(0,k-1));
ae_v_move(&state->x.ptr.p_double[0], 1, &state->taskx.ptr.pp_double[i][0], 1, ae_v_len(0,m-1));
state->pointindex = i;
lsfit_lsfitclearrequestfields(state, _state);
state->needf = ae_true;
state->rstate.stage = 4;
goto lbl_rcomm;
lbl_4:
state->needf = ae_false;
vv = state->wcur.ptr.p_double[i];
state->optstate.f = state->optstate.f+ae_sqr(vv*(state->f-state->tasky.ptr.p_double[i]), _state);
i = i+1;
goto lbl_31;
lbl_33:
goto lbl_22;
lbl_29:
if( !state->optstate.needfg )
{
goto lbl_34;
}
/*
* calculate F/gradF
*/
state->optstate.f = 0;
for(i=0; i<=k-1; i++)
{
state->optstate.g.ptr.p_double[i] = 0;
}
i = 0;
lbl_36:
if( i>n-1 )
{
goto lbl_38;
}
ae_v_move(&state->c.ptr.p_double[0], 1, &state->optstate.x.ptr.p_double[0], 1, ae_v_len(0,k-1));
ae_v_move(&state->x.ptr.p_double[0], 1, &state->taskx.ptr.pp_double[i][0], 1, ae_v_len(0,m-1));
state->pointindex = i;
lsfit_lsfitclearrequestfields(state, _state);
state->needfg = ae_true;
state->rstate.stage = 5;
goto lbl_rcomm;
lbl_5:
state->needfg = ae_false;
vv = state->wcur.ptr.p_double[i];
state->optstate.f = state->optstate.f+ae_sqr(vv*(state->f-state->tasky.ptr.p_double[i]), _state);
v = ae_sqr(vv, _state)*2*(state->f-state->tasky.ptr.p_double[i]);
ae_v_addd(&state->optstate.g.ptr.p_double[0], 1, &state->g.ptr.p_double[0], 1, ae_v_len(0,k-1), v);
i = i+1;
goto lbl_36;
lbl_38:
goto lbl_22;
lbl_34:
if( !state->optstate.needfij )
{
goto lbl_39;
}
/*
* calculate Fi/jac(Fi)
*/
i = 0;
lbl_41:
if( i>n-1 )
{
goto lbl_43;
}
ae_v_move(&state->c.ptr.p_double[0], 1, &state->optstate.x.ptr.p_double[0], 1, ae_v_len(0,k-1));
ae_v_move(&state->x.ptr.p_double[0], 1, &state->taskx.ptr.pp_double[i][0], 1, ae_v_len(0,m-1));
state->pointindex = i;
lsfit_lsfitclearrequestfields(state, _state);
state->needfg = ae_true;
state->rstate.stage = 6;
goto lbl_rcomm;
lbl_6:
state->needfg = ae_false;
vv = state->wcur.ptr.p_double[i];
state->optstate.fi.ptr.p_double[i] = vv*(state->f-state->tasky.ptr.p_double[i]);
ae_v_moved(&state->optstate.j.ptr.pp_double[i][0], 1, &state->g.ptr.p_double[0], 1, ae_v_len(0,k-1), vv);
i = i+1;
goto lbl_41;
lbl_43:
goto lbl_22;
lbl_39:
if( !state->optstate.needfgh )
{
goto lbl_44;
}
/*
* calculate F/grad(F)/hess(F)
*/
state->optstate.f = 0;
for(i=0; i<=k-1; i++)
{
state->optstate.g.ptr.p_double[i] = 0;
}
for(i=0; i<=k-1; i++)
{
for(j=0; j<=k-1; j++)
{
state->optstate.h.ptr.pp_double[i][j] = 0;
}
}
i = 0;
lbl_46:
if( i>n-1 )
{
goto lbl_48;
}
ae_v_move(&state->c.ptr.p_double[0], 1, &state->optstate.x.ptr.p_double[0], 1, ae_v_len(0,k-1));
ae_v_move(&state->x.ptr.p_double[0], 1, &state->taskx.ptr.pp_double[i][0], 1, ae_v_len(0,m-1));
state->pointindex = i;
lsfit_lsfitclearrequestfields(state, _state);
state->needfgh = ae_true;
state->rstate.stage = 7;
goto lbl_rcomm;
lbl_7:
state->needfgh = ae_false;
vv = state->wcur.ptr.p_double[i];
state->optstate.f = state->optstate.f+ae_sqr(vv*(state->f-state->tasky.ptr.p_double[i]), _state);
v = ae_sqr(vv, _state)*2*(state->f-state->tasky.ptr.p_double[i]);
ae_v_addd(&state->optstate.g.ptr.p_double[0], 1, &state->g.ptr.p_double[0], 1, ae_v_len(0,k-1), v);
for(j=0; j<=k-1; j++)
{
v = 2*ae_sqr(vv, _state)*state->g.ptr.p_double[j];
ae_v_addd(&state->optstate.h.ptr.pp_double[j][0], 1, &state->g.ptr.p_double[0], 1, ae_v_len(0,k-1), v);
v = 2*ae_sqr(vv, _state)*(state->f-state->tasky.ptr.p_double[i]);
ae_v_addd(&state->optstate.h.ptr.pp_double[j][0], 1, &state->h.ptr.pp_double[j][0], 1, ae_v_len(0,k-1), v);
}
i = i+1;
goto lbl_46;
lbl_48:
goto lbl_22;
lbl_44:
if( !state->optstate.xupdated )
{
goto lbl_49;
}
/*
* Report new iteration
*/
ae_v_move(&state->c.ptr.p_double[0], 1, &state->optstate.x.ptr.p_double[0], 1, ae_v_len(0,k-1));
state->f = state->optstate.f;
lsfit_lsfitclearrequestfields(state, _state);
state->xupdated = ae_true;
state->rstate.stage = 8;
goto lbl_rcomm;
lbl_8:
state->xupdated = ae_false;
goto lbl_22;
lbl_49:
goto lbl_22;
lbl_23:
minlmresults(&state->optstate, &state->c, &state->optrep, _state);
state->repterminationtype = state->optrep.terminationtype;
state->repiterationscount = state->optrep.iterationscount;
/*
* calculate errors
*/
if( state->repterminationtype<=0 )
{
goto lbl_51;
}
/*
* Calculate RMS/Avg/Max/... errors
*/
state->reprmserror = 0;
state->repwrmserror = 0;
state->repavgerror = 0;
state->repavgrelerror = 0;
state->repmaxerror = 0;
relcnt = 0;
i = 0;
lbl_53:
if( i>n-1 )
{
goto lbl_55;
}
ae_v_move(&state->c.ptr.p_double[0], 1, &state->c.ptr.p_double[0], 1, ae_v_len(0,k-1));
ae_v_move(&state->x.ptr.p_double[0], 1, &state->taskx.ptr.pp_double[i][0], 1, ae_v_len(0,m-1));
state->pointindex = i;
lsfit_lsfitclearrequestfields(state, _state);
state->needf = ae_true;
state->rstate.stage = 9;
goto lbl_rcomm;
lbl_9:
state->needf = ae_false;
v = state->f;
vv = state->wcur.ptr.p_double[i];
state->reprmserror = state->reprmserror+ae_sqr(v-state->tasky.ptr.p_double[i], _state);
state->repwrmserror = state->repwrmserror+ae_sqr(vv*(v-state->tasky.ptr.p_double[i]), _state);
state->repavgerror = state->repavgerror+ae_fabs(v-state->tasky.ptr.p_double[i], _state);
if( ae_fp_neq(state->tasky.ptr.p_double[i],0) )
{
state->repavgrelerror = state->repavgrelerror+ae_fabs(v-state->tasky.ptr.p_double[i], _state)/ae_fabs(state->tasky.ptr.p_double[i], _state);
relcnt = relcnt+1;
}
state->repmaxerror = ae_maxreal(state->repmaxerror, ae_fabs(v-state->tasky.ptr.p_double[i], _state), _state);
i = i+1;
goto lbl_53;
lbl_55:
state->reprmserror = ae_sqrt(state->reprmserror/n, _state);
state->repwrmserror = ae_sqrt(state->repwrmserror/n, _state);
state->repavgerror = state->repavgerror/n;
if( ae_fp_neq(relcnt,0) )
{
state->repavgrelerror = state->repavgrelerror/relcnt;
}
/*
* Calculate covariance matrix
*/
rmatrixsetlengthatleast(&state->tmpjac, n, k, _state);
rvectorsetlengthatleast(&state->tmpf, n, _state);
rvectorsetlengthatleast(&state->tmp, k, _state);
if( ae_fp_less_eq(state->diffstep,0) )
{
goto lbl_56;
}
/*
* Compute Jacobian by means of numerical differentiation
*/
lsfit_lsfitclearrequestfields(state, _state);
state->needf = ae_true;
i = 0;
lbl_58:
if( i>n-1 )
{
goto lbl_60;
}
ae_v_move(&state->x.ptr.p_double[0], 1, &state->taskx.ptr.pp_double[i][0], 1, ae_v_len(0,m-1));
state->pointindex = i;
state->rstate.stage = 10;
goto lbl_rcomm;
lbl_10:
state->tmpf.ptr.p_double[i] = state->f;
j = 0;
lbl_61:
if( j>k-1 )
{
goto lbl_63;
}
v = state->c.ptr.p_double[j];
lx = v-state->diffstep*state->s.ptr.p_double[j];
state->c.ptr.p_double[j] = lx;
if( ae_isfinite(state->bndl.ptr.p_double[j], _state) )
{
state->c.ptr.p_double[j] = ae_maxreal(state->c.ptr.p_double[j], state->bndl.ptr.p_double[j], _state);
}
state->rstate.stage = 11;
goto lbl_rcomm;
lbl_11:
lf = state->f;
rx = v+state->diffstep*state->s.ptr.p_double[j];
state->c.ptr.p_double[j] = rx;
if( ae_isfinite(state->bndu.ptr.p_double[j], _state) )
{
state->c.ptr.p_double[j] = ae_minreal(state->c.ptr.p_double[j], state->bndu.ptr.p_double[j], _state);
}
state->rstate.stage = 12;
goto lbl_rcomm;
lbl_12:
rf = state->f;
state->c.ptr.p_double[j] = v;
if( ae_fp_neq(rx,lx) )
{
state->tmpjac.ptr.pp_double[i][j] = (rf-lf)/(rx-lx);
}
else
{
state->tmpjac.ptr.pp_double[i][j] = 0;
}
j = j+1;
goto lbl_61;
lbl_63:
i = i+1;
goto lbl_58;
lbl_60:
state->needf = ae_false;
goto lbl_57;
lbl_56:
/*
* Jacobian is calculated with user-provided analytic gradient
*/
lsfit_lsfitclearrequestfields(state, _state);
state->needfg = ae_true;
i = 0;
lbl_64:
if( i>n-1 )
{
goto lbl_66;
}
ae_v_move(&state->x.ptr.p_double[0], 1, &state->taskx.ptr.pp_double[i][0], 1, ae_v_len(0,m-1));
state->pointindex = i;
state->rstate.stage = 13;
goto lbl_rcomm;
lbl_13:
state->tmpf.ptr.p_double[i] = state->f;
for(j=0; j<=k-1; j++)
{
state->tmpjac.ptr.pp_double[i][j] = state->g.ptr.p_double[j];
}
i = i+1;
goto lbl_64;
lbl_66:
state->needfg = ae_false;
lbl_57:
for(i=0; i<=k-1; i++)
{
state->tmp.ptr.p_double[i] = 0.0;
}
lsfit_estimateerrors(&state->tmpjac, &state->tmpf, &state->tasky, &state->wcur, &state->tmp, &state->s, n, k, &state->rep, &state->tmpjacw, 0, _state);
lbl_51:
result = ae_false;
return result;
/*
* Saving state
*/
lbl_rcomm:
result = ae_true;
state->rstate.ia.ptr.p_int[0] = n;
state->rstate.ia.ptr.p_int[1] = m;
state->rstate.ia.ptr.p_int[2] = k;
state->rstate.ia.ptr.p_int[3] = i;
state->rstate.ia.ptr.p_int[4] = j;
state->rstate.ia.ptr.p_int[5] = j1;
state->rstate.ia.ptr.p_int[6] = info;
state->rstate.ra.ptr.p_double[0] = lx;
state->rstate.ra.ptr.p_double[1] = lf;
state->rstate.ra.ptr.p_double[2] = ld;
state->rstate.ra.ptr.p_double[3] = rx;
state->rstate.ra.ptr.p_double[4] = rf;
state->rstate.ra.ptr.p_double[5] = rd;
state->rstate.ra.ptr.p_double[6] = v;
state->rstate.ra.ptr.p_double[7] = vv;
state->rstate.ra.ptr.p_double[8] = relcnt;
return result;
}
/*************************************************************************
Nonlinear least squares fitting results.
Called after return from LSFitFit().
INPUT PARAMETERS:
State - algorithm state
OUTPUT PARAMETERS:
Info - completion code:
* -7 gradient verification failed.
See LSFitSetGradientCheck() for more information.
* 1 relative function improvement is no more than
EpsF.
* 2 relative step is no more than EpsX.
* 4 gradient norm is no more than EpsG
* 5 MaxIts steps was taken
* 7 stopping conditions are too stringent,
further improvement is impossible
C - array[0..K-1], solution
Rep - optimization report. On success following fields are set:
* R2 non-adjusted coefficient of determination
(non-weighted)
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
* WRMSError weighted rms error on the (X,Y).
ERRORS IN PARAMETERS
This solver also calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar covariance matrix for parameters, array[K,K].
* Rep.ErrPar errors in parameters, array[K],
errpar = sqrt(diag(CovPar))
* Rep.ErrCurve vector of fit errors - standard deviations of empirical
best-fit curve from "ideal" best-fit curve built with
infinite number of samples, array[N].
errcurve = sqrt(diag(J*CovPar*J')),
where J is Jacobian matrix.
* Rep.Noise vector of per-point estimates of noise, array[N]
IMPORTANT: errors in parameters are calculated without taking into
account boundary/linear constraints! Presence of constraints
changes distribution of errors, but there is no easy way to
account for constraints when you calculate covariance matrix.
NOTE: noise in the data is estimated as follows:
* for fitting without user-supplied weights all points are
assumed to have same level of noise, which is estimated from
the data
* for fitting with user-supplied weights we assume that noise
level in I-th point is inversely proportional to Ith weight.
Coefficient of proportionality is estimated from the data.
NOTE: we apply small amount of regularization when we invert squared
Jacobian and calculate covariance matrix. It guarantees that
algorithm won't divide by zero during inversion, but skews
error estimates a bit (fractional error is about 10^-9).
However, we believe that this difference is insignificant for
all practical purposes except for the situation when you want
to compare ALGLIB results with "reference" implementation up
to the last significant digit.
NOTE: covariance matrix is estimated using correction for degrees
of freedom (covariances are divided by N-M instead of dividing
by N).
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitresults(lsfitstate* state,
ae_int_t* info,
/* Real */ ae_vector* c,
lsfitreport* rep,
ae_state *_state)
{
ae_int_t i;
ae_int_t j;
*info = 0;
ae_vector_clear(c);
_lsfitreport_clear(rep);
lsfit_clearreport(rep, _state);
*info = state->repterminationtype;
rep->varidx = state->repvaridx;
if( *info>0 )
{
ae_vector_set_length(c, state->k, _state);
ae_v_move(&c->ptr.p_double[0], 1, &state->c.ptr.p_double[0], 1, ae_v_len(0,state->k-1));
rep->rmserror = state->reprmserror;
rep->wrmserror = state->repwrmserror;
rep->avgerror = state->repavgerror;
rep->avgrelerror = state->repavgrelerror;
rep->maxerror = state->repmaxerror;
rep->iterationscount = state->repiterationscount;
ae_matrix_set_length(&rep->covpar, state->k, state->k, _state);
ae_vector_set_length(&rep->errpar, state->k, _state);
ae_vector_set_length(&rep->errcurve, state->npoints, _state);
ae_vector_set_length(&rep->noise, state->npoints, _state);
rep->r2 = state->rep.r2;
for(i=0; i<=state->k-1; i++)
{
for(j=0; j<=state->k-1; j++)
{
rep->covpar.ptr.pp_double[i][j] = state->rep.covpar.ptr.pp_double[i][j];
}
rep->errpar.ptr.p_double[i] = state->rep.errpar.ptr.p_double[i];
}
for(i=0; i<=state->npoints-1; i++)
{
rep->errcurve.ptr.p_double[i] = state->rep.errcurve.ptr.p_double[i];
rep->noise.ptr.p_double[i] = state->rep.noise.ptr.p_double[i];
}
}
}
/*************************************************************************
This subroutine turns on verification of the user-supplied analytic
gradient:
* user calls this subroutine before fitting begins
* LSFitFit() is called
* prior to actual fitting, for each point in data set X_i and each
component of parameters being fited C_j algorithm performs following
steps:
* two trial steps are made to C_j-TestStep*S[j] and C_j+TestStep*S[j],
where C_j is j-th parameter and S[j] is a scale of j-th parameter
* if needed, steps are bounded with respect to constraints on C[]
* F(X_i|C) is evaluated at these trial points
* we perform one more evaluation in the middle point of the interval
* we build cubic model using function values and derivatives at trial
points and we compare its prediction with actual value in the middle
point
* in case difference between prediction and actual value is higher than
some predetermined threshold, algorithm stops with completion code -7;
Rep.VarIdx is set to index of the parameter with incorrect derivative.
* after verification is over, algorithm proceeds to the actual optimization.
NOTE 1: verification needs N*K (points count * parameters count) gradient
evaluations. It is very costly and you should use it only for low
dimensional problems, when you want to be sure that you've
correctly calculated analytic derivatives. You should not use it
in the production code (unless you want to check derivatives
provided by some third party).
NOTE 2: you should carefully choose TestStep. Value which is too large
(so large that function behaviour is significantly non-cubic) will
lead to false alarms. You may use different step for different
parameters by means of setting scale with LSFitSetScale().
NOTE 3: this function may lead to false positives. In case it reports that
I-th derivative was calculated incorrectly, you may decrease test
step and try one more time - maybe your function changes too
sharply and your step is too large for such rapidly chanding
function.
NOTE 4: this function works only for optimizers created with LSFitCreateWFG()
or LSFitCreateFG() constructors.
INPUT PARAMETERS:
State - structure used to store algorithm state
TestStep - verification step:
* TestStep=0 turns verification off
* TestStep>0 activates verification
-- ALGLIB --
Copyright 15.06.2012 by Bochkanov Sergey
*************************************************************************/
void lsfitsetgradientcheck(lsfitstate* state,
double teststep,
ae_state *_state)
{
ae_assert(ae_isfinite(teststep, _state), "LSFitSetGradientCheck: TestStep contains NaN or Infinite", _state);
ae_assert(ae_fp_greater_eq(teststep,0), "LSFitSetGradientCheck: invalid argument TestStep(TestStep<0)", _state);
state->teststep = teststep;
}
/*************************************************************************
Internal subroutine: automatic scaling for LLS tasks.
NEVER CALL IT DIRECTLY!
Maps abscissas to [-1,1], standartizes ordinates and correspondingly scales
constraints. It also scales weights so that max(W[i])=1
Transformations performed:
* X, XC [XA,XB] => [-1,+1]
transformation makes min(X)=-1, max(X)=+1
* Y [SA,SB] => [0,1]
transformation makes mean(Y)=0, stddev(Y)=1
* YC transformed accordingly to SA, SB, DC[I]
-- ALGLIB PROJECT --
Copyright 08.09.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitscalexy(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
ae_int_t n,
/* Real */ ae_vector* xc,
/* Real */ ae_vector* yc,
/* Integer */ ae_vector* dc,
ae_int_t k,
double* xa,
double* xb,
double* sa,
double* sb,
/* Real */ ae_vector* xoriginal,
/* Real */ ae_vector* yoriginal,
ae_state *_state)
{
double xmin;
double xmax;
ae_int_t i;
double mx;
*xa = 0;
*xb = 0;
*sa = 0;
*sb = 0;
ae_vector_clear(xoriginal);
ae_vector_clear(yoriginal);
ae_assert(n>=1, "LSFitScaleXY: incorrect N", _state);
ae_assert(k>=0, "LSFitScaleXY: incorrect K", _state);
/*
* Calculate xmin/xmax.
* Force xmin<>xmax.
*/
xmin = x->ptr.p_double[0];
xmax = x->ptr.p_double[0];
for(i=1; i<=n-1; i++)
{
xmin = ae_minreal(xmin, x->ptr.p_double[i], _state);
xmax = ae_maxreal(xmax, x->ptr.p_double[i], _state);
}
for(i=0; i<=k-1; i++)
{
xmin = ae_minreal(xmin, xc->ptr.p_double[i], _state);
xmax = ae_maxreal(xmax, xc->ptr.p_double[i], _state);
}
if( ae_fp_eq(xmin,xmax) )
{
if( ae_fp_eq(xmin,0) )
{
xmin = -1;
xmax = 1;
}
else
{
if( ae_fp_greater(xmin,0) )
{
xmin = 0.5*xmin;
}
else
{
xmax = 0.5*xmax;
}
}
}
/*
* Transform abscissas: map [XA,XB] to [0,1]
*
* Store old X[] in XOriginal[] (it will be used
* to calculate relative error).
*/
ae_vector_set_length(xoriginal, n, _state);
ae_v_move(&xoriginal->ptr.p_double[0], 1, &x->ptr.p_double[0], 1, ae_v_len(0,n-1));
*xa = xmin;
*xb = xmax;
for(i=0; i<=n-1; i++)
{
x->ptr.p_double[i] = 2*(x->ptr.p_double[i]-0.5*(*xa+(*xb)))/(*xb-(*xa));
}
for(i=0; i<=k-1; i++)
{
ae_assert(dc->ptr.p_int[i]>=0, "LSFitScaleXY: internal error!", _state);
xc->ptr.p_double[i] = 2*(xc->ptr.p_double[i]-0.5*(*xa+(*xb)))/(*xb-(*xa));
yc->ptr.p_double[i] = yc->ptr.p_double[i]*ae_pow(0.5*(*xb-(*xa)), dc->ptr.p_int[i], _state);
}
/*
* Transform function values: map [SA,SB] to [0,1]
* SA = mean(Y),
* SB = SA+stddev(Y).
*
* Store old Y[] in YOriginal[] (it will be used
* to calculate relative error).
*/
ae_vector_set_length(yoriginal, n, _state);
ae_v_move(&yoriginal->ptr.p_double[0], 1, &y->ptr.p_double[0], 1, ae_v_len(0,n-1));
*sa = 0;
for(i=0; i<=n-1; i++)
{
*sa = *sa+y->ptr.p_double[i];
}
*sa = *sa/n;
*sb = 0;
for(i=0; i<=n-1; i++)
{
*sb = *sb+ae_sqr(y->ptr.p_double[i]-(*sa), _state);
}
*sb = ae_sqrt(*sb/n, _state)+(*sa);
if( ae_fp_eq(*sb,*sa) )
{
*sb = 2*(*sa);
}
if( ae_fp_eq(*sb,*sa) )
{
*sb = *sa+1;
}
for(i=0; i<=n-1; i++)
{
y->ptr.p_double[i] = (y->ptr.p_double[i]-(*sa))/(*sb-(*sa));
}
for(i=0; i<=k-1; i++)
{
if( dc->ptr.p_int[i]==0 )
{
yc->ptr.p_double[i] = (yc->ptr.p_double[i]-(*sa))/(*sb-(*sa));
}
else
{
yc->ptr.p_double[i] = yc->ptr.p_double[i]/(*sb-(*sa));
}
}
/*
* Scale weights
*/
mx = 0;
for(i=0; i<=n-1; i++)
{
mx = ae_maxreal(mx, ae_fabs(w->ptr.p_double[i], _state), _state);
}
if( ae_fp_neq(mx,0) )
{
for(i=0; i<=n-1; i++)
{
w->ptr.p_double[i] = w->ptr.p_double[i]/mx;
}
}
}
/*************************************************************************
Internal spline fitting subroutine
-- ALGLIB PROJECT --
Copyright 08.09.2009 by Bochkanov Sergey
*************************************************************************/
static void lsfit_spline1dfitinternal(ae_int_t st,
/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
ae_int_t n,
/* Real */ ae_vector* xc,
/* Real */ ae_vector* yc,
/* Integer */ ae_vector* dc,
ae_int_t k,
ae_int_t m,
ae_int_t* info,
spline1dinterpolant* s,
spline1dfitreport* rep,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _x;
ae_vector _y;
ae_vector _w;
ae_vector _xc;
ae_vector _yc;
ae_matrix fmatrix;
ae_matrix cmatrix;
ae_vector y2;
ae_vector w2;
ae_vector sx;
ae_vector sy;
ae_vector sd;
ae_vector tmp;
ae_vector xoriginal;
ae_vector yoriginal;
lsfitreport lrep;
double v0;
double v1;
double v2;
double mx;
spline1dinterpolant s2;
ae_int_t i;
ae_int_t j;
ae_int_t relcnt;
double xa;
double xb;
double sa;
double sb;
double bl;
double br;
double decay;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_x, x, _state, ae_true);
x = &_x;
ae_vector_init_copy(&_y, y, _state, ae_true);
y = &_y;
ae_vector_init_copy(&_w, w, _state, ae_true);
w = &_w;
ae_vector_init_copy(&_xc, xc, _state, ae_true);
xc = &_xc;
ae_vector_init_copy(&_yc, yc, _state, ae_true);
yc = &_yc;
*info = 0;
_spline1dinterpolant_clear(s);
_spline1dfitreport_clear(rep);
ae_matrix_init(&fmatrix, 0, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&cmatrix, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&y2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&w2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&sx, 0, DT_REAL, _state, ae_true);
ae_vector_init(&sy, 0, DT_REAL, _state, ae_true);
ae_vector_init(&sd, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmp, 0, DT_REAL, _state, ae_true);
ae_vector_init(&xoriginal, 0, DT_REAL, _state, ae_true);
ae_vector_init(&yoriginal, 0, DT_REAL, _state, ae_true);
_lsfitreport_init(&lrep, _state, ae_true);
_spline1dinterpolant_init(&s2, _state, ae_true);
ae_assert(st==0||st==1, "Spline1DFit: internal error!", _state);
if( st==0&&m<4 )
{
*info = -1;
ae_frame_leave(_state);
return;
}
if( st==1&&m<4 )
{
*info = -1;
ae_frame_leave(_state);
return;
}
if( (n<1||k<0)||k>=m )
{
*info = -1;
ae_frame_leave(_state);
return;
}
for(i=0; i<=k-1; i++)
{
*info = 0;
if( dc->ptr.p_int[i]<0 )
{
*info = -1;
}
if( dc->ptr.p_int[i]>1 )
{
*info = -1;
}
if( *info<0 )
{
ae_frame_leave(_state);
return;
}
}
if( st==1&&m%2!=0 )
{
/*
* Hermite fitter must have even number of basis functions
*/
*info = -2;
ae_frame_leave(_state);
return;
}
/*
* weight decay for correct handling of task which becomes
* degenerate after constraints are applied
*/
decay = 10000*ae_machineepsilon;
/*
* Scale X, Y, XC, YC
*/
lsfitscalexy(x, y, w, n, xc, yc, dc, k, &xa, &xb, &sa, &sb, &xoriginal, &yoriginal, _state);
/*
* allocate space, initialize:
* * SX - grid for basis functions
* * SY - values of basis functions at grid points
* * FMatrix- values of basis functions at X[]
* * CMatrix- values (derivatives) of basis functions at XC[]
*/
ae_vector_set_length(&y2, n+m, _state);
ae_vector_set_length(&w2, n+m, _state);
ae_matrix_set_length(&fmatrix, n+m, m, _state);
if( k>0 )
{
ae_matrix_set_length(&cmatrix, k, m+1, _state);
}
if( st==0 )
{
/*
* allocate space for cubic spline
*/
ae_vector_set_length(&sx, m-2, _state);
ae_vector_set_length(&sy, m-2, _state);
for(j=0; j<=m-2-1; j++)
{
sx.ptr.p_double[j] = (double)(2*j)/(double)(m-2-1)-1;
}
}
if( st==1 )
{
/*
* allocate space for Hermite spline
*/
ae_vector_set_length(&sx, m/2, _state);
ae_vector_set_length(&sy, m/2, _state);
ae_vector_set_length(&sd, m/2, _state);
for(j=0; j<=m/2-1; j++)
{
sx.ptr.p_double[j] = (double)(2*j)/(double)(m/2-1)-1;
}
}
/*
* Prepare design and constraints matrices:
* * fill constraints matrix
* * fill first N rows of design matrix with values
* * fill next M rows of design matrix with regularizing term
* * append M zeros to Y
* * append M elements, mean(abs(W)) each, to W
*/
for(j=0; j<=m-1; j++)
{
/*
* prepare Jth basis function
*/
if( st==0 )
{
/*
* cubic spline basis
*/
for(i=0; i<=m-2-1; i++)
{
sy.ptr.p_double[i] = 0;
}
bl = 0;
br = 0;
if( j<m-2 )
{
sy.ptr.p_double[j] = 1;
}
if( j==m-2 )
{
bl = 1;
}
if( j==m-1 )
{
br = 1;
}
spline1dbuildcubic(&sx, &sy, m-2, 1, bl, 1, br, &s2, _state);
}
if( st==1 )
{
/*
* Hermite basis
*/
for(i=0; i<=m/2-1; i++)
{
sy.ptr.p_double[i] = 0;
sd.ptr.p_double[i] = 0;
}
if( j%2==0 )
{
sy.ptr.p_double[j/2] = 1;
}
else
{
sd.ptr.p_double[j/2] = 1;
}
spline1dbuildhermite(&sx, &sy, &sd, m/2, &s2, _state);
}
/*
* values at X[], XC[]
*/
for(i=0; i<=n-1; i++)
{
fmatrix.ptr.pp_double[i][j] = spline1dcalc(&s2, x->ptr.p_double[i], _state);
}
for(i=0; i<=k-1; i++)
{
ae_assert(dc->ptr.p_int[i]>=0&&dc->ptr.p_int[i]<=2, "Spline1DFit: internal error!", _state);
spline1ddiff(&s2, xc->ptr.p_double[i], &v0, &v1, &v2, _state);
if( dc->ptr.p_int[i]==0 )
{
cmatrix.ptr.pp_double[i][j] = v0;
}
if( dc->ptr.p_int[i]==1 )
{
cmatrix.ptr.pp_double[i][j] = v1;
}
if( dc->ptr.p_int[i]==2 )
{
cmatrix.ptr.pp_double[i][j] = v2;
}
}
}
for(i=0; i<=k-1; i++)
{
cmatrix.ptr.pp_double[i][m] = yc->ptr.p_double[i];
}
for(i=0; i<=m-1; i++)
{
for(j=0; j<=m-1; j++)
{
if( i==j )
{
fmatrix.ptr.pp_double[n+i][j] = decay;
}
else
{
fmatrix.ptr.pp_double[n+i][j] = 0;
}
}
}
ae_vector_set_length(&y2, n+m, _state);
ae_vector_set_length(&w2, n+m, _state);
ae_v_move(&y2.ptr.p_double[0], 1, &y->ptr.p_double[0], 1, ae_v_len(0,n-1));
ae_v_move(&w2.ptr.p_double[0], 1, &w->ptr.p_double[0], 1, ae_v_len(0,n-1));
mx = 0;
for(i=0; i<=n-1; i++)
{
mx = mx+ae_fabs(w->ptr.p_double[i], _state);
}
mx = mx/n;
for(i=0; i<=m-1; i++)
{
y2.ptr.p_double[n+i] = 0;
w2.ptr.p_double[n+i] = mx;
}
/*
* Solve constrained task
*/
if( k>0 )
{
/*
* solve using regularization
*/
lsfitlinearwc(&y2, &w2, &fmatrix, &cmatrix, n+m, m, k, info, &tmp, &lrep, _state);
}
else
{
/*
* no constraints, no regularization needed
*/
lsfitlinearwc(y, w, &fmatrix, &cmatrix, n, m, k, info, &tmp, &lrep, _state);
}
if( *info<0 )
{
ae_frame_leave(_state);
return;
}
/*
* Generate spline and scale it
*/
if( st==0 )
{
/*
* cubic spline basis
*/
ae_v_move(&sy.ptr.p_double[0], 1, &tmp.ptr.p_double[0], 1, ae_v_len(0,m-2-1));
spline1dbuildcubic(&sx, &sy, m-2, 1, tmp.ptr.p_double[m-2], 1, tmp.ptr.p_double[m-1], s, _state);
}
if( st==1 )
{
/*
* Hermite basis
*/
for(i=0; i<=m/2-1; i++)
{
sy.ptr.p_double[i] = tmp.ptr.p_double[2*i];
sd.ptr.p_double[i] = tmp.ptr.p_double[2*i+1];
}
spline1dbuildhermite(&sx, &sy, &sd, m/2, s, _state);
}
spline1dlintransx(s, 2/(xb-xa), -(xa+xb)/(xb-xa), _state);
spline1dlintransy(s, sb-sa, sa, _state);
/*
* Scale absolute errors obtained from LSFitLinearW.
* Relative error should be calculated separately
* (because of shifting/scaling of the task)
*/
rep->taskrcond = lrep.taskrcond;
rep->rmserror = lrep.rmserror*(sb-sa);
rep->avgerror = lrep.avgerror*(sb-sa);
rep->maxerror = lrep.maxerror*(sb-sa);
rep->avgrelerror = 0;
relcnt = 0;
for(i=0; i<=n-1; i++)
{
if( ae_fp_neq(yoriginal.ptr.p_double[i],0) )
{
rep->avgrelerror = rep->avgrelerror+ae_fabs(spline1dcalc(s, xoriginal.ptr.p_double[i], _state)-yoriginal.ptr.p_double[i], _state)/ae_fabs(yoriginal.ptr.p_double[i], _state);
relcnt = relcnt+1;
}
}
if( relcnt!=0 )
{
rep->avgrelerror = rep->avgrelerror/relcnt;
}
ae_frame_leave(_state);
}
/*************************************************************************
Internal fitting subroutine
*************************************************************************/
static void lsfit_lsfitlinearinternal(/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
/* Real */ ae_matrix* fmatrix,
ae_int_t n,
ae_int_t m,
ae_int_t* info,
/* Real */ ae_vector* c,
lsfitreport* rep,
ae_state *_state)
{
ae_frame _frame_block;
double threshold;
ae_matrix ft;
ae_matrix q;
ae_matrix l;
ae_matrix r;
ae_vector b;
ae_vector wmod;
ae_vector tau;
ae_vector nzeros;
ae_vector s;
ae_int_t i;
ae_int_t j;
double v;
ae_vector sv;
ae_matrix u;
ae_matrix vt;
ae_vector tmp;
ae_vector utb;
ae_vector sutb;
ae_int_t relcnt;
ae_frame_make(_state, &_frame_block);
*info = 0;
ae_vector_clear(c);
_lsfitreport_clear(rep);
ae_matrix_init(&ft, 0, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&q, 0, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&l, 0, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&r, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&b, 0, DT_REAL, _state, ae_true);
ae_vector_init(&wmod, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tau, 0, DT_REAL, _state, ae_true);
ae_vector_init(&nzeros, 0, DT_REAL, _state, ae_true);
ae_vector_init(&s, 0, DT_REAL, _state, ae_true);
ae_vector_init(&sv, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&u, 0, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&vt, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmp, 0, DT_REAL, _state, ae_true);
ae_vector_init(&utb, 0, DT_REAL, _state, ae_true);
ae_vector_init(&sutb, 0, DT_REAL, _state, ae_true);
lsfit_clearreport(rep, _state);
if( n<1||m<1 )
{
*info = -1;
ae_frame_leave(_state);
return;
}
*info = 1;
threshold = ae_sqrt(ae_machineepsilon, _state);
/*
* Degenerate case, needs special handling
*/
if( n<m )
{
/*
* Create design matrix.
*/
ae_matrix_set_length(&ft, n, m, _state);
ae_vector_set_length(&b, n, _state);
ae_vector_set_length(&wmod, n, _state);
for(j=0; j<=n-1; j++)
{
v = w->ptr.p_double[j];
ae_v_moved(&ft.ptr.pp_double[j][0], 1, &fmatrix->ptr.pp_double[j][0], 1, ae_v_len(0,m-1), v);
b.ptr.p_double[j] = w->ptr.p_double[j]*y->ptr.p_double[j];
wmod.ptr.p_double[j] = 1;
}
/*
* LQ decomposition and reduction to M=N
*/
ae_vector_set_length(c, m, _state);
for(i=0; i<=m-1; i++)
{
c->ptr.p_double[i] = 0;
}
rep->taskrcond = 0;
rmatrixlq(&ft, n, m, &tau, _state);
rmatrixlqunpackq(&ft, n, m, &tau, n, &q, _state);
rmatrixlqunpackl(&ft, n, m, &l, _state);
lsfit_lsfitlinearinternal(&b, &wmod, &l, n, n, info, &tmp, rep, _state);
if( *info<=0 )
{
ae_frame_leave(_state);
return;
}
for(i=0; i<=n-1; i++)
{
v = tmp.ptr.p_double[i];
ae_v_addd(&c->ptr.p_double[0], 1, &q.ptr.pp_double[i][0], 1, ae_v_len(0,m-1), v);
}
ae_frame_leave(_state);
return;
}
/*
* N>=M. Generate design matrix and reduce to N=M using
* QR decomposition.
*/
ae_matrix_set_length(&ft, n, m, _state);
ae_vector_set_length(&b, n, _state);
for(j=0; j<=n-1; j++)
{
v = w->ptr.p_double[j];
ae_v_moved(&ft.ptr.pp_double[j][0], 1, &fmatrix->ptr.pp_double[j][0], 1, ae_v_len(0,m-1), v);
b.ptr.p_double[j] = w->ptr.p_double[j]*y->ptr.p_double[j];
}
rmatrixqr(&ft, n, m, &tau, _state);
rmatrixqrunpackq(&ft, n, m, &tau, m, &q, _state);
rmatrixqrunpackr(&ft, n, m, &r, _state);
ae_vector_set_length(&tmp, m, _state);
for(i=0; i<=m-1; i++)
{
tmp.ptr.p_double[i] = 0;
}
for(i=0; i<=n-1; i++)
{
v = b.ptr.p_double[i];
ae_v_addd(&tmp.ptr.p_double[0], 1, &q.ptr.pp_double[i][0], 1, ae_v_len(0,m-1), v);
}
ae_vector_set_length(&b, m, _state);
ae_v_move(&b.ptr.p_double[0], 1, &tmp.ptr.p_double[0], 1, ae_v_len(0,m-1));
/*
* R contains reduced MxM design upper triangular matrix,
* B contains reduced Mx1 right part.
*
* Determine system condition number and decide
* should we use triangular solver (faster) or
* SVD-based solver (more stable).
*
* We can use LU-based RCond estimator for this task.
*/
rep->taskrcond = rmatrixlurcondinf(&r, m, _state);
if( ae_fp_greater(rep->taskrcond,threshold) )
{
/*
* use QR-based solver
*/
ae_vector_set_length(c, m, _state);
c->ptr.p_double[m-1] = b.ptr.p_double[m-1]/r.ptr.pp_double[m-1][m-1];
for(i=m-2; i>=0; i--)
{
v = ae_v_dotproduct(&r.ptr.pp_double[i][i+1], 1, &c->ptr.p_double[i+1], 1, ae_v_len(i+1,m-1));
c->ptr.p_double[i] = (b.ptr.p_double[i]-v)/r.ptr.pp_double[i][i];
}
}
else
{
/*
* use SVD-based solver
*/
if( !rmatrixsvd(&r, m, m, 1, 1, 2, &sv, &u, &vt, _state) )
{
*info = -4;
ae_frame_leave(_state);
return;
}
ae_vector_set_length(&utb, m, _state);
ae_vector_set_length(&sutb, m, _state);
for(i=0; i<=m-1; i++)
{
utb.ptr.p_double[i] = 0;
}
for(i=0; i<=m-1; i++)
{
v = b.ptr.p_double[i];
ae_v_addd(&utb.ptr.p_double[0], 1, &u.ptr.pp_double[i][0], 1, ae_v_len(0,m-1), v);
}
if( ae_fp_greater(sv.ptr.p_double[0],0) )
{
rep->taskrcond = sv.ptr.p_double[m-1]/sv.ptr.p_double[0];
for(i=0; i<=m-1; i++)
{
if( ae_fp_greater(sv.ptr.p_double[i],threshold*sv.ptr.p_double[0]) )
{
sutb.ptr.p_double[i] = utb.ptr.p_double[i]/sv.ptr.p_double[i];
}
else
{
sutb.ptr.p_double[i] = 0;
}
}
}
else
{
rep->taskrcond = 0;
for(i=0; i<=m-1; i++)
{
sutb.ptr.p_double[i] = 0;
}
}
ae_vector_set_length(c, m, _state);
for(i=0; i<=m-1; i++)
{
c->ptr.p_double[i] = 0;
}
for(i=0; i<=m-1; i++)
{
v = sutb.ptr.p_double[i];
ae_v_addd(&c->ptr.p_double[0], 1, &vt.ptr.pp_double[i][0], 1, ae_v_len(0,m-1), v);
}
}
/*
* calculate errors
*/
rep->rmserror = 0;
rep->avgerror = 0;
rep->avgrelerror = 0;
rep->maxerror = 0;
relcnt = 0;
for(i=0; i<=n-1; i++)
{
v = ae_v_dotproduct(&fmatrix->ptr.pp_double[i][0], 1, &c->ptr.p_double[0], 1, ae_v_len(0,m-1));
rep->rmserror = rep->rmserror+ae_sqr(v-y->ptr.p_double[i], _state);
rep->avgerror = rep->avgerror+ae_fabs(v-y->ptr.p_double[i], _state);
if( ae_fp_neq(y->ptr.p_double[i],0) )
{
rep->avgrelerror = rep->avgrelerror+ae_fabs(v-y->ptr.p_double[i], _state)/ae_fabs(y->ptr.p_double[i], _state);
relcnt = relcnt+1;
}
rep->maxerror = ae_maxreal(rep->maxerror, ae_fabs(v-y->ptr.p_double[i], _state), _state);
}
rep->rmserror = ae_sqrt(rep->rmserror/n, _state);
rep->avgerror = rep->avgerror/n;
if( relcnt!=0 )
{
rep->avgrelerror = rep->avgrelerror/relcnt;
}
ae_vector_set_length(&nzeros, n, _state);
ae_vector_set_length(&s, m, _state);
for(i=0; i<=m-1; i++)
{
s.ptr.p_double[i] = 0;
}
for(i=0; i<=n-1; i++)
{
for(j=0; j<=m-1; j++)
{
s.ptr.p_double[j] = s.ptr.p_double[j]+ae_sqr(fmatrix->ptr.pp_double[i][j], _state);
}
nzeros.ptr.p_double[i] = 0;
}
for(i=0; i<=m-1; i++)
{
if( ae_fp_neq(s.ptr.p_double[i],0) )
{
s.ptr.p_double[i] = ae_sqrt(1/s.ptr.p_double[i], _state);
}
else
{
s.ptr.p_double[i] = 1;
}
}
lsfit_estimateerrors(fmatrix, &nzeros, y, w, c, &s, n, m, rep, &r, 1, _state);
ae_frame_leave(_state);
}
/*************************************************************************
Internal subroutine
*************************************************************************/
static void lsfit_lsfitclearrequestfields(lsfitstate* state,
ae_state *_state)
{
state->needf = ae_false;
state->needfg = ae_false;
state->needfgh = ae_false;
state->xupdated = ae_false;
}
/*************************************************************************
Internal subroutine, calculates barycentric basis functions.
Used for efficient simultaneous calculation of N basis functions.
-- ALGLIB --
Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
static void lsfit_barycentriccalcbasis(barycentricinterpolant* b,
double t,
/* Real */ ae_vector* y,
ae_state *_state)
{
double s2;
double s;
double v;
ae_int_t i;
ae_int_t j;
/*
* special case: N=1
*/
if( b->n==1 )
{
y->ptr.p_double[0] = 1;
return;
}
/*
* Here we assume that task is normalized, i.e.:
* 1. abs(Y[i])<=1
* 2. abs(W[i])<=1
* 3. X[] is ordered
*
* First, we decide: should we use "safe" formula (guarded
* against overflow) or fast one?
*/
s = ae_fabs(t-b->x.ptr.p_double[0], _state);
for(i=0; i<=b->n-1; i++)
{
v = b->x.ptr.p_double[i];
if( ae_fp_eq(v,t) )
{
for(j=0; j<=b->n-1; j++)
{
y->ptr.p_double[j] = 0;
}
y->ptr.p_double[i] = 1;
return;
}
v = ae_fabs(t-v, _state);
if( ae_fp_less(v,s) )
{
s = v;
}
}
s2 = 0;
for(i=0; i<=b->n-1; i++)
{
v = s/(t-b->x.ptr.p_double[i]);
v = v*b->w.ptr.p_double[i];
y->ptr.p_double[i] = v;
s2 = s2+v;
}
v = 1/s2;
ae_v_muld(&y->ptr.p_double[0], 1, ae_v_len(0,b->n-1), v);
}
/*************************************************************************
This is internal function for Chebyshev fitting.
It assumes that input data are normalized:
* X/XC belong to [-1,+1],
* mean(Y)=0, stddev(Y)=1.
It does not checks inputs for errors.
This function is used to fit general (shifted) Chebyshev models, power
basis models or barycentric models.
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
W - weights, array[0..N-1]
N - number of points, N>0.
XC - points where polynomial values/derivatives are constrained,
array[0..K-1].
YC - values of constraints, array[0..K-1]
DC - array[0..K-1], types of constraints:
* DC[i]=0 means that P(XC[i])=YC[i]
* DC[i]=1 means that P'(XC[i])=YC[i]
K - number of constraints, 0<=K<M.
K=0 means no constraints (XC/YC/DC are not used in such cases)
M - number of basis functions (= polynomial_degree + 1), M>=1
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearW() subroutine:
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
-3 means inconsistent constraints
C - interpolant in Chebyshev form; [-1,+1] is used as base interval
Rep - report, same format as in LSFitLinearW() subroutine.
Following fields are set:
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
-- ALGLIB PROJECT --
Copyright 10.12.2009 by Bochkanov Sergey
*************************************************************************/
static void lsfit_internalchebyshevfit(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
ae_int_t n,
/* Real */ ae_vector* xc,
/* Real */ ae_vector* yc,
/* Integer */ ae_vector* dc,
ae_int_t k,
ae_int_t m,
ae_int_t* info,
/* Real */ ae_vector* c,
lsfitreport* rep,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _xc;
ae_vector _yc;
ae_vector y2;
ae_vector w2;
ae_vector tmp;
ae_vector tmp2;
ae_vector tmpdiff;
ae_vector bx;
ae_vector by;
ae_vector bw;
ae_matrix fmatrix;
ae_matrix cmatrix;
ae_int_t i;
ae_int_t j;
double mx;
double decay;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_xc, xc, _state, ae_true);
xc = &_xc;
ae_vector_init_copy(&_yc, yc, _state, ae_true);
yc = &_yc;
*info = 0;
ae_vector_clear(c);
_lsfitreport_clear(rep);
ae_vector_init(&y2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&w2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmp, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmp2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmpdiff, 0, DT_REAL, _state, ae_true);
ae_vector_init(&bx, 0, DT_REAL, _state, ae_true);
ae_vector_init(&by, 0, DT_REAL, _state, ae_true);
ae_vector_init(&bw, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&fmatrix, 0, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&cmatrix, 0, 0, DT_REAL, _state, ae_true);
lsfit_clearreport(rep, _state);
/*
* weight decay for correct handling of task which becomes
* degenerate after constraints are applied
*/
decay = 10000*ae_machineepsilon;
/*
* allocate space, initialize/fill:
* * FMatrix- values of basis functions at X[]
* * CMatrix- values (derivatives) of basis functions at XC[]
* * fill constraints matrix
* * fill first N rows of design matrix with values
* * fill next M rows of design matrix with regularizing term
* * append M zeros to Y
* * append M elements, mean(abs(W)) each, to W
*/
ae_vector_set_length(&y2, n+m, _state);
ae_vector_set_length(&w2, n+m, _state);
ae_vector_set_length(&tmp, m, _state);
ae_vector_set_length(&tmpdiff, m, _state);
ae_matrix_set_length(&fmatrix, n+m, m, _state);
if( k>0 )
{
ae_matrix_set_length(&cmatrix, k, m+1, _state);
}
/*
* Fill design matrix, Y2, W2:
* * first N rows with basis functions for original points
* * next M rows with decay terms
*/
for(i=0; i<=n-1; i++)
{
/*
* prepare Ith row
* use Tmp for calculations to avoid multidimensional arrays overhead
*/
for(j=0; j<=m-1; j++)
{
if( j==0 )
{
tmp.ptr.p_double[j] = 1;
}
else
{
if( j==1 )
{
tmp.ptr.p_double[j] = x->ptr.p_double[i];
}
else
{
tmp.ptr.p_double[j] = 2*x->ptr.p_double[i]*tmp.ptr.p_double[j-1]-tmp.ptr.p_double[j-2];
}
}
}
ae_v_move(&fmatrix.ptr.pp_double[i][0], 1, &tmp.ptr.p_double[0], 1, ae_v_len(0,m-1));
}
for(i=0; i<=m-1; i++)
{
for(j=0; j<=m-1; j++)
{
if( i==j )
{
fmatrix.ptr.pp_double[n+i][j] = decay;
}
else
{
fmatrix.ptr.pp_double[n+i][j] = 0;
}
}
}
ae_v_move(&y2.ptr.p_double[0], 1, &y->ptr.p_double[0], 1, ae_v_len(0,n-1));
ae_v_move(&w2.ptr.p_double[0], 1, &w->ptr.p_double[0], 1, ae_v_len(0,n-1));
mx = 0;
for(i=0; i<=n-1; i++)
{
mx = mx+ae_fabs(w->ptr.p_double[i], _state);
}
mx = mx/n;
for(i=0; i<=m-1; i++)
{
y2.ptr.p_double[n+i] = 0;
w2.ptr.p_double[n+i] = mx;
}
/*
* fill constraints matrix
*/
for(i=0; i<=k-1; i++)
{
/*
* prepare Ith row
* use Tmp for basis function values,
* TmpDiff for basos function derivatives
*/
for(j=0; j<=m-1; j++)
{
if( j==0 )
{
tmp.ptr.p_double[j] = 1;
tmpdiff.ptr.p_double[j] = 0;
}
else
{
if( j==1 )
{
tmp.ptr.p_double[j] = xc->ptr.p_double[i];
tmpdiff.ptr.p_double[j] = 1;
}
else
{
tmp.ptr.p_double[j] = 2*xc->ptr.p_double[i]*tmp.ptr.p_double[j-1]-tmp.ptr.p_double[j-2];
tmpdiff.ptr.p_double[j] = 2*(tmp.ptr.p_double[j-1]+xc->ptr.p_double[i]*tmpdiff.ptr.p_double[j-1])-tmpdiff.ptr.p_double[j-2];
}
}
}
if( dc->ptr.p_int[i]==0 )
{
ae_v_move(&cmatrix.ptr.pp_double[i][0], 1, &tmp.ptr.p_double[0], 1, ae_v_len(0,m-1));
}
if( dc->ptr.p_int[i]==1 )
{
ae_v_move(&cmatrix.ptr.pp_double[i][0], 1, &tmpdiff.ptr.p_double[0], 1, ae_v_len(0,m-1));
}
cmatrix.ptr.pp_double[i][m] = yc->ptr.p_double[i];
}
/*
* Solve constrained task
*/
if( k>0 )
{
/*
* solve using regularization
*/
lsfitlinearwc(&y2, &w2, &fmatrix, &cmatrix, n+m, m, k, info, c, rep, _state);
}
else
{
/*
* no constraints, no regularization needed
*/
lsfitlinearwc(y, w, &fmatrix, &cmatrix, n, m, 0, info, c, rep, _state);
}
if( *info<0 )
{
ae_frame_leave(_state);
return;
}
ae_frame_leave(_state);
}
/*************************************************************************
Internal Floater-Hormann fitting subroutine for fixed D
*************************************************************************/
static void lsfit_barycentricfitwcfixedd(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
ae_int_t n,
/* Real */ ae_vector* xc,
/* Real */ ae_vector* yc,
/* Integer */ ae_vector* dc,
ae_int_t k,
ae_int_t m,
ae_int_t d,
ae_int_t* info,
barycentricinterpolant* b,
barycentricfitreport* rep,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _x;
ae_vector _y;
ae_vector _w;
ae_vector _xc;
ae_vector _yc;
ae_matrix fmatrix;
ae_matrix cmatrix;
ae_vector y2;
ae_vector w2;
ae_vector sx;
ae_vector sy;
ae_vector sbf;
ae_vector xoriginal;
ae_vector yoriginal;
ae_vector tmp;
lsfitreport lrep;
double v0;
double v1;
double mx;
barycentricinterpolant b2;
ae_int_t i;
ae_int_t j;
ae_int_t relcnt;
double xa;
double xb;
double sa;
double sb;
double decay;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_x, x, _state, ae_true);
x = &_x;
ae_vector_init_copy(&_y, y, _state, ae_true);
y = &_y;
ae_vector_init_copy(&_w, w, _state, ae_true);
w = &_w;
ae_vector_init_copy(&_xc, xc, _state, ae_true);
xc = &_xc;
ae_vector_init_copy(&_yc, yc, _state, ae_true);
yc = &_yc;
*info = 0;
_barycentricinterpolant_clear(b);
_barycentricfitreport_clear(rep);
ae_matrix_init(&fmatrix, 0, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&cmatrix, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&y2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&w2, 0, DT_REAL, _state, ae_true);
ae_vector_init(&sx, 0, DT_REAL, _state, ae_true);
ae_vector_init(&sy, 0, DT_REAL, _state, ae_true);
ae_vector_init(&sbf, 0, DT_REAL, _state, ae_true);
ae_vector_init(&xoriginal, 0, DT_REAL, _state, ae_true);
ae_vector_init(&yoriginal, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmp, 0, DT_REAL, _state, ae_true);
_lsfitreport_init(&lrep, _state, ae_true);
_barycentricinterpolant_init(&b2, _state, ae_true);
if( ((n<1||m<2)||k<0)||k>=m )
{
*info = -1;
ae_frame_leave(_state);
return;
}
for(i=0; i<=k-1; i++)
{
*info = 0;
if( dc->ptr.p_int[i]<0 )
{
*info = -1;
}
if( dc->ptr.p_int[i]>1 )
{
*info = -1;
}
if( *info<0 )
{
ae_frame_leave(_state);
return;
}
}
/*
* weight decay for correct handling of task which becomes
* degenerate after constraints are applied
*/
decay = 10000*ae_machineepsilon;
/*
* Scale X, Y, XC, YC
*/
lsfitscalexy(x, y, w, n, xc, yc, dc, k, &xa, &xb, &sa, &sb, &xoriginal, &yoriginal, _state);
/*
* allocate space, initialize:
* * FMatrix- values of basis functions at X[]
* * CMatrix- values (derivatives) of basis functions at XC[]
*/
ae_vector_set_length(&y2, n+m, _state);
ae_vector_set_length(&w2, n+m, _state);
ae_matrix_set_length(&fmatrix, n+m, m, _state);
if( k>0 )
{
ae_matrix_set_length(&cmatrix, k, m+1, _state);
}
ae_vector_set_length(&y2, n+m, _state);
ae_vector_set_length(&w2, n+m, _state);
/*
* Prepare design and constraints matrices:
* * fill constraints matrix
* * fill first N rows of design matrix with values
* * fill next M rows of design matrix with regularizing term
* * append M zeros to Y
* * append M elements, mean(abs(W)) each, to W
*/
ae_vector_set_length(&sx, m, _state);
ae_vector_set_length(&sy, m, _state);
ae_vector_set_length(&sbf, m, _state);
for(j=0; j<=m-1; j++)
{
sx.ptr.p_double[j] = (double)(2*j)/(double)(m-1)-1;
}
for(i=0; i<=m-1; i++)
{
sy.ptr.p_double[i] = 1;
}
barycentricbuildfloaterhormann(&sx, &sy, m, d, &b2, _state);
mx = 0;
for(i=0; i<=n-1; i++)
{
lsfit_barycentriccalcbasis(&b2, x->ptr.p_double[i], &sbf, _state);
ae_v_move(&fmatrix.ptr.pp_double[i][0], 1, &sbf.ptr.p_double[0], 1, ae_v_len(0,m-1));
y2.ptr.p_double[i] = y->ptr.p_double[i];
w2.ptr.p_double[i] = w->ptr.p_double[i];
mx = mx+ae_fabs(w->ptr.p_double[i], _state)/n;
}
for(i=0; i<=m-1; i++)
{
for(j=0; j<=m-1; j++)
{
if( i==j )
{
fmatrix.ptr.pp_double[n+i][j] = decay;
}
else
{
fmatrix.ptr.pp_double[n+i][j] = 0;
}
}
y2.ptr.p_double[n+i] = 0;
w2.ptr.p_double[n+i] = mx;
}
if( k>0 )
{
for(j=0; j<=m-1; j++)
{
for(i=0; i<=m-1; i++)
{
sy.ptr.p_double[i] = 0;
}
sy.ptr.p_double[j] = 1;
barycentricbuildfloaterhormann(&sx, &sy, m, d, &b2, _state);
for(i=0; i<=k-1; i++)
{
ae_assert(dc->ptr.p_int[i]>=0&&dc->ptr.p_int[i]<=1, "BarycentricFit: internal error!", _state);
barycentricdiff1(&b2, xc->ptr.p_double[i], &v0, &v1, _state);
if( dc->ptr.p_int[i]==0 )
{
cmatrix.ptr.pp_double[i][j] = v0;
}
if( dc->ptr.p_int[i]==1 )
{
cmatrix.ptr.pp_double[i][j] = v1;
}
}
}
for(i=0; i<=k-1; i++)
{
cmatrix.ptr.pp_double[i][m] = yc->ptr.p_double[i];
}
}
/*
* Solve constrained task
*/
if( k>0 )
{
/*
* solve using regularization
*/
lsfitlinearwc(&y2, &w2, &fmatrix, &cmatrix, n+m, m, k, info, &tmp, &lrep, _state);
}
else
{
/*
* no constraints, no regularization needed
*/
lsfitlinearwc(y, w, &fmatrix, &cmatrix, n, m, k, info, &tmp, &lrep, _state);
}
if( *info<0 )
{
ae_frame_leave(_state);
return;
}
/*
* Generate interpolant and scale it
*/
ae_v_move(&sy.ptr.p_double[0], 1, &tmp.ptr.p_double[0], 1, ae_v_len(0,m-1));
barycentricbuildfloaterhormann(&sx, &sy, m, d, b, _state);
barycentriclintransx(b, 2/(xb-xa), -(xa+xb)/(xb-xa), _state);
barycentriclintransy(b, sb-sa, sa, _state);
/*
* Scale absolute errors obtained from LSFitLinearW.
* Relative error should be calculated separately
* (because of shifting/scaling of the task)
*/
rep->taskrcond = lrep.taskrcond;
rep->rmserror = lrep.rmserror*(sb-sa);
rep->avgerror = lrep.avgerror*(sb-sa);
rep->maxerror = lrep.maxerror*(sb-sa);
rep->avgrelerror = 0;
relcnt = 0;
for(i=0; i<=n-1; i++)
{
if( ae_fp_neq(yoriginal.ptr.p_double[i],0) )
{
rep->avgrelerror = rep->avgrelerror+ae_fabs(barycentriccalc(b, xoriginal.ptr.p_double[i], _state)-yoriginal.ptr.p_double[i], _state)/ae_fabs(yoriginal.ptr.p_double[i], _state);
relcnt = relcnt+1;
}
}
if( relcnt!=0 )
{
rep->avgrelerror = rep->avgrelerror/relcnt;
}
ae_frame_leave(_state);
}
static void lsfit_clearreport(lsfitreport* rep, ae_state *_state)
{
rep->taskrcond = 0;
rep->iterationscount = 0;
rep->varidx = -1;
rep->rmserror = 0;
rep->avgerror = 0;
rep->avgrelerror = 0;
rep->maxerror = 0;
rep->wrmserror = 0;
rep->r2 = 0;
ae_matrix_set_length(&rep->covpar, 0, 0, _state);
ae_vector_set_length(&rep->errpar, 0, _state);
ae_vector_set_length(&rep->errcurve, 0, _state);
ae_vector_set_length(&rep->noise, 0, _state);
}
/*************************************************************************
This internal function estimates covariance matrix and other error-related
information for linear/nonlinear least squares model.
It has a bit awkward interface, but it can be used for both linear and
nonlinear problems.
INPUT PARAMETERS:
F1 - array[0..N-1,0..K-1]:
* for linear problems - matrix of function values
* for nonlinear problems - Jacobian matrix
F0 - array[0..N-1]:
* for linear problems - must be filled with zeros
* for nonlinear problems - must store values of function being
fitted
Y - array[0..N-1]:
* for linear and nonlinear problems - must store target values
W - weights, array[0..N-1]:
* for linear and nonlinear problems - weights
X - array[0..K-1]:
* for linear and nonlinear problems - current solution
S - array[0..K-1]:
* its components should be strictly positive
* squared inverse of this diagonal matrix is used as damping
factor for covariance matrix (linear and nonlinear problems)
* for nonlinear problems, when scale of the variables is usually
explicitly given by user, you may use scale vector for this
parameter
* for linear problems you may set this parameter to
S=sqrt(1/diag(F'*F))
* this parameter is automatically rescaled by this function,
only relative magnitudes of its components (with respect to
each other) matter.
N - number of points, N>0.
K - number of dimensions
Rep - structure which is used to store results
Z - additional matrix which, depending on ZKind, may contain some
information used to accelerate calculations - or just can be
temporary buffer:
* for ZKind=0 Z contains no information, just temporary
buffer which can be resized and used as needed
* for ZKind=1 Z contains triangular matrix from QR
decomposition of W*F1. This matrix can be used
to speedup calculation of covariance matrix.
It should not be changed by algorithm.
ZKind- contents of Z
OUTPUT PARAMETERS:
* Rep.CovPar covariance matrix for parameters, array[K,K].
* Rep.ErrPar errors in parameters, array[K],
errpar = sqrt(diag(CovPar))
* Rep.ErrCurve vector of fit errors - standard deviations of empirical
best-fit curve from "ideal" best-fit curve built with
infinite number of samples, array[N].
errcurve = sqrt(diag(J*CovPar*J')),
where J is Jacobian matrix.
* Rep.Noise vector of per-point estimates of noise, array[N]
* Rep.R2 coefficient of determination (non-weighted)
Other fields of Rep are not changed.
IMPORTANT: errors in parameters are calculated without taking into
account boundary/linear constraints! Presence of constraints
changes distribution of errors, but there is no easy way to
account for constraints when you calculate covariance matrix.
NOTE: noise in the data is estimated as follows:
* for fitting without user-supplied weights all points are
assumed to have same level of noise, which is estimated from
the data
* for fitting with user-supplied weights we assume that noise
level in I-th point is inversely proportional to Ith weight.
Coefficient of proportionality is estimated from the data.
NOTE: we apply small amount of regularization when we invert squared
Jacobian and calculate covariance matrix. It guarantees that
algorithm won't divide by zero during inversion, but skews
error estimates a bit (fractional error is about 10^-9).
However, we believe that this difference is insignificant for
all practical purposes except for the situation when you want
to compare ALGLIB results with "reference" implementation up
to the last significant digit.
-- ALGLIB PROJECT --
Copyright 10.12.2009 by Bochkanov Sergey
*************************************************************************/
static void lsfit_estimateerrors(/* Real */ ae_matrix* f1,
/* Real */ ae_vector* f0,
/* Real */ ae_vector* y,
/* Real */ ae_vector* w,
/* Real */ ae_vector* x,
/* Real */ ae_vector* s,
ae_int_t n,
ae_int_t k,
lsfitreport* rep,
/* Real */ ae_matrix* z,
ae_int_t zkind,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _s;
ae_int_t i;
ae_int_t j;
ae_int_t j1;
double v;
double noisec;
ae_int_t info;
matinvreport invrep;
ae_int_t nzcnt;
double avg;
double rss;
double tss;
double sz;
double ss;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_s, s, _state, ae_true);
s = &_s;
_matinvreport_init(&invrep, _state, ae_true);
/*
* Compute NZCnt - count of non-zero weights
*/
nzcnt = 0;
for(i=0; i<=n-1; i++)
{
if( ae_fp_neq(w->ptr.p_double[i],0) )
{
nzcnt = nzcnt+1;
}
}
/*
* Compute R2
*/
if( nzcnt>0 )
{
avg = 0.0;
for(i=0; i<=n-1; i++)
{
if( ae_fp_neq(w->ptr.p_double[i],0) )
{
avg = avg+y->ptr.p_double[i];
}
}
avg = avg/nzcnt;
rss = 0.0;
tss = 0.0;
for(i=0; i<=n-1; i++)
{
if( ae_fp_neq(w->ptr.p_double[i],0) )
{
v = ae_v_dotproduct(&f1->ptr.pp_double[i][0], 1, &x->ptr.p_double[0], 1, ae_v_len(0,k-1));
v = v+f0->ptr.p_double[i];
rss = rss+ae_sqr(v-y->ptr.p_double[i], _state);
tss = tss+ae_sqr(y->ptr.p_double[i]-avg, _state);
}
}
if( ae_fp_neq(tss,0) )
{
rep->r2 = ae_maxreal(1.0-rss/tss, 0.0, _state);
}
else
{
rep->r2 = 1.0;
}
}
else
{
rep->r2 = 0;
}
/*
* Compute estimate of proportionality between noise in the data and weights:
* NoiseC = mean(per-point-noise*per-point-weight)
* Noise level (standard deviation) at each point is equal to NoiseC/W[I].
*/
if( nzcnt>k )
{
noisec = 0.0;
for(i=0; i<=n-1; i++)
{
if( ae_fp_neq(w->ptr.p_double[i],0) )
{
v = ae_v_dotproduct(&f1->ptr.pp_double[i][0], 1, &x->ptr.p_double[0], 1, ae_v_len(0,k-1));
v = v+f0->ptr.p_double[i];
noisec = noisec+ae_sqr((v-y->ptr.p_double[i])*w->ptr.p_double[i], _state);
}
}
noisec = ae_sqrt(noisec/(nzcnt-k), _state);
}
else
{
noisec = 0.0;
}
/*
* Two branches on noise level:
* * NoiseC>0 normal situation
* * NoiseC=0 degenerate case CovPar is filled by zeros
*/
rmatrixsetlengthatleast(&rep->covpar, k, k, _state);
if( ae_fp_greater(noisec,0) )
{
/*
* Normal situation: non-zero noise level
*/
ae_assert(zkind==0||zkind==1, "LSFit: internal error in EstimateErrors() function", _state);
if( zkind==0 )
{
/*
* Z contains no additional information which can be used to speed up
* calculations. We have to calculate covariance matrix on our own:
* * Compute scaled Jacobian N*J, where N[i,i]=WCur[I]/NoiseC, store in Z
* * Compute Z'*Z, store in CovPar
* * Apply moderate regularization to CovPar and compute matrix inverse.
* In case inverse failed, increase regularization parameter and try
* again.
*/
rmatrixsetlengthatleast(z, n, k, _state);
for(i=0; i<=n-1; i++)
{
v = w->ptr.p_double[i]/noisec;
ae_v_moved(&z->ptr.pp_double[i][0], 1, &f1->ptr.pp_double[i][0], 1, ae_v_len(0,k-1), v);
}
/*
* Convert S to automatically scaled damped matrix:
* * calculate SZ - sum of diagonal elements of Z'*Z
* * calculate SS - sum of diagonal elements of S^(-2)
* * overwrite S by (SZ/SS)*S^(-2)
* * now S has approximately same magnitude as giagonal of Z'*Z
*/
sz = 0;
for(i=0; i<=n-1; i++)
{
for(j=0; j<=k-1; j++)
{
sz = sz+z->ptr.pp_double[i][j]*z->ptr.pp_double[i][j];
}
}
if( ae_fp_eq(sz,0) )
{
sz = 1;
}
ss = 0;
for(j=0; j<=k-1; j++)
{
ss = ss+1/ae_sqr(s->ptr.p_double[j], _state);
}
for(j=0; j<=k-1; j++)
{
s->ptr.p_double[j] = sz/ss/ae_sqr(s->ptr.p_double[j], _state);
}
/*
* Calculate damped inverse inv(Z'*Z+S).
* We increase damping factor V until Z'*Z become well-conditioned.
*/
v = 1.0E3*ae_machineepsilon;
do
{
rmatrixsyrk(k, n, 1.0, z, 0, 0, 2, 0.0, &rep->covpar, 0, 0, ae_true, _state);
for(i=0; i<=k-1; i++)
{
rep->covpar.ptr.pp_double[i][i] = rep->covpar.ptr.pp_double[i][i]+v*s->ptr.p_double[i];
}
spdmatrixinverse(&rep->covpar, k, ae_true, &info, &invrep, _state);
v = 10*v;
}
while(info<=0);
for(i=0; i<=k-1; i++)
{
for(j=i+1; j<=k-1; j++)
{
rep->covpar.ptr.pp_double[j][i] = rep->covpar.ptr.pp_double[i][j];
}
}
}
if( zkind==1 )
{
/*
* We can reuse additional information:
* * Z contains R matrix from QR decomposition of W*F1
* * After multiplication by 1/NoiseC we get Z_mod = N*F1, where diag(N)=w[i]/NoiseC
* * Such triangular Z_mod is a Cholesky factor from decomposition of J'*N'*N*J.
* Thus, we can calculate covariance matrix as inverse of the matrix given by
* its Cholesky decomposition. It allow us to avoid time-consuming calculation
* of J'*N'*N*J in CovPar - complexity is reduced from O(N*K^2) to O(K^3), which
* is quite good because K is usually orders of magnitude smaller than N.
*
* First, convert S to automatically scaled damped matrix:
* * calculate SZ - sum of magnitudes of diagonal elements of Z/NoiseC
* * calculate SS - sum of diagonal elements of S^(-1)
* * overwrite S by (SZ/SS)*S^(-1)
* * now S has approximately same magnitude as giagonal of Z'*Z
*/
sz = 0;
for(j=0; j<=k-1; j++)
{
sz = sz+ae_fabs(z->ptr.pp_double[j][j]/noisec, _state);
}
if( ae_fp_eq(sz,0) )
{
sz = 1;
}
ss = 0;
for(j=0; j<=k-1; j++)
{
ss = ss+1/s->ptr.p_double[j];
}
for(j=0; j<=k-1; j++)
{
s->ptr.p_double[j] = sz/ss/s->ptr.p_double[j];
}
/*
* Calculate damped inverse of inv((Z+v*S)'*(Z+v*S))
* We increase damping factor V until matrix become well-conditioned.
*/
v = 1.0E3*ae_machineepsilon;
do
{
for(i=0; i<=k-1; i++)
{
for(j=i; j<=k-1; j++)
{
rep->covpar.ptr.pp_double[i][j] = z->ptr.pp_double[i][j]/noisec;
}
rep->covpar.ptr.pp_double[i][i] = rep->covpar.ptr.pp_double[i][i]+v*s->ptr.p_double[i];
}
spdmatrixcholeskyinverse(&rep->covpar, k, ae_true, &info, &invrep, _state);
v = 10*v;
}
while(info<=0);
for(i=0; i<=k-1; i++)
{
for(j=i+1; j<=k-1; j++)
{
rep->covpar.ptr.pp_double[j][i] = rep->covpar.ptr.pp_double[i][j];
}
}
}
}
else
{
/*
* Degenerate situation: zero noise level, covariance matrix is zero.
*/
for(i=0; i<=k-1; i++)
{
for(j=0; j<=k-1; j++)
{
rep->covpar.ptr.pp_double[j][i] = 0;
}
}
}
/*
* Estimate erorrs in parameters, curve and per-point noise
*/
rvectorsetlengthatleast(&rep->errpar, k, _state);
rvectorsetlengthatleast(&rep->errcurve, n, _state);
rvectorsetlengthatleast(&rep->noise, n, _state);
for(i=0; i<=k-1; i++)
{
rep->errpar.ptr.p_double[i] = ae_sqrt(rep->covpar.ptr.pp_double[i][i], _state);
}
for(i=0; i<=n-1; i++)
{
/*
* ErrCurve[I] is sqrt(P[i,i]) where P=J*CovPar*J'
*/
v = 0.0;
for(j=0; j<=k-1; j++)
{
for(j1=0; j1<=k-1; j1++)
{
v = v+f1->ptr.pp_double[i][j]*rep->covpar.ptr.pp_double[j][j1]*f1->ptr.pp_double[i][j1];
}
}
rep->errcurve.ptr.p_double[i] = ae_sqrt(v, _state);
/*
* Noise[i] is filled using weights and current estimate of noise level
*/
if( ae_fp_neq(w->ptr.p_double[i],0) )
{
rep->noise.ptr.p_double[i] = noisec/w->ptr.p_double[i];
}
else
{
rep->noise.ptr.p_double[i] = 0;
}
}
ae_frame_leave(_state);
}
ae_bool _polynomialfitreport_init(void* _p, ae_state *_state, ae_bool make_automatic)
{
polynomialfitreport *p = (polynomialfitreport*)_p;
ae_touch_ptr((void*)p);
return ae_true;
}
ae_bool _polynomialfitreport_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic)
{
polynomialfitreport *dst = (polynomialfitreport*)_dst;
polynomialfitreport *src = (polynomialfitreport*)_src;
dst->taskrcond = src->taskrcond;
dst->rmserror = src->rmserror;
dst->avgerror = src->avgerror;
dst->avgrelerror = src->avgrelerror;
dst->maxerror = src->maxerror;
return ae_true;
}
void _polynomialfitreport_clear(void* _p)
{
polynomialfitreport *p = (polynomialfitreport*)_p;
ae_touch_ptr((void*)p);
}
void _polynomialfitreport_destroy(void* _p)
{
polynomialfitreport *p = (polynomialfitreport*)_p;
ae_touch_ptr((void*)p);
}
ae_bool _barycentricfitreport_init(void* _p, ae_state *_state, ae_bool make_automatic)
{
barycentricfitreport *p = (barycentricfitreport*)_p;
ae_touch_ptr((void*)p);
return ae_true;
}
ae_bool _barycentricfitreport_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic)
{
barycentricfitreport *dst = (barycentricfitreport*)_dst;
barycentricfitreport *src = (barycentricfitreport*)_src;
dst->taskrcond = src->taskrcond;
dst->dbest = src->dbest;
dst->rmserror = src->rmserror;
dst->avgerror = src->avgerror;
dst->avgrelerror = src->avgrelerror;
dst->maxerror = src->maxerror;
return ae_true;
}
void _barycentricfitreport_clear(void* _p)
{
barycentricfitreport *p = (barycentricfitreport*)_p;
ae_touch_ptr((void*)p);
}
void _barycentricfitreport_destroy(void* _p)
{
barycentricfitreport *p = (barycentricfitreport*)_p;
ae_touch_ptr((void*)p);
}
ae_bool _spline1dfitreport_init(void* _p, ae_state *_state, ae_bool make_automatic)
{
spline1dfitreport *p = (spline1dfitreport*)_p;
ae_touch_ptr((void*)p);
return ae_true;
}
ae_bool _spline1dfitreport_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic)
{
spline1dfitreport *dst = (spline1dfitreport*)_dst;
spline1dfitreport *src = (spline1dfitreport*)_src;
dst->taskrcond = src->taskrcond;
dst->rmserror = src->rmserror;
dst->avgerror = src->avgerror;
dst->avgrelerror = src->avgrelerror;
dst->maxerror = src->maxerror;
return ae_true;
}
void _spline1dfitreport_clear(void* _p)
{
spline1dfitreport *p = (spline1dfitreport*)_p;
ae_touch_ptr((void*)p);
}
void _spline1dfitreport_destroy(void* _p)
{
spline1dfitreport *p = (spline1dfitreport*)_p;
ae_touch_ptr((void*)p);
}
ae_bool _lsfitreport_init(void* _p, ae_state *_state, ae_bool make_automatic)
{
lsfitreport *p = (lsfitreport*)_p;
ae_touch_ptr((void*)p);
if( !ae_matrix_init(&p->covpar, 0, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->errpar, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->errcurve, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->noise, 0, DT_REAL, _state, make_automatic) )
return ae_false;
return ae_true;
}
ae_bool _lsfitreport_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic)
{
lsfitreport *dst = (lsfitreport*)_dst;
lsfitreport *src = (lsfitreport*)_src;
dst->taskrcond = src->taskrcond;
dst->iterationscount = src->iterationscount;
dst->varidx = src->varidx;
dst->rmserror = src->rmserror;
dst->avgerror = src->avgerror;
dst->avgrelerror = src->avgrelerror;
dst->maxerror = src->maxerror;
dst->wrmserror = src->wrmserror;
if( !ae_matrix_init_copy(&dst->covpar, &src->covpar, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->errpar, &src->errpar, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->errcurve, &src->errcurve, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->noise, &src->noise, _state, make_automatic) )
return ae_false;
dst->r2 = src->r2;
return ae_true;
}
void _lsfitreport_clear(void* _p)
{
lsfitreport *p = (lsfitreport*)_p;
ae_touch_ptr((void*)p);
ae_matrix_clear(&p->covpar);
ae_vector_clear(&p->errpar);
ae_vector_clear(&p->errcurve);
ae_vector_clear(&p->noise);
}
void _lsfitreport_destroy(void* _p)
{
lsfitreport *p = (lsfitreport*)_p;
ae_touch_ptr((void*)p);
ae_matrix_destroy(&p->covpar);
ae_vector_destroy(&p->errpar);
ae_vector_destroy(&p->errcurve);
ae_vector_destroy(&p->noise);
}
ae_bool _lsfitstate_init(void* _p, ae_state *_state, ae_bool make_automatic)
{
lsfitstate *p = (lsfitstate*)_p;
ae_touch_ptr((void*)p);
if( !ae_vector_init(&p->s, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->bndl, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->bndu, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init(&p->taskx, 0, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->tasky, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->taskw, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->x, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->c, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->g, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init(&p->h, 0, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->wcur, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->tmp, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->tmpf, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init(&p->tmpjac, 0, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init(&p->tmpjacw, 0, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !_matinvreport_init(&p->invrep, _state, make_automatic) )
return ae_false;
if( !_lsfitreport_init(&p->rep, _state, make_automatic) )
return ae_false;
if( !_minlmstate_init(&p->optstate, _state, make_automatic) )
return ae_false;
if( !_minlmreport_init(&p->optrep, _state, make_automatic) )
return ae_false;
if( !_rcommstate_init(&p->rstate, _state, make_automatic) )
return ae_false;
return ae_true;
}
ae_bool _lsfitstate_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic)
{
lsfitstate *dst = (lsfitstate*)_dst;
lsfitstate *src = (lsfitstate*)_src;
dst->optalgo = src->optalgo;
dst->m = src->m;
dst->k = src->k;
dst->epsf = src->epsf;
dst->epsx = src->epsx;
dst->maxits = src->maxits;
dst->stpmax = src->stpmax;
dst->xrep = src->xrep;
if( !ae_vector_init_copy(&dst->s, &src->s, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->bndl, &src->bndl, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->bndu, &src->bndu, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init_copy(&dst->taskx, &src->taskx, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->tasky, &src->tasky, _state, make_automatic) )
return ae_false;
dst->npoints = src->npoints;
if( !ae_vector_init_copy(&dst->taskw, &src->taskw, _state, make_automatic) )
return ae_false;
dst->nweights = src->nweights;
dst->wkind = src->wkind;
dst->wits = src->wits;
dst->diffstep = src->diffstep;
dst->teststep = src->teststep;
dst->xupdated = src->xupdated;
dst->needf = src->needf;
dst->needfg = src->needfg;
dst->needfgh = src->needfgh;
dst->pointindex = src->pointindex;
if( !ae_vector_init_copy(&dst->x, &src->x, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->c, &src->c, _state, make_automatic) )
return ae_false;
dst->f = src->f;
if( !ae_vector_init_copy(&dst->g, &src->g, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init_copy(&dst->h, &src->h, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->wcur, &src->wcur, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->tmp, &src->tmp, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->tmpf, &src->tmpf, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init_copy(&dst->tmpjac, &src->tmpjac, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init_copy(&dst->tmpjacw, &src->tmpjacw, _state, make_automatic) )
return ae_false;
dst->tmpnoise = src->tmpnoise;
if( !_matinvreport_init_copy(&dst->invrep, &src->invrep, _state, make_automatic) )
return ae_false;
dst->repiterationscount = src->repiterationscount;
dst->repterminationtype = src->repterminationtype;
dst->repvaridx = src->repvaridx;
dst->reprmserror = src->reprmserror;
dst->repavgerror = src->repavgerror;
dst->repavgrelerror = src->repavgrelerror;
dst->repmaxerror = src->repmaxerror;
dst->repwrmserror = src->repwrmserror;
if( !_lsfitreport_init_copy(&dst->rep, &src->rep, _state, make_automatic) )
return ae_false;
if( !_minlmstate_init_copy(&dst->optstate, &src->optstate, _state, make_automatic) )
return ae_false;
if( !_minlmreport_init_copy(&dst->optrep, &src->optrep, _state, make_automatic) )
return ae_false;
dst->prevnpt = src->prevnpt;
dst->prevalgo = src->prevalgo;
if( !_rcommstate_init_copy(&dst->rstate, &src->rstate, _state, make_automatic) )
return ae_false;
return ae_true;
}
void _lsfitstate_clear(void* _p)
{
lsfitstate *p = (lsfitstate*)_p;
ae_touch_ptr((void*)p);
ae_vector_clear(&p->s);
ae_vector_clear(&p->bndl);
ae_vector_clear(&p->bndu);
ae_matrix_clear(&p->taskx);
ae_vector_clear(&p->tasky);
ae_vector_clear(&p->taskw);
ae_vector_clear(&p->x);
ae_vector_clear(&p->c);
ae_vector_clear(&p->g);
ae_matrix_clear(&p->h);
ae_vector_clear(&p->wcur);
ae_vector_clear(&p->tmp);
ae_vector_clear(&p->tmpf);
ae_matrix_clear(&p->tmpjac);
ae_matrix_clear(&p->tmpjacw);
_matinvreport_clear(&p->invrep);
_lsfitreport_clear(&p->rep);
_minlmstate_clear(&p->optstate);
_minlmreport_clear(&p->optrep);
_rcommstate_clear(&p->rstate);
}
void _lsfitstate_destroy(void* _p)
{
lsfitstate *p = (lsfitstate*)_p;
ae_touch_ptr((void*)p);
ae_vector_destroy(&p->s);
ae_vector_destroy(&p->bndl);
ae_vector_destroy(&p->bndu);
ae_matrix_destroy(&p->taskx);
ae_vector_destroy(&p->tasky);
ae_vector_destroy(&p->taskw);
ae_vector_destroy(&p->x);
ae_vector_destroy(&p->c);
ae_vector_destroy(&p->g);
ae_matrix_destroy(&p->h);
ae_vector_destroy(&p->wcur);
ae_vector_destroy(&p->tmp);
ae_vector_destroy(&p->tmpf);
ae_matrix_destroy(&p->tmpjac);
ae_matrix_destroy(&p->tmpjacw);
_matinvreport_destroy(&p->invrep);
_lsfitreport_destroy(&p->rep);
_minlmstate_destroy(&p->optstate);
_minlmreport_destroy(&p->optrep);
_rcommstate_destroy(&p->rstate);
}
/*************************************************************************
This function builds non-periodic 2-dimensional parametric spline which
starts at (X[0],Y[0]) and ends at (X[N-1],Y[N-1]).
INPUT PARAMETERS:
XY - points, array[0..N-1,0..1].
XY[I,0:1] corresponds to the Ith point.
Order of points is important!
N - points count, N>=5 for Akima splines, N>=2 for other types of
splines.
ST - spline type:
* 0 Akima spline
* 1 parabolically terminated Catmull-Rom spline (Tension=0)
* 2 parabolically terminated cubic spline
PT - parameterization type:
* 0 uniform
* 1 chord length
* 2 centripetal
OUTPUT PARAMETERS:
P - parametric spline interpolant
NOTES:
* this function assumes that there all consequent points are distinct.
I.e. (x0,y0)<>(x1,y1), (x1,y1)<>(x2,y2), (x2,y2)<>(x3,y3) and so on.
However, non-consequent points may coincide, i.e. we can have (x0,y0)=
=(x2,y2).
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2build(/* Real */ ae_matrix* xy,
ae_int_t n,
ae_int_t st,
ae_int_t pt,
pspline2interpolant* p,
ae_state *_state)
{
ae_frame _frame_block;
ae_matrix _xy;
ae_vector tmp;
ae_frame_make(_state, &_frame_block);
ae_matrix_init_copy(&_xy, xy, _state, ae_true);
xy = &_xy;
_pspline2interpolant_clear(p);
ae_vector_init(&tmp, 0, DT_REAL, _state, ae_true);
ae_assert(st>=0&&st<=2, "PSpline2Build: incorrect spline type!", _state);
ae_assert(pt>=0&&pt<=2, "PSpline2Build: incorrect parameterization type!", _state);
if( st==0 )
{
ae_assert(n>=5, "PSpline2Build: N<5 (minimum value for Akima splines)!", _state);
}
else
{
ae_assert(n>=2, "PSpline2Build: N<2!", _state);
}
/*
* Prepare
*/
p->n = n;
p->periodic = ae_false;
ae_vector_set_length(&tmp, n, _state);
/*
* Build parameterization, check that all parameters are distinct
*/
pspline_pspline2par(xy, n, pt, &p->p, _state);
ae_assert(aredistinct(&p->p, n, _state), "PSpline2Build: consequent points are too close!", _state);
/*
* Build splines
*/
if( st==0 )
{
ae_v_move(&tmp.ptr.p_double[0], 1, &xy->ptr.pp_double[0][0], xy->stride, ae_v_len(0,n-1));
spline1dbuildakima(&p->p, &tmp, n, &p->x, _state);
ae_v_move(&tmp.ptr.p_double[0], 1, &xy->ptr.pp_double[0][1], xy->stride, ae_v_len(0,n-1));
spline1dbuildakima(&p->p, &tmp, n, &p->y, _state);
}
if( st==1 )
{
ae_v_move(&tmp.ptr.p_double[0], 1, &xy->ptr.pp_double[0][0], xy->stride, ae_v_len(0,n-1));
spline1dbuildcatmullrom(&p->p, &tmp, n, 0, 0.0, &p->x, _state);
ae_v_move(&tmp.ptr.p_double[0], 1, &xy->ptr.pp_double[0][1], xy->stride, ae_v_len(0,n-1));
spline1dbuildcatmullrom(&p->p, &tmp, n, 0, 0.0, &p->y, _state);
}
if( st==2 )
{
ae_v_move(&tmp.ptr.p_double[0], 1, &xy->ptr.pp_double[0][0], xy->stride, ae_v_len(0,n-1));
spline1dbuildcubic(&p->p, &tmp, n, 0, 0.0, 0, 0.0, &p->x, _state);
ae_v_move(&tmp.ptr.p_double[0], 1, &xy->ptr.pp_double[0][1], xy->stride, ae_v_len(0,n-1));
spline1dbuildcubic(&p->p, &tmp, n, 0, 0.0, 0, 0.0, &p->y, _state);
}
ae_frame_leave(_state);
}
/*************************************************************************
This function builds non-periodic 3-dimensional parametric spline which
starts at (X[0],Y[0],Z[0]) and ends at (X[N-1],Y[N-1],Z[N-1]).
Same as PSpline2Build() function, but for 3D, so we won't duplicate its
description here.
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3build(/* Real */ ae_matrix* xy,
ae_int_t n,
ae_int_t st,
ae_int_t pt,
pspline3interpolant* p,
ae_state *_state)
{
ae_frame _frame_block;
ae_matrix _xy;
ae_vector tmp;
ae_frame_make(_state, &_frame_block);
ae_matrix_init_copy(&_xy, xy, _state, ae_true);
xy = &_xy;
_pspline3interpolant_clear(p);
ae_vector_init(&tmp, 0, DT_REAL, _state, ae_true);
ae_assert(st>=0&&st<=2, "PSpline3Build: incorrect spline type!", _state);
ae_assert(pt>=0&&pt<=2, "PSpline3Build: incorrect parameterization type!", _state);
if( st==0 )
{
ae_assert(n>=5, "PSpline3Build: N<5 (minimum value for Akima splines)!", _state);
}
else
{
ae_assert(n>=2, "PSpline3Build: N<2!", _state);
}
/*
* Prepare
*/
p->n = n;
p->periodic = ae_false;
ae_vector_set_length(&tmp, n, _state);
/*
* Build parameterization, check that all parameters are distinct
*/
pspline_pspline3par(xy, n, pt, &p->p, _state);
ae_assert(aredistinct(&p->p, n, _state), "PSpline3Build: consequent points are too close!", _state);
/*
* Build splines
*/
if( st==0 )
{
ae_v_move(&tmp.ptr.p_double[0], 1, &xy->ptr.pp_double[0][0], xy->stride, ae_v_len(0,n-1));
spline1dbuildakima(&p->p, &tmp, n, &p->x, _state);
ae_v_move(&tmp.ptr.p_double[0], 1, &xy->ptr.pp_double[0][1], xy->stride, ae_v_len(0,n-1));
spline1dbuildakima(&p->p, &tmp, n, &p->y, _state);
ae_v_move(&tmp.ptr.p_double[0], 1, &xy->ptr.pp_double[0][2], xy->stride, ae_v_len(0,n-1));
spline1dbuildakima(&p->p, &tmp, n, &p->z, _state);
}
if( st==1 )
{
ae_v_move(&tmp.ptr.p_double[0], 1, &xy->ptr.pp_double[0][0], xy->stride, ae_v_len(0,n-1));
spline1dbuildcatmullrom(&p->p, &tmp, n, 0, 0.0, &p->x, _state);
ae_v_move(&tmp.ptr.p_double[0], 1, &xy->ptr.pp_double[0][1], xy->stride, ae_v_len(0,n-1));
spline1dbuildcatmullrom(&p->p, &tmp, n, 0, 0.0, &p->y, _state);
ae_v_move(&tmp.ptr.p_double[0], 1, &xy->ptr.pp_double[0][2], xy->stride, ae_v_len(0,n-1));
spline1dbuildcatmullrom(&p->p, &tmp, n, 0, 0.0, &p->z, _state);
}
if( st==2 )
{
ae_v_move(&tmp.ptr.p_double[0], 1, &xy->ptr.pp_double[0][0], xy->stride, ae_v_len(0,n-1));
spline1dbuildcubic(&p->p, &tmp, n, 0, 0.0, 0, 0.0, &p->x, _state);
ae_v_move(&tmp.ptr.p_double[0], 1, &xy->ptr.pp_double[0][1], xy->stride, ae_v_len(0,n-1));
spline1dbuildcubic(&p->p, &tmp, n, 0, 0.0, 0, 0.0, &p->y, _state);
ae_v_move(&tmp.ptr.p_double[0], 1, &xy->ptr.pp_double[0][2], xy->stride, ae_v_len(0,n-1));
spline1dbuildcubic(&p->p, &tmp, n, 0, 0.0, 0, 0.0, &p->z, _state);
}
ae_frame_leave(_state);
}
/*************************************************************************
This function builds periodic 2-dimensional parametric spline which
starts at (X[0],Y[0]), goes through all points to (X[N-1],Y[N-1]) and then
back to (X[0],Y[0]).
INPUT PARAMETERS:
XY - points, array[0..N-1,0..1].
XY[I,0:1] corresponds to the Ith point.
XY[N-1,0:1] must be different from XY[0,0:1].
Order of points is important!
N - points count, N>=3 for other types of splines.
ST - spline type:
* 1 Catmull-Rom spline (Tension=0) with cyclic boundary conditions
* 2 cubic spline with cyclic boundary conditions
PT - parameterization type:
* 0 uniform
* 1 chord length
* 2 centripetal
OUTPUT PARAMETERS:
P - parametric spline interpolant
NOTES:
* this function assumes that there all consequent points are distinct.
I.e. (x0,y0)<>(x1,y1), (x1,y1)<>(x2,y2), (x2,y2)<>(x3,y3) and so on.
However, non-consequent points may coincide, i.e. we can have (x0,y0)=
=(x2,y2).
* last point of sequence is NOT equal to the first point. You shouldn't
make curve "explicitly periodic" by making them equal.
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2buildperiodic(/* Real */ ae_matrix* xy,
ae_int_t n,
ae_int_t st,
ae_int_t pt,
pspline2interpolant* p,
ae_state *_state)
{
ae_frame _frame_block;
ae_matrix _xy;
ae_matrix xyp;
ae_vector tmp;
ae_frame_make(_state, &_frame_block);
ae_matrix_init_copy(&_xy, xy, _state, ae_true);
xy = &_xy;
_pspline2interpolant_clear(p);
ae_matrix_init(&xyp, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmp, 0, DT_REAL, _state, ae_true);
ae_assert(st>=1&&st<=2, "PSpline2BuildPeriodic: incorrect spline type!", _state);
ae_assert(pt>=0&&pt<=2, "PSpline2BuildPeriodic: incorrect parameterization type!", _state);
ae_assert(n>=3, "PSpline2BuildPeriodic: N<3!", _state);
/*
* Prepare
*/
p->n = n;
p->periodic = ae_true;
ae_vector_set_length(&tmp, n+1, _state);
ae_matrix_set_length(&xyp, n+1, 2, _state);
ae_v_move(&xyp.ptr.pp_double[0][0], xyp.stride, &xy->ptr.pp_double[0][0], xy->stride, ae_v_len(0,n-1));
ae_v_move(&xyp.ptr.pp_double[0][1], xyp.stride, &xy->ptr.pp_double[0][1], xy->stride, ae_v_len(0,n-1));
ae_v_move(&xyp.ptr.pp_double[n][0], 1, &xy->ptr.pp_double[0][0], 1, ae_v_len(0,1));
/*
* Build parameterization, check that all parameters are distinct
*/
pspline_pspline2par(&xyp, n+1, pt, &p->p, _state);
ae_assert(aredistinct(&p->p, n+1, _state), "PSpline2BuildPeriodic: consequent (or first and last) points are too close!", _state);
/*
* Build splines
*/
if( st==1 )
{
ae_v_move(&tmp.ptr.p_double[0], 1, &xyp.ptr.pp_double[0][0], xyp.stride, ae_v_len(0,n));
spline1dbuildcatmullrom(&p->p, &tmp, n+1, -1, 0.0, &p->x, _state);
ae_v_move(&tmp.ptr.p_double[0], 1, &xyp.ptr.pp_double[0][1], xyp.stride, ae_v_len(0,n));
spline1dbuildcatmullrom(&p->p, &tmp, n+1, -1, 0.0, &p->y, _state);
}
if( st==2 )
{
ae_v_move(&tmp.ptr.p_double[0], 1, &xyp.ptr.pp_double[0][0], xyp.stride, ae_v_len(0,n));
spline1dbuildcubic(&p->p, &tmp, n+1, -1, 0.0, -1, 0.0, &p->x, _state);
ae_v_move(&tmp.ptr.p_double[0], 1, &xyp.ptr.pp_double[0][1], xyp.stride, ae_v_len(0,n));
spline1dbuildcubic(&p->p, &tmp, n+1, -1, 0.0, -1, 0.0, &p->y, _state);
}
ae_frame_leave(_state);
}
/*************************************************************************
This function builds periodic 3-dimensional parametric spline which
starts at (X[0],Y[0],Z[0]), goes through all points to (X[N-1],Y[N-1],Z[N-1])
and then back to (X[0],Y[0],Z[0]).
Same as PSpline2Build() function, but for 3D, so we won't duplicate its
description here.
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3buildperiodic(/* Real */ ae_matrix* xy,
ae_int_t n,
ae_int_t st,
ae_int_t pt,
pspline3interpolant* p,
ae_state *_state)
{
ae_frame _frame_block;
ae_matrix _xy;
ae_matrix xyp;
ae_vector tmp;
ae_frame_make(_state, &_frame_block);
ae_matrix_init_copy(&_xy, xy, _state, ae_true);
xy = &_xy;
_pspline3interpolant_clear(p);
ae_matrix_init(&xyp, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmp, 0, DT_REAL, _state, ae_true);
ae_assert(st>=1&&st<=2, "PSpline3BuildPeriodic: incorrect spline type!", _state);
ae_assert(pt>=0&&pt<=2, "PSpline3BuildPeriodic: incorrect parameterization type!", _state);
ae_assert(n>=3, "PSpline3BuildPeriodic: N<3!", _state);
/*
* Prepare
*/
p->n = n;
p->periodic = ae_true;
ae_vector_set_length(&tmp, n+1, _state);
ae_matrix_set_length(&xyp, n+1, 3, _state);
ae_v_move(&xyp.ptr.pp_double[0][0], xyp.stride, &xy->ptr.pp_double[0][0], xy->stride, ae_v_len(0,n-1));
ae_v_move(&xyp.ptr.pp_double[0][1], xyp.stride, &xy->ptr.pp_double[0][1], xy->stride, ae_v_len(0,n-1));
ae_v_move(&xyp.ptr.pp_double[0][2], xyp.stride, &xy->ptr.pp_double[0][2], xy->stride, ae_v_len(0,n-1));
ae_v_move(&xyp.ptr.pp_double[n][0], 1, &xy->ptr.pp_double[0][0], 1, ae_v_len(0,2));
/*
* Build parameterization, check that all parameters are distinct
*/
pspline_pspline3par(&xyp, n+1, pt, &p->p, _state);
ae_assert(aredistinct(&p->p, n+1, _state), "PSplineBuild2Periodic: consequent (or first and last) points are too close!", _state);
/*
* Build splines
*/
if( st==1 )
{
ae_v_move(&tmp.ptr.p_double[0], 1, &xyp.ptr.pp_double[0][0], xyp.stride, ae_v_len(0,n));
spline1dbuildcatmullrom(&p->p, &tmp, n+1, -1, 0.0, &p->x, _state);
ae_v_move(&tmp.ptr.p_double[0], 1, &xyp.ptr.pp_double[0][1], xyp.stride, ae_v_len(0,n));
spline1dbuildcatmullrom(&p->p, &tmp, n+1, -1, 0.0, &p->y, _state);
ae_v_move(&tmp.ptr.p_double[0], 1, &xyp.ptr.pp_double[0][2], xyp.stride, ae_v_len(0,n));
spline1dbuildcatmullrom(&p->p, &tmp, n+1, -1, 0.0, &p->z, _state);
}
if( st==2 )
{
ae_v_move(&tmp.ptr.p_double[0], 1, &xyp.ptr.pp_double[0][0], xyp.stride, ae_v_len(0,n));
spline1dbuildcubic(&p->p, &tmp, n+1, -1, 0.0, -1, 0.0, &p->x, _state);
ae_v_move(&tmp.ptr.p_double[0], 1, &xyp.ptr.pp_double[0][1], xyp.stride, ae_v_len(0,n));
spline1dbuildcubic(&p->p, &tmp, n+1, -1, 0.0, -1, 0.0, &p->y, _state);
ae_v_move(&tmp.ptr.p_double[0], 1, &xyp.ptr.pp_double[0][2], xyp.stride, ae_v_len(0,n));
spline1dbuildcubic(&p->p, &tmp, n+1, -1, 0.0, -1, 0.0, &p->z, _state);
}
ae_frame_leave(_state);
}
/*************************************************************************
This function returns vector of parameter values correspoding to points.
I.e. for P created from (X[0],Y[0])...(X[N-1],Y[N-1]) and U=TValues(P) we
have
(X[0],Y[0]) = PSpline2Calc(P,U[0]),
(X[1],Y[1]) = PSpline2Calc(P,U[1]),
(X[2],Y[2]) = PSpline2Calc(P,U[2]),
...
INPUT PARAMETERS:
P - parametric spline interpolant
OUTPUT PARAMETERS:
N - array size
T - array[0..N-1]
NOTES:
* for non-periodic splines U[0]=0, U[0]<U[1]<...<U[N-1], U[N-1]=1
* for periodic splines U[0]=0, U[0]<U[1]<...<U[N-1], U[N-1]<1
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2parametervalues(pspline2interpolant* p,
ae_int_t* n,
/* Real */ ae_vector* t,
ae_state *_state)
{
*n = 0;
ae_vector_clear(t);
ae_assert(p->n>=2, "PSpline2ParameterValues: internal error!", _state);
*n = p->n;
ae_vector_set_length(t, *n, _state);
ae_v_move(&t->ptr.p_double[0], 1, &p->p.ptr.p_double[0], 1, ae_v_len(0,*n-1));
t->ptr.p_double[0] = 0;
if( !p->periodic )
{
t->ptr.p_double[*n-1] = 1;
}
}
/*************************************************************************
This function returns vector of parameter values correspoding to points.
Same as PSpline2ParameterValues(), but for 3D.
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3parametervalues(pspline3interpolant* p,
ae_int_t* n,
/* Real */ ae_vector* t,
ae_state *_state)
{
*n = 0;
ae_vector_clear(t);
ae_assert(p->n>=2, "PSpline3ParameterValues: internal error!", _state);
*n = p->n;
ae_vector_set_length(t, *n, _state);
ae_v_move(&t->ptr.p_double[0], 1, &p->p.ptr.p_double[0], 1, ae_v_len(0,*n-1));
t->ptr.p_double[0] = 0;
if( !p->periodic )
{
t->ptr.p_double[*n-1] = 1;
}
}
/*************************************************************************
This function calculates the value of the parametric spline for a given
value of parameter T
INPUT PARAMETERS:
P - parametric spline interpolant
T - point:
* T in [0,1] corresponds to interval spanned by points
* for non-periodic splines T<0 (or T>1) correspond to parts of
the curve before the first (after the last) point
* for periodic splines T<0 (or T>1) are projected into [0,1]
by making T=T-floor(T).
OUTPUT PARAMETERS:
X - X-position
Y - Y-position
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2calc(pspline2interpolant* p,
double t,
double* x,
double* y,
ae_state *_state)
{
*x = 0;
*y = 0;
if( p->periodic )
{
t = t-ae_ifloor(t, _state);
}
*x = spline1dcalc(&p->x, t, _state);
*y = spline1dcalc(&p->y, t, _state);
}
/*************************************************************************
This function calculates the value of the parametric spline for a given
value of parameter T.
INPUT PARAMETERS:
P - parametric spline interpolant
T - point:
* T in [0,1] corresponds to interval spanned by points
* for non-periodic splines T<0 (or T>1) correspond to parts of
the curve before the first (after the last) point
* for periodic splines T<0 (or T>1) are projected into [0,1]
by making T=T-floor(T).
OUTPUT PARAMETERS:
X - X-position
Y - Y-position
Z - Z-position
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3calc(pspline3interpolant* p,
double t,
double* x,
double* y,
double* z,
ae_state *_state)
{
*x = 0;
*y = 0;
*z = 0;
if( p->periodic )
{
t = t-ae_ifloor(t, _state);
}
*x = spline1dcalc(&p->x, t, _state);
*y = spline1dcalc(&p->y, t, _state);
*z = spline1dcalc(&p->z, t, _state);
}
/*************************************************************************
This function calculates tangent vector for a given value of parameter T
INPUT PARAMETERS:
P - parametric spline interpolant
T - point:
* T in [0,1] corresponds to interval spanned by points
* for non-periodic splines T<0 (or T>1) correspond to parts of
the curve before the first (after the last) point
* for periodic splines T<0 (or T>1) are projected into [0,1]
by making T=T-floor(T).
OUTPUT PARAMETERS:
X - X-component of tangent vector (normalized)
Y - Y-component of tangent vector (normalized)
NOTE:
X^2+Y^2 is either 1 (for non-zero tangent vector) or 0.
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2tangent(pspline2interpolant* p,
double t,
double* x,
double* y,
ae_state *_state)
{
double v;
double v0;
double v1;
*x = 0;
*y = 0;
if( p->periodic )
{
t = t-ae_ifloor(t, _state);
}
pspline2diff(p, t, &v0, x, &v1, y, _state);
if( ae_fp_neq(*x,0)||ae_fp_neq(*y,0) )
{
/*
* this code is a bit more complex than X^2+Y^2 to avoid
* overflow for large values of X and Y.
*/
v = safepythag2(*x, *y, _state);
*x = *x/v;
*y = *y/v;
}
}
/*************************************************************************
This function calculates tangent vector for a given value of parameter T
INPUT PARAMETERS:
P - parametric spline interpolant
T - point:
* T in [0,1] corresponds to interval spanned by points
* for non-periodic splines T<0 (or T>1) correspond to parts of
the curve before the first (after the last) point
* for periodic splines T<0 (or T>1) are projected into [0,1]
by making T=T-floor(T).
OUTPUT PARAMETERS:
X - X-component of tangent vector (normalized)
Y - Y-component of tangent vector (normalized)
Z - Z-component of tangent vector (normalized)
NOTE:
X^2+Y^2+Z^2 is either 1 (for non-zero tangent vector) or 0.
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3tangent(pspline3interpolant* p,
double t,
double* x,
double* y,
double* z,
ae_state *_state)
{
double v;
double v0;
double v1;
double v2;
*x = 0;
*y = 0;
*z = 0;
if( p->periodic )
{
t = t-ae_ifloor(t, _state);
}
pspline3diff(p, t, &v0, x, &v1, y, &v2, z, _state);
if( (ae_fp_neq(*x,0)||ae_fp_neq(*y,0))||ae_fp_neq(*z,0) )
{
v = safepythag3(*x, *y, *z, _state);
*x = *x/v;
*y = *y/v;
*z = *z/v;
}
}
/*************************************************************************
This function calculates derivative, i.e. it returns (dX/dT,dY/dT).
INPUT PARAMETERS:
P - parametric spline interpolant
T - point:
* T in [0,1] corresponds to interval spanned by points
* for non-periodic splines T<0 (or T>1) correspond to parts of
the curve before the first (after the last) point
* for periodic splines T<0 (or T>1) are projected into [0,1]
by making T=T-floor(T).
OUTPUT PARAMETERS:
X - X-value
DX - X-derivative
Y - Y-value
DY - Y-derivative
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2diff(pspline2interpolant* p,
double t,
double* x,
double* dx,
double* y,
double* dy,
ae_state *_state)
{
double d2s;
*x = 0;
*dx = 0;
*y = 0;
*dy = 0;
if( p->periodic )
{
t = t-ae_ifloor(t, _state);
}
spline1ddiff(&p->x, t, x, dx, &d2s, _state);
spline1ddiff(&p->y, t, y, dy, &d2s, _state);
}
/*************************************************************************
This function calculates derivative, i.e. it returns (dX/dT,dY/dT,dZ/dT).
INPUT PARAMETERS:
P - parametric spline interpolant
T - point:
* T in [0,1] corresponds to interval spanned by points
* for non-periodic splines T<0 (or T>1) correspond to parts of
the curve before the first (after the last) point
* for periodic splines T<0 (or T>1) are projected into [0,1]
by making T=T-floor(T).
OUTPUT PARAMETERS:
X - X-value
DX - X-derivative
Y - Y-value
DY - Y-derivative
Z - Z-value
DZ - Z-derivative
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3diff(pspline3interpolant* p,
double t,
double* x,
double* dx,
double* y,
double* dy,
double* z,
double* dz,
ae_state *_state)
{
double d2s;
*x = 0;
*dx = 0;
*y = 0;
*dy = 0;
*z = 0;
*dz = 0;
if( p->periodic )
{
t = t-ae_ifloor(t, _state);
}
spline1ddiff(&p->x, t, x, dx, &d2s, _state);
spline1ddiff(&p->y, t, y, dy, &d2s, _state);
spline1ddiff(&p->z, t, z, dz, &d2s, _state);
}
/*************************************************************************
This function calculates first and second derivative with respect to T.
INPUT PARAMETERS:
P - parametric spline interpolant
T - point:
* T in [0,1] corresponds to interval spanned by points
* for non-periodic splines T<0 (or T>1) correspond to parts of
the curve before the first (after the last) point
* for periodic splines T<0 (or T>1) are projected into [0,1]
by making T=T-floor(T).
OUTPUT PARAMETERS:
X - X-value
DX - derivative
D2X - second derivative
Y - Y-value
DY - derivative
D2Y - second derivative
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2diff2(pspline2interpolant* p,
double t,
double* x,
double* dx,
double* d2x,
double* y,
double* dy,
double* d2y,
ae_state *_state)
{
*x = 0;
*dx = 0;
*d2x = 0;
*y = 0;
*dy = 0;
*d2y = 0;
if( p->periodic )
{
t = t-ae_ifloor(t, _state);
}
spline1ddiff(&p->x, t, x, dx, d2x, _state);
spline1ddiff(&p->y, t, y, dy, d2y, _state);
}
/*************************************************************************
This function calculates first and second derivative with respect to T.
INPUT PARAMETERS:
P - parametric spline interpolant
T - point:
* T in [0,1] corresponds to interval spanned by points
* for non-periodic splines T<0 (or T>1) correspond to parts of
the curve before the first (after the last) point
* for periodic splines T<0 (or T>1) are projected into [0,1]
by making T=T-floor(T).
OUTPUT PARAMETERS:
X - X-value
DX - derivative
D2X - second derivative
Y - Y-value
DY - derivative
D2Y - second derivative
Z - Z-value
DZ - derivative
D2Z - second derivative
-- ALGLIB PROJECT --
Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3diff2(pspline3interpolant* p,
double t,
double* x,
double* dx,
double* d2x,
double* y,
double* dy,
double* d2y,
double* z,
double* dz,
double* d2z,
ae_state *_state)
{
*x = 0;
*dx = 0;
*d2x = 0;
*y = 0;
*dy = 0;
*d2y = 0;
*z = 0;
*dz = 0;
*d2z = 0;
if( p->periodic )
{
t = t-ae_ifloor(t, _state);
}
spline1ddiff(&p->x, t, x, dx, d2x, _state);
spline1ddiff(&p->y, t, y, dy, d2y, _state);
spline1ddiff(&p->z, t, z, dz, d2z, _state);
}
/*************************************************************************
This function calculates arc length, i.e. length of curve between t=a
and t=b.
INPUT PARAMETERS:
P - parametric spline interpolant
A,B - parameter values corresponding to arc ends:
* B>A will result in positive length returned
* B<A will result in negative length returned
RESULT:
length of arc starting at T=A and ending at T=B.
-- ALGLIB PROJECT --
Copyright 30.05.2010 by Bochkanov Sergey
*************************************************************************/
double pspline2arclength(pspline2interpolant* p,
double a,
double b,
ae_state *_state)
{
ae_frame _frame_block;
autogkstate state;
autogkreport rep;
double sx;
double dsx;
double d2sx;
double sy;
double dsy;
double d2sy;
double result;
ae_frame_make(_state, &_frame_block);
_autogkstate_init(&state, _state, ae_true);
_autogkreport_init(&rep, _state, ae_true);
autogksmooth(a, b, &state, _state);
while(autogkiteration(&state, _state))
{
spline1ddiff(&p->x, state.x, &sx, &dsx, &d2sx, _state);
spline1ddiff(&p->y, state.x, &sy, &dsy, &d2sy, _state);
state.f = safepythag2(dsx, dsy, _state);
}
autogkresults(&state, &result, &rep, _state);
ae_assert(rep.terminationtype>0, "PSpline2ArcLength: internal error!", _state);
ae_frame_leave(_state);
return result;
}
/*************************************************************************
This function calculates arc length, i.e. length of curve between t=a
and t=b.
INPUT PARAMETERS:
P - parametric spline interpolant
A,B - parameter values corresponding to arc ends:
* B>A will result in positive length returned
* B<A will result in negative length returned
RESULT:
length of arc starting at T=A and ending at T=B.
-- ALGLIB PROJECT --
Copyright 30.05.2010 by Bochkanov Sergey
*************************************************************************/
double pspline3arclength(pspline3interpolant* p,
double a,
double b,
ae_state *_state)
{
ae_frame _frame_block;
autogkstate state;
autogkreport rep;
double sx;
double dsx;
double d2sx;
double sy;
double dsy;
double d2sy;
double sz;
double dsz;
double d2sz;
double result;
ae_frame_make(_state, &_frame_block);
_autogkstate_init(&state, _state, ae_true);
_autogkreport_init(&rep, _state, ae_true);
autogksmooth(a, b, &state, _state);
while(autogkiteration(&state, _state))
{
spline1ddiff(&p->x, state.x, &sx, &dsx, &d2sx, _state);
spline1ddiff(&p->y, state.x, &sy, &dsy, &d2sy, _state);
spline1ddiff(&p->z, state.x, &sz, &dsz, &d2sz, _state);
state.f = safepythag3(dsx, dsy, dsz, _state);
}
autogkresults(&state, &result, &rep, _state);
ae_assert(rep.terminationtype>0, "PSpline3ArcLength: internal error!", _state);
ae_frame_leave(_state);
return result;
}
/*************************************************************************
Builds non-periodic parameterization for 2-dimensional spline
*************************************************************************/
static void pspline_pspline2par(/* Real */ ae_matrix* xy,
ae_int_t n,
ae_int_t pt,
/* Real */ ae_vector* p,
ae_state *_state)
{
double v;
ae_int_t i;
ae_vector_clear(p);
ae_assert(pt>=0&&pt<=2, "PSpline2Par: internal error!", _state);
/*
* Build parameterization:
* * fill by non-normalized values
* * normalize them so we have P[0]=0, P[N-1]=1.
*/
ae_vector_set_length(p, n, _state);
if( pt==0 )
{
for(i=0; i<=n-1; i++)
{
p->ptr.p_double[i] = i;
}
}
if( pt==1 )
{
p->ptr.p_double[0] = 0;
for(i=1; i<=n-1; i++)
{
p->ptr.p_double[i] = p->ptr.p_double[i-1]+safepythag2(xy->ptr.pp_double[i][0]-xy->ptr.pp_double[i-1][0], xy->ptr.pp_double[i][1]-xy->ptr.pp_double[i-1][1], _state);
}
}
if( pt==2 )
{
p->ptr.p_double[0] = 0;
for(i=1; i<=n-1; i++)
{
p->ptr.p_double[i] = p->ptr.p_double[i-1]+ae_sqrt(safepythag2(xy->ptr.pp_double[i][0]-xy->ptr.pp_double[i-1][0], xy->ptr.pp_double[i][1]-xy->ptr.pp_double[i-1][1], _state), _state);
}
}
v = 1/p->ptr.p_double[n-1];
ae_v_muld(&p->ptr.p_double[0], 1, ae_v_len(0,n-1), v);
}
/*************************************************************************
Builds non-periodic parameterization for 3-dimensional spline
*************************************************************************/
static void pspline_pspline3par(/* Real */ ae_matrix* xy,
ae_int_t n,
ae_int_t pt,
/* Real */ ae_vector* p,
ae_state *_state)
{
double v;
ae_int_t i;
ae_vector_clear(p);
ae_assert(pt>=0&&pt<=2, "PSpline3Par: internal error!", _state);
/*
* Build parameterization:
* * fill by non-normalized values
* * normalize them so we have P[0]=0, P[N-1]=1.
*/
ae_vector_set_length(p, n, _state);
if( pt==0 )
{
for(i=0; i<=n-1; i++)
{
p->ptr.p_double[i] = i;
}
}
if( pt==1 )
{
p->ptr.p_double[0] = 0;
for(i=1; i<=n-1; i++)
{
p->ptr.p_double[i] = p->ptr.p_double[i-1]+safepythag3(xy->ptr.pp_double[i][0]-xy->ptr.pp_double[i-1][0], xy->ptr.pp_double[i][1]-xy->ptr.pp_double[i-1][1], xy->ptr.pp_double[i][2]-xy->ptr.pp_double[i-1][2], _state);
}
}
if( pt==2 )
{
p->ptr.p_double[0] = 0;
for(i=1; i<=n-1; i++)
{
p->ptr.p_double[i] = p->ptr.p_double[i-1]+ae_sqrt(safepythag3(xy->ptr.pp_double[i][0]-xy->ptr.pp_double[i-1][0], xy->ptr.pp_double[i][1]-xy->ptr.pp_double[i-1][1], xy->ptr.pp_double[i][2]-xy->ptr.pp_double[i-1][2], _state), _state);
}
}
v = 1/p->ptr.p_double[n-1];
ae_v_muld(&p->ptr.p_double[0], 1, ae_v_len(0,n-1), v);
}
ae_bool _pspline2interpolant_init(void* _p, ae_state *_state, ae_bool make_automatic)
{
pspline2interpolant *p = (pspline2interpolant*)_p;
ae_touch_ptr((void*)p);
if( !ae_vector_init(&p->p, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !_spline1dinterpolant_init(&p->x, _state, make_automatic) )
return ae_false;
if( !_spline1dinterpolant_init(&p->y, _state, make_automatic) )
return ae_false;
return ae_true;
}
ae_bool _pspline2interpolant_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic)
{
pspline2interpolant *dst = (pspline2interpolant*)_dst;
pspline2interpolant *src = (pspline2interpolant*)_src;
dst->n = src->n;
dst->periodic = src->periodic;
if( !ae_vector_init_copy(&dst->p, &src->p, _state, make_automatic) )
return ae_false;
if( !_spline1dinterpolant_init_copy(&dst->x, &src->x, _state, make_automatic) )
return ae_false;
if( !_spline1dinterpolant_init_copy(&dst->y, &src->y, _state, make_automatic) )
return ae_false;
return ae_true;
}
void _pspline2interpolant_clear(void* _p)
{
pspline2interpolant *p = (pspline2interpolant*)_p;
ae_touch_ptr((void*)p);
ae_vector_clear(&p->p);
_spline1dinterpolant_clear(&p->x);
_spline1dinterpolant_clear(&p->y);
}
void _pspline2interpolant_destroy(void* _p)
{
pspline2interpolant *p = (pspline2interpolant*)_p;
ae_touch_ptr((void*)p);
ae_vector_destroy(&p->p);
_spline1dinterpolant_destroy(&p->x);
_spline1dinterpolant_destroy(&p->y);
}
ae_bool _pspline3interpolant_init(void* _p, ae_state *_state, ae_bool make_automatic)
{
pspline3interpolant *p = (pspline3interpolant*)_p;
ae_touch_ptr((void*)p);
if( !ae_vector_init(&p->p, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !_spline1dinterpolant_init(&p->x, _state, make_automatic) )
return ae_false;
if( !_spline1dinterpolant_init(&p->y, _state, make_automatic) )
return ae_false;
if( !_spline1dinterpolant_init(&p->z, _state, make_automatic) )
return ae_false;
return ae_true;
}
ae_bool _pspline3interpolant_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic)
{
pspline3interpolant *dst = (pspline3interpolant*)_dst;
pspline3interpolant *src = (pspline3interpolant*)_src;
dst->n = src->n;
dst->periodic = src->periodic;
if( !ae_vector_init_copy(&dst->p, &src->p, _state, make_automatic) )
return ae_false;
if( !_spline1dinterpolant_init_copy(&dst->x, &src->x, _state, make_automatic) )
return ae_false;
if( !_spline1dinterpolant_init_copy(&dst->y, &src->y, _state, make_automatic) )
return ae_false;
if( !_spline1dinterpolant_init_copy(&dst->z, &src->z, _state, make_automatic) )
return ae_false;
return ae_true;
}
void _pspline3interpolant_clear(void* _p)
{
pspline3interpolant *p = (pspline3interpolant*)_p;
ae_touch_ptr((void*)p);
ae_vector_clear(&p->p);
_spline1dinterpolant_clear(&p->x);
_spline1dinterpolant_clear(&p->y);
_spline1dinterpolant_clear(&p->z);
}
void _pspline3interpolant_destroy(void* _p)
{
pspline3interpolant *p = (pspline3interpolant*)_p;
ae_touch_ptr((void*)p);
ae_vector_destroy(&p->p);
_spline1dinterpolant_destroy(&p->x);
_spline1dinterpolant_destroy(&p->y);
_spline1dinterpolant_destroy(&p->z);
}
/*************************************************************************
This function creates RBF model for a scalar (NY=1) or vector (NY>1)
function in a NX-dimensional space (NX=2 or NX=3).
Newly created model is empty. It can be used for interpolation right after
creation, but it just returns zeros. You have to add points to the model,
tune interpolation settings, and then call model construction function
RBFBuildModel() which will update model according to your specification.
USAGE:
1. User creates model with RBFCreate()
2. User adds dataset with RBFSetPoints() (points do NOT have to be on a
regular grid)
3. (OPTIONAL) User chooses polynomial term by calling:
* RBFLinTerm() to set linear term
* RBFConstTerm() to set constant term
* RBFZeroTerm() to set zero term
By default, linear term is used.
4. User chooses specific RBF algorithm to use: either QNN (RBFSetAlgoQNN)
or ML (RBFSetAlgoMultiLayer).
5. User calls RBFBuildModel() function which rebuilds model according to
the specification
6. User may call RBFCalc() to calculate model value at the specified point,
RBFGridCalc() to calculate model values at the points of the regular
grid. User may extract model coefficients with RBFUnpack() call.
INPUT PARAMETERS:
NX - dimension of the space, NX=2 or NX=3
NY - function dimension, NY>=1
OUTPUT PARAMETERS:
S - RBF model (initially equals to zero)
NOTE 1: memory requirements. RBF models require amount of memory which is
proportional to the number of data points. Memory is allocated
during model construction, but most of this memory is freed after
model coefficients are calculated.
Some approximate estimates for N centers with default settings are
given below:
* about 250*N*(sizeof(double)+2*sizeof(int)) bytes of memory is
needed during model construction stage.
* about 15*N*sizeof(double) bytes is needed after model is built.
For example, for N=100000 we may need 0.6 GB of memory to build
model, but just about 0.012 GB to store it.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfcreate(ae_int_t nx, ae_int_t ny, rbfmodel* s, ae_state *_state)
{
ae_int_t i;
ae_int_t j;
_rbfmodel_clear(s);
ae_assert(nx==2||nx==3, "RBFCreate: NX<>2 and NX<>3", _state);
ae_assert(ny>=1, "RBFCreate: NY<1", _state);
s->nx = nx;
s->ny = ny;
s->nl = 0;
s->nc = 0;
ae_matrix_set_length(&s->v, ny, rbf_mxnx+1, _state);
for(i=0; i<=ny-1; i++)
{
for(j=0; j<=rbf_mxnx; j++)
{
s->v.ptr.pp_double[i][j] = 0;
}
}
s->n = 0;
s->rmax = 0;
s->gridtype = 2;
s->fixrad = ae_false;
s->radvalue = 1;
s->radzvalue = 5;
s->aterm = 1;
s->algorithmtype = 1;
/*
* stopping criteria
*/
s->epsort = rbf_eps;
s->epserr = rbf_eps;
s->maxits = 0;
}
/*************************************************************************
This function adds dataset.
This function overrides results of the previous calls, i.e. multiple calls
of this function will result in only the last set being added.
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call.
XY - points, array[N,NX+NY]. One row corresponds to one point
in the dataset. First NX elements are coordinates, next
NY elements are function values. Array may be larger than
specific, in this case only leading [N,NX+NY] elements
will be used.
N - number of points in the dataset
After you've added dataset and (optionally) tuned algorithm settings you
should call RBFBuildModel() in order to build a model for you.
NOTE: this function has some serialization-related subtleties. We
recommend you to study serialization examples from ALGLIB Reference
Manual if you want to perform serialization of your models.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfsetpoints(rbfmodel* s,
/* Real */ ae_matrix* xy,
ae_int_t n,
ae_state *_state)
{
ae_int_t i;
ae_int_t j;
ae_assert(n>0, "RBFSetPoints: N<0", _state);
ae_assert(xy->rows>=n, "RBFSetPoints: Rows(XY)<N", _state);
ae_assert(xy->cols>=s->nx+s->ny, "RBFSetPoints: Cols(XY)<NX+NY", _state);
s->n = n;
ae_matrix_set_length(&s->x, s->n, rbf_mxnx, _state);
ae_matrix_set_length(&s->y, s->n, s->ny, _state);
for(i=0; i<=s->n-1; i++)
{
for(j=0; j<=rbf_mxnx-1; j++)
{
s->x.ptr.pp_double[i][j] = 0;
}
for(j=0; j<=s->nx-1; j++)
{
s->x.ptr.pp_double[i][j] = xy->ptr.pp_double[i][j];
}
for(j=0; j<=s->ny-1; j++)
{
s->y.ptr.pp_double[i][j] = xy->ptr.pp_double[i][j+s->nx];
}
}
}
/*************************************************************************
This function sets RBF interpolation algorithm. ALGLIB supports several
RBF algorithms with different properties.
This algorithm is called RBF-QNN and it is good for point sets with
following properties:
a) all points are distinct
b) all points are well separated.
c) points distribution is approximately uniform. There is no "contour
lines", clusters of points, or other small-scale structures.
Algorithm description:
1) interpolation centers are allocated to data points
2) interpolation radii are calculated as distances to the nearest centers
times Q coefficient (where Q is a value from [0.75,1.50]).
3) after performing (2) radii are transformed in order to avoid situation
when single outlier has very large radius and influences many points
across all dataset. Transformation has following form:
new_r[i] = min(r[i],Z*median(r[]))
where r[i] is I-th radius, median() is a median radius across entire
dataset, Z is user-specified value which controls amount of deviation
from median radius.
When (a) is violated, we will be unable to build RBF model. When (b) or
(c) are violated, model will be built, but interpolation quality will be
low. See http://www.alglib.net/interpolation/ for more information on this
subject.
This algorithm is used by default.
Additional Q parameter controls smoothness properties of the RBF basis:
* Q<0.75 will give perfectly conditioned basis, but terrible smoothness
properties (RBF interpolant will have sharp peaks around function values)
* Q around 1.0 gives good balance between smoothness and condition number
* Q>1.5 will lead to badly conditioned systems and slow convergence of the
underlying linear solver (although smoothness will be very good)
* Q>2.0 will effectively make optimizer useless because it won't converge
within reasonable amount of iterations. It is possible to set such large
Q, but it is advised not to do so.
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call
Q - Q parameter, Q>0, recommended value - 1.0
Z - Z parameter, Z>0, recommended value - 5.0
NOTE: this function has some serialization-related subtleties. We
recommend you to study serialization examples from ALGLIB Reference
Manual if you want to perform serialization of your models.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfsetalgoqnn(rbfmodel* s, double q, double z, ae_state *_state)
{
ae_assert(ae_isfinite(q, _state), "RBFSetAlgoQNN: Q is infinite or NAN", _state);
ae_assert(ae_fp_greater(q,0), "RBFSetAlgoQNN: Q<=0", _state);
rbf_rbfgridpoints(s, _state);
rbf_rbfradnn(s, q, z, _state);
s->algorithmtype = 1;
}
/*************************************************************************
This function sets RBF interpolation algorithm. ALGLIB supports several
RBF algorithms with different properties.
This algorithm is called RBF-ML. It builds multilayer RBF model, i.e.
model with subsequently decreasing radii, which allows us to combine
smoothness (due to large radii of the first layers) with exactness (due
to small radii of the last layers) and fast convergence.
Internally RBF-ML uses many different means of acceleration, from sparse
matrices to KD-trees, which results in algorithm whose working time is
roughly proportional to N*log(N)*Density*RBase^2*NLayers, where N is a
number of points, Density is an average density if points per unit of the
interpolation space, RBase is an initial radius, NLayers is a number of
layers.
RBF-ML is good for following kinds of interpolation problems:
1. "exact" problems (perfect fit) with well separated points
2. least squares problems with arbitrary distribution of points (algorithm
gives perfect fit where it is possible, and resorts to least squares
fit in the hard areas).
3. noisy problems where we want to apply some controlled amount of
smoothing.
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call
RBase - RBase parameter, RBase>0
NLayers - NLayers parameter, NLayers>0, recommended value to start
with - about 5.
LambdaV - regularization value, can be useful when solving problem
in the least squares sense. Optimal lambda is problem-
dependent and require trial and error. In our experience,
good lambda can be as large as 0.1, and you can use 0.001
as initial guess.
Default value - 0.01, which is used when LambdaV is not
given. You can specify zero value, but it is not
recommended to do so.
TUNING ALGORITHM
In order to use this algorithm you have to choose three parameters:
* initial radius RBase
* number of layers in the model NLayers
* regularization coefficient LambdaV
Initial radius is easy to choose - you can pick any number several times
larger than the average distance between points. Algorithm won't break
down if you choose radius which is too large (model construction time will
increase, but model will be built correctly).
Choose such number of layers that RLast=RBase/2^(NLayers-1) (radius used
by the last layer) will be smaller than the typical distance between
points. In case model error is too large, you can increase number of
layers. Having more layers will make model construction and evaluation
proportionally slower, but it will allow you to have model which precisely
fits your data. From the other side, if you want to suppress noise, you
can DECREASE number of layers to make your model less flexible.
Regularization coefficient LambdaV controls smoothness of the individual
models built for each layer. We recommend you to use default value in case
you don't want to tune this parameter, because having non-zero LambdaV
accelerates and stabilizes internal iterative algorithm. In case you want
to suppress noise you can use LambdaV as additional parameter (larger
value = more smoothness) to tune.
TYPICAL ERRORS
1. Using initial radius which is too large. Memory requirements of the
RBF-ML are roughly proportional to N*Density*RBase^2 (where Density is
an average density of points per unit of the interpolation space). In
the extreme case of the very large RBase we will need O(N^2) units of
memory - and many layers in order to decrease radius to some reasonably
small value.
2. Using too small number of layers - RBF models with large radius are not
flexible enough to reproduce small variations in the target function.
You need many layers with different radii, from large to small, in
order to have good model.
3. Using initial radius which is too small. You will get model with
"holes" in the areas which are too far away from interpolation centers.
However, algorithm will work correctly (and quickly) in this case.
4. Using too many layers - you will get too large and too slow model. This
model will perfectly reproduce your function, but maybe you will be
able to achieve similar results with less layers (and less memory).
-- ALGLIB --
Copyright 02.03.2012 by Bochkanov Sergey
*************************************************************************/
void rbfsetalgomultilayer(rbfmodel* s,
double rbase,
ae_int_t nlayers,
double lambdav,
ae_state *_state)
{
ae_assert(ae_isfinite(rbase, _state), "RBFSetAlgoMultiLayer: RBase is infinite or NaN", _state);
ae_assert(ae_fp_greater(rbase,0), "RBFSetAlgoMultiLayer: RBase<=0", _state);
ae_assert(nlayers>=0, "RBFSetAlgoMultiLayer: NLayers<0", _state);
ae_assert(ae_isfinite(lambdav, _state), "RBFSetAlgoMultiLayer: LambdaV is infinite or NAN", _state);
ae_assert(ae_fp_greater_eq(lambdav,0), "RBFSetAlgoMultiLayer: LambdaV<0", _state);
s->radvalue = rbase;
s->nlayers = nlayers;
s->algorithmtype = 2;
s->lambdav = lambdav;
}
/*************************************************************************
This function sets linear term (model is a sum of radial basis functions
plus linear polynomial). This function won't have effect until next call
to RBFBuildModel().
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call
NOTE: this function has some serialization-related subtleties. We
recommend you to study serialization examples from ALGLIB Reference
Manual if you want to perform serialization of your models.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfsetlinterm(rbfmodel* s, ae_state *_state)
{
s->aterm = 1;
}
/*************************************************************************
This function sets constant term (model is a sum of radial basis functions
plus constant). This function won't have effect until next call to
RBFBuildModel().
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call
NOTE: this function has some serialization-related subtleties. We
recommend you to study serialization examples from ALGLIB Reference
Manual if you want to perform serialization of your models.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfsetconstterm(rbfmodel* s, ae_state *_state)
{
s->aterm = 2;
}
/*************************************************************************
This function sets zero term (model is a sum of radial basis functions
without polynomial term). This function won't have effect until next call
to RBFBuildModel().
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call
NOTE: this function has some serialization-related subtleties. We
recommend you to study serialization examples from ALGLIB Reference
Manual if you want to perform serialization of your models.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfsetzeroterm(rbfmodel* s, ae_state *_state)
{
s->aterm = 3;
}
/*************************************************************************
This function sets stopping criteria of the underlying linear solver.
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call
EpsOrt - orthogonality stopping criterion, EpsOrt>=0. Algorithm will
stop when ||A'*r||<=EpsOrt where A' is a transpose of the
system matrix, r is a residual vector.
Recommended value of EpsOrt is equal to 1E-6.
This criterion will stop algorithm when we have "bad fit"
situation, i.e. when we should stop in a point with large,
nonzero residual.
EpsErr - residual stopping criterion. Algorithm will stop when
||r||<=EpsErr*||b||, where r is a residual vector, b is a
right part of the system (function values).
Recommended value of EpsErr is equal to 1E-3 or 1E-6.
This criterion will stop algorithm in a "good fit"
situation when we have near-zero residual near the desired
solution.
MaxIts - this criterion will stop algorithm after MaxIts iterations.
It should be used for debugging purposes only!
Zero MaxIts means that no limit is placed on the number of
iterations.
We recommend to set moderate non-zero values EpsOrt and EpsErr
simultaneously. Values equal to 10E-6 are good to start with. In case you
need high performance and do not need high precision , you may decrease
EpsErr down to 0.001. However, we do not recommend decreasing EpsOrt.
As for MaxIts, we recommend to leave it zero unless you know what you do.
NOTE: this function has some serialization-related subtleties. We
recommend you to study serialization examples from ALGLIB Reference
Manual if you want to perform serialization of your models.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfsetcond(rbfmodel* s,
double epsort,
double epserr,
ae_int_t maxits,
ae_state *_state)
{
ae_assert(ae_isfinite(epsort, _state)&&ae_fp_greater_eq(epsort,0), "RBFSetCond: EpsOrt is negative, INF or NAN", _state);
ae_assert(ae_isfinite(epserr, _state)&&ae_fp_greater_eq(epserr,0), "RBFSetCond: EpsB is negative, INF or NAN", _state);
ae_assert(maxits>=0, "RBFSetCond: MaxIts is negative", _state);
if( (ae_fp_eq(epsort,0)&&ae_fp_eq(epserr,0))&&maxits==0 )
{
s->epsort = rbf_eps;
s->epserr = rbf_eps;
s->maxits = 0;
}
else
{
s->epsort = epsort;
s->epserr = epserr;
s->maxits = maxits;
}
}
/*************************************************************************
This function builds RBF model and returns report (contains some
information which can be used for evaluation of the algorithm properties).
Call to this function modifies RBF model by calculating its centers/radii/
weights and saving them into RBFModel structure. Initially RBFModel
contain zero coefficients, but after call to this function we will have
coefficients which were calculated in order to fit our dataset.
After you called this function you can call RBFCalc(), RBFGridCalc() and
other model calculation functions.
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call
Rep - report:
* Rep.TerminationType:
* -5 - non-distinct basis function centers were detected,
interpolation aborted
* -4 - nonconvergence of the internal SVD solver
* 1 - successful termination
Fields are used for debugging purposes:
* Rep.IterationsCount - iterations count of the LSQR solver
* Rep.NMV - number of matrix-vector products
* Rep.ARows - rows count for the system matrix
* Rep.ACols - columns count for the system matrix
* Rep.ANNZ - number of significantly non-zero elements
(elements above some algorithm-determined threshold)
NOTE: failure to build model will leave current state of the structure
unchanged.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfbuildmodel(rbfmodel* s, rbfreport* rep, ae_state *_state)
{
ae_frame _frame_block;
kdtree tree;
kdtree ctree;
ae_vector dist;
ae_vector xcx;
ae_matrix a;
ae_matrix v;
ae_matrix omega;
ae_vector y;
ae_matrix residualy;
ae_vector radius;
ae_matrix xc;
ae_vector mnx;
ae_vector mxx;
ae_vector edge;
ae_vector mxsteps;
ae_int_t nc;
double rmax;
ae_vector tags;
ae_vector ctags;
ae_int_t i;
ae_int_t j;
ae_int_t k;
ae_int_t k2;
ae_int_t snnz;
ae_vector tmp0;
ae_vector tmp1;
ae_int_t layerscnt;
ae_frame_make(_state, &_frame_block);
_rbfreport_clear(rep);
_kdtree_init(&tree, _state, ae_true);
_kdtree_init(&ctree, _state, ae_true);
ae_vector_init(&dist, 0, DT_REAL, _state, ae_true);
ae_vector_init(&xcx, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&a, 0, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&v, 0, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&omega, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&y, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&residualy, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&radius, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&xc, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&mnx, 0, DT_REAL, _state, ae_true);
ae_vector_init(&mxx, 0, DT_REAL, _state, ae_true);
ae_vector_init(&edge, 0, DT_REAL, _state, ae_true);
ae_vector_init(&mxsteps, 0, DT_INT, _state, ae_true);
ae_vector_init(&tags, 0, DT_INT, _state, ae_true);
ae_vector_init(&ctags, 0, DT_INT, _state, ae_true);
ae_vector_init(&tmp0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmp1, 0, DT_REAL, _state, ae_true);
ae_assert(s->nx==2||s->nx==3, "RBFBuildModel: S.NX<>2 or S.NX<>3!", _state);
/*
* Quick exit when we have no points
*/
if( s->n==0 )
{
rep->terminationtype = 1;
rep->iterationscount = 0;
rep->nmv = 0;
rep->arows = 0;
rep->acols = 0;
kdtreebuildtagged(&s->xc, &tags, 0, rbf_mxnx, 0, 2, &s->tree, _state);
ae_matrix_set_length(&s->xc, 0, 0, _state);
ae_matrix_set_length(&s->wr, 0, 0, _state);
s->nc = 0;
s->rmax = 0;
ae_matrix_set_length(&s->v, s->ny, rbf_mxnx+1, _state);
for(i=0; i<=s->ny-1; i++)
{
for(j=0; j<=rbf_mxnx; j++)
{
s->v.ptr.pp_double[i][j] = 0;
}
}
ae_frame_leave(_state);
return;
}
/*
* General case, N>0
*/
rep->annz = 0;
rep->iterationscount = 0;
rep->nmv = 0;
ae_vector_set_length(&xcx, rbf_mxnx, _state);
/*
* First model in a sequence - linear model.
* Residuals from linear regression are stored in the ResidualY variable
* (used later to build RBF models).
*/
ae_matrix_set_length(&residualy, s->n, s->ny, _state);
for(i=0; i<=s->n-1; i++)
{
for(j=0; j<=s->ny-1; j++)
{
residualy.ptr.pp_double[i][j] = s->y.ptr.pp_double[i][j];
}
}
if( !rbf_buildlinearmodel(&s->x, &residualy, s->n, s->ny, s->aterm, &v, _state) )
{
rep->terminationtype = -5;
ae_frame_leave(_state);
return;
}
/*
* Handle special case: multilayer model with NLayers=0.
* Quick exit.
*/
if( s->algorithmtype==2&&s->nlayers==0 )
{
rep->terminationtype = 1;
rep->iterationscount = 0;
rep->nmv = 0;
rep->arows = 0;
rep->acols = 0;
kdtreebuildtagged(&s->xc, &tags, 0, rbf_mxnx, 0, 2, &s->tree, _state);
ae_matrix_set_length(&s->xc, 0, 0, _state);
ae_matrix_set_length(&s->wr, 0, 0, _state);
s->nc = 0;
s->rmax = 0;
ae_matrix_set_length(&s->v, s->ny, rbf_mxnx+1, _state);
for(i=0; i<=s->ny-1; i++)
{
for(j=0; j<=rbf_mxnx; j++)
{
s->v.ptr.pp_double[i][j] = v.ptr.pp_double[i][j];
}
}
ae_frame_leave(_state);
return;
}
/*
* Second model in a sequence - RBF term.
*
* NOTE: assignments below are not necessary, but without them
* MSVC complains about unitialized variables.
*/
nc = 0;
rmax = 0;
layerscnt = 0;
if( s->algorithmtype==1 )
{
/*
* Add RBF model.
* This model uses local KD-trees to speed-up nearest neighbor searches.
*/
if( s->gridtype==1 )
{
ae_vector_set_length(&mxx, s->nx, _state);
ae_vector_set_length(&mnx, s->nx, _state);
ae_vector_set_length(&mxsteps, s->nx, _state);
ae_vector_set_length(&edge, s->nx, _state);
for(i=0; i<=s->nx-1; i++)
{
mxx.ptr.p_double[i] = s->x.ptr.pp_double[0][i];
mnx.ptr.p_double[i] = s->x.ptr.pp_double[0][i];
}
for(i=0; i<=s->n-1; i++)
{
for(j=0; j<=s->nx-1; j++)
{
if( ae_fp_less(mxx.ptr.p_double[j],s->x.ptr.pp_double[i][j]) )
{
mxx.ptr.p_double[j] = s->x.ptr.pp_double[i][j];
}
if( ae_fp_greater(mnx.ptr.p_double[j],s->x.ptr.pp_double[i][j]) )
{
mnx.ptr.p_double[j] = s->x.ptr.pp_double[i][j];
}
}
}
for(i=0; i<=s->nx-1; i++)
{
mxsteps.ptr.p_int[i] = ae_trunc((mxx.ptr.p_double[i]-mnx.ptr.p_double[i])/(2*s->h), _state)+1;
edge.ptr.p_double[i] = (mxx.ptr.p_double[i]+mnx.ptr.p_double[i])/2-s->h*mxsteps.ptr.p_int[i];
}
nc = 1;
for(i=0; i<=s->nx-1; i++)
{
mxsteps.ptr.p_int[i] = 2*mxsteps.ptr.p_int[i]+1;
nc = nc*mxsteps.ptr.p_int[i];
}
ae_matrix_set_length(&xc, nc, rbf_mxnx, _state);
if( s->nx==2 )
{
for(i=0; i<=mxsteps.ptr.p_int[0]-1; i++)
{
for(j=0; j<=mxsteps.ptr.p_int[1]-1; j++)
{
for(k2=0; k2<=rbf_mxnx-1; k2++)
{
xc.ptr.pp_double[i*mxsteps.ptr.p_int[1]+j][k2] = 0;
}
xc.ptr.pp_double[i*mxsteps.ptr.p_int[1]+j][0] = edge.ptr.p_double[0]+s->h*i;
xc.ptr.pp_double[i*mxsteps.ptr.p_int[1]+j][1] = edge.ptr.p_double[1]+s->h*j;
}
}
}
if( s->nx==3 )
{
for(i=0; i<=mxsteps.ptr.p_int[0]-1; i++)
{
for(j=0; j<=mxsteps.ptr.p_int[1]-1; j++)
{
for(k=0; k<=mxsteps.ptr.p_int[2]-1; k++)
{
for(k2=0; k2<=rbf_mxnx-1; k2++)
{
xc.ptr.pp_double[i*mxsteps.ptr.p_int[1]+j][k2] = 0;
}
xc.ptr.pp_double[(i*mxsteps.ptr.p_int[1]+j)*mxsteps.ptr.p_int[2]+k][0] = edge.ptr.p_double[0]+s->h*i;
xc.ptr.pp_double[(i*mxsteps.ptr.p_int[1]+j)*mxsteps.ptr.p_int[2]+k][1] = edge.ptr.p_double[1]+s->h*j;
xc.ptr.pp_double[(i*mxsteps.ptr.p_int[1]+j)*mxsteps.ptr.p_int[2]+k][2] = edge.ptr.p_double[2]+s->h*k;
}
}
}
}
}
else
{
if( s->gridtype==2 )
{
nc = s->n;
ae_matrix_set_length(&xc, nc, rbf_mxnx, _state);
for(i=0; i<=nc-1; i++)
{
for(j=0; j<=rbf_mxnx-1; j++)
{
xc.ptr.pp_double[i][j] = s->x.ptr.pp_double[i][j];
}
}
}
else
{
if( s->gridtype==3 )
{
nc = s->nc;
ae_matrix_set_length(&xc, nc, rbf_mxnx, _state);
for(i=0; i<=nc-1; i++)
{
for(j=0; j<=rbf_mxnx-1; j++)
{
xc.ptr.pp_double[i][j] = s->xc.ptr.pp_double[i][j];
}
}
}
else
{
ae_assert(ae_false, "RBFBuildModel: either S.GridType<1 or S.GridType>3!", _state);
}
}
}
rmax = 0;
ae_vector_set_length(&radius, nc, _state);
ae_vector_set_length(&ctags, nc, _state);
for(i=0; i<=nc-1; i++)
{
ctags.ptr.p_int[i] = i;
}
kdtreebuildtagged(&xc, &ctags, nc, rbf_mxnx, 0, 2, &ctree, _state);
if( s->fixrad )
{
/*
* Fixed radius
*/
for(i=0; i<=nc-1; i++)
{
radius.ptr.p_double[i] = s->radvalue;
}
rmax = radius.ptr.p_double[0];
}
else
{
/*
* Dynamic radius
*/
if( nc==0 )
{
rmax = 1;
}
else
{
if( nc==1 )
{
radius.ptr.p_double[0] = s->radvalue;
rmax = radius.ptr.p_double[0];
}
else
{
/*
* NC>1, calculate radii using distances to nearest neigbors
*/
for(i=0; i<=nc-1; i++)
{
for(j=0; j<=rbf_mxnx-1; j++)
{
xcx.ptr.p_double[j] = xc.ptr.pp_double[i][j];
}
if( kdtreequeryknn(&ctree, &xcx, 1, ae_false, _state)>0 )
{
kdtreequeryresultsdistances(&ctree, &dist, _state);
radius.ptr.p_double[i] = s->radvalue*dist.ptr.p_double[0];
}
else
{
/*
* No neighbors found (it will happen when we have only one center).
* Initialize radius with default value.
*/
radius.ptr.p_double[i] = 1.0;
}
}
/*
* Apply filtering
*/
rvectorsetlengthatleast(&tmp0, nc, _state);
for(i=0; i<=nc-1; i++)
{
tmp0.ptr.p_double[i] = radius.ptr.p_double[i];
}
tagsortfast(&tmp0, &tmp1, nc, _state);
for(i=0; i<=nc-1; i++)
{
radius.ptr.p_double[i] = ae_minreal(radius.ptr.p_double[i], s->radzvalue*tmp0.ptr.p_double[nc/2], _state);
}
/*
* Calculate RMax, check that all radii are non-zero
*/
for(i=0; i<=nc-1; i++)
{
rmax = ae_maxreal(rmax, radius.ptr.p_double[i], _state);
}
for(i=0; i<=nc-1; i++)
{
if( ae_fp_eq(radius.ptr.p_double[i],0) )
{
rep->terminationtype = -5;
ae_frame_leave(_state);
return;
}
}
}
}
}
ivectorsetlengthatleast(&tags, s->n, _state);
for(i=0; i<=s->n-1; i++)
{
tags.ptr.p_int[i] = i;
}
kdtreebuildtagged(&s->x, &tags, s->n, rbf_mxnx, 0, 2, &tree, _state);
rbf_buildrbfmodellsqr(&s->x, &residualy, &xc, &radius, s->n, nc, s->ny, &tree, &ctree, s->epsort, s->epserr, s->maxits, &rep->annz, &snnz, &omega, &rep->terminationtype, &rep->iterationscount, &rep->nmv, _state);
layerscnt = 1;
}
else
{
if( s->algorithmtype==2 )
{
rmax = s->radvalue;
rbf_buildrbfmlayersmodellsqr(&s->x, &residualy, &xc, s->radvalue, &radius, s->n, &nc, s->ny, s->nlayers, &ctree, 1.0E-6, 1.0E-6, 50, s->lambdav, &rep->annz, &omega, &rep->terminationtype, &rep->iterationscount, &rep->nmv, _state);
layerscnt = s->nlayers;
}
else
{
ae_assert(ae_false, "RBFBuildModel: internal error(AlgorithmType neither 1 nor 2)", _state);
}
}
if( rep->terminationtype<=0 )
{
ae_frame_leave(_state);
return;
}
/*
* Model is built
*/
s->nc = nc/layerscnt;
s->rmax = rmax;
s->nl = layerscnt;
ae_matrix_set_length(&s->xc, s->nc, rbf_mxnx, _state);
ae_matrix_set_length(&s->wr, s->nc, 1+s->nl*s->ny, _state);
ae_matrix_set_length(&s->v, s->ny, rbf_mxnx+1, _state);
for(i=0; i<=s->nc-1; i++)
{
for(j=0; j<=rbf_mxnx-1; j++)
{
s->xc.ptr.pp_double[i][j] = xc.ptr.pp_double[i][j];
}
}
ivectorsetlengthatleast(&tags, s->nc, _state);
for(i=0; i<=s->nc-1; i++)
{
tags.ptr.p_int[i] = i;
}
kdtreebuildtagged(&s->xc, &tags, s->nc, rbf_mxnx, 0, 2, &s->tree, _state);
for(i=0; i<=s->nc-1; i++)
{
s->wr.ptr.pp_double[i][0] = radius.ptr.p_double[i];
for(k=0; k<=layerscnt-1; k++)
{
for(j=0; j<=s->ny-1; j++)
{
s->wr.ptr.pp_double[i][1+k*s->ny+j] = omega.ptr.pp_double[k*s->nc+i][j];
}
}
}
for(i=0; i<=s->ny-1; i++)
{
for(j=0; j<=rbf_mxnx; j++)
{
s->v.ptr.pp_double[i][j] = v.ptr.pp_double[i][j];
}
}
rep->terminationtype = 1;
rep->arows = s->n;
rep->acols = s->nc;
ae_frame_leave(_state);
}
/*************************************************************************
This function calculates values of the RBF model in the given point.
This function should be used when we have NY=1 (scalar function) and NX=2
(2-dimensional space). If you have 3-dimensional space, use RBFCalc3(). If
you have general situation (NX-dimensional space, NY-dimensional function)
you should use general, less efficient implementation RBFCalc().
If you want to calculate function values many times, consider using
RBFGridCalc2(), which is far more efficient than many subsequent calls to
RBFCalc2().
This function returns 0.0 when:
* model is not initialized
* NX<>2
*NY<>1
INPUT PARAMETERS:
S - RBF model
X0 - first coordinate, finite number
X1 - second coordinate, finite number
RESULT:
value of the model or 0.0 (as defined above)
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
double rbfcalc2(rbfmodel* s, double x0, double x1, ae_state *_state)
{
ae_int_t i;
ae_int_t j;
ae_int_t lx;
ae_int_t tg;
double d2;
double t;
double bfcur;
double rcur;
double result;
ae_assert(ae_isfinite(x0, _state), "RBFCalc2: invalid value for X0 (X0 is Inf)!", _state);
ae_assert(ae_isfinite(x1, _state), "RBFCalc2: invalid value for X1 (X1 is Inf)!", _state);
if( s->ny!=1||s->nx!=2 )
{
result = 0;
return result;
}
result = s->v.ptr.pp_double[0][0]*x0+s->v.ptr.pp_double[0][1]*x1+s->v.ptr.pp_double[0][rbf_mxnx];
if( s->nc==0 )
{
return result;
}
rvectorsetlengthatleast(&s->calcbufxcx, rbf_mxnx, _state);
for(i=0; i<=rbf_mxnx-1; i++)
{
s->calcbufxcx.ptr.p_double[i] = 0.0;
}
s->calcbufxcx.ptr.p_double[0] = x0;
s->calcbufxcx.ptr.p_double[1] = x1;
lx = kdtreequeryrnn(&s->tree, &s->calcbufxcx, s->rmax*rbf_rbffarradius, ae_true, _state);
kdtreequeryresultsx(&s->tree, &s->calcbufx, _state);
kdtreequeryresultstags(&s->tree, &s->calcbuftags, _state);
for(i=0; i<=lx-1; i++)
{
tg = s->calcbuftags.ptr.p_int[i];
d2 = ae_sqr(x0-s->calcbufx.ptr.pp_double[i][0], _state)+ae_sqr(x1-s->calcbufx.ptr.pp_double[i][1], _state);
rcur = s->wr.ptr.pp_double[tg][0];
bfcur = ae_exp(-d2/(rcur*rcur), _state);
for(j=0; j<=s->nl-1; j++)
{
result = result+bfcur*s->wr.ptr.pp_double[tg][1+j];
rcur = 0.5*rcur;
t = bfcur*bfcur;
bfcur = t*t;
}
}
return result;
}
/*************************************************************************
This function calculates values of the RBF model in the given point.
This function should be used when we have NY=1 (scalar function) and NX=3
(3-dimensional space). If you have 2-dimensional space, use RBFCalc2(). If
you have general situation (NX-dimensional space, NY-dimensional function)
you should use general, less efficient implementation RBFCalc().
This function returns 0.0 when:
* model is not initialized
* NX<>3
*NY<>1
INPUT PARAMETERS:
S - RBF model
X0 - first coordinate, finite number
X1 - second coordinate, finite number
X2 - third coordinate, finite number
RESULT:
value of the model or 0.0 (as defined above)
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
double rbfcalc3(rbfmodel* s,
double x0,
double x1,
double x2,
ae_state *_state)
{
ae_int_t i;
ae_int_t j;
ae_int_t lx;
ae_int_t tg;
double t;
double rcur;
double bf;
double result;
ae_assert(ae_isfinite(x0, _state), "RBFCalc3: invalid value for X0 (X0 is Inf or NaN)!", _state);
ae_assert(ae_isfinite(x1, _state), "RBFCalc3: invalid value for X1 (X1 is Inf or NaN)!", _state);
ae_assert(ae_isfinite(x2, _state), "RBFCalc3: invalid value for X2 (X2 is Inf or NaN)!", _state);
if( s->ny!=1||s->nx!=3 )
{
result = 0;
return result;
}
result = s->v.ptr.pp_double[0][0]*x0+s->v.ptr.pp_double[0][1]*x1+s->v.ptr.pp_double[0][2]*x2+s->v.ptr.pp_double[0][rbf_mxnx];
if( s->nc==0 )
{
return result;
}
/*
* calculating value for F(X)
*/
rvectorsetlengthatleast(&s->calcbufxcx, rbf_mxnx, _state);
for(i=0; i<=rbf_mxnx-1; i++)
{
s->calcbufxcx.ptr.p_double[i] = 0.0;
}
s->calcbufxcx.ptr.p_double[0] = x0;
s->calcbufxcx.ptr.p_double[1] = x1;
s->calcbufxcx.ptr.p_double[2] = x2;
lx = kdtreequeryrnn(&s->tree, &s->calcbufxcx, s->rmax*rbf_rbffarradius, ae_true, _state);
kdtreequeryresultsx(&s->tree, &s->calcbufx, _state);
kdtreequeryresultstags(&s->tree, &s->calcbuftags, _state);
for(i=0; i<=lx-1; i++)
{
tg = s->calcbuftags.ptr.p_int[i];
rcur = s->wr.ptr.pp_double[tg][0];
bf = ae_exp(-(ae_sqr(x0-s->calcbufx.ptr.pp_double[i][0], _state)+ae_sqr(x1-s->calcbufx.ptr.pp_double[i][1], _state)+ae_sqr(x2-s->calcbufx.ptr.pp_double[i][2], _state))/ae_sqr(rcur, _state), _state);
for(j=0; j<=s->nl-1; j++)
{
result = result+bf*s->wr.ptr.pp_double[tg][1+j];
t = bf*bf;
bf = t*t;
}
}
return result;
}
/*************************************************************************
This function calculates values of the RBF model at the given point.
This is general function which can be used for arbitrary NX (dimension of
the space of arguments) and NY (dimension of the function itself). However
when you have NY=1 you may find more convenient to use RBFCalc2() or
RBFCalc3().
This function returns 0.0 when model is not initialized.
INPUT PARAMETERS:
S - RBF model
X - coordinates, array[NX].
X may have more than NX elements, in this case only
leading NX will be used.
OUTPUT PARAMETERS:
Y - function value, array[NY]. Y is out-parameter and
reallocated after call to this function. In case you want
to reuse previously allocated Y, you may use RBFCalcBuf(),
which reallocates Y only when it is too small.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfcalc(rbfmodel* s,
/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_state *_state)
{
ae_vector_clear(y);
ae_assert(x->cnt>=s->nx, "RBFCalc: Length(X)<NX", _state);
ae_assert(isfinitevector(x, s->nx, _state), "RBFCalc: X contains infinite or NaN values", _state);
rbfcalcbuf(s, x, y, _state);
}
/*************************************************************************
This function calculates values of the RBF model at the given point.
Same as RBFCalc(), but does not reallocate Y when in is large enough to
store function values.
INPUT PARAMETERS:
S - RBF model
X - coordinates, array[NX].
X may have more than NX elements, in this case only
leading NX will be used.
Y - possibly preallocated array
OUTPUT PARAMETERS:
Y - function value, array[NY]. Y is not reallocated when it
is larger than NY.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfcalcbuf(rbfmodel* s,
/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_state *_state)
{
ae_int_t i;
ae_int_t j;
ae_int_t k;
ae_int_t lx;
ae_int_t tg;
double t;
double rcur;
double bf;
ae_assert(x->cnt>=s->nx, "RBFCalcBuf: Length(X)<NX", _state);
ae_assert(isfinitevector(x, s->nx, _state), "RBFCalcBuf: X contains infinite or NaN values", _state);
if( y->cnt<s->ny )
{
ae_vector_set_length(y, s->ny, _state);
}
for(i=0; i<=s->ny-1; i++)
{
y->ptr.p_double[i] = s->v.ptr.pp_double[i][rbf_mxnx];
for(j=0; j<=s->nx-1; j++)
{
y->ptr.p_double[i] = y->ptr.p_double[i]+s->v.ptr.pp_double[i][j]*x->ptr.p_double[j];
}
}
if( s->nc==0 )
{
return;
}
rvectorsetlengthatleast(&s->calcbufxcx, rbf_mxnx, _state);
for(i=0; i<=rbf_mxnx-1; i++)
{
s->calcbufxcx.ptr.p_double[i] = 0.0;
}
for(i=0; i<=s->nx-1; i++)
{
s->calcbufxcx.ptr.p_double[i] = x->ptr.p_double[i];
}
lx = kdtreequeryrnn(&s->tree, &s->calcbufxcx, s->rmax*rbf_rbffarradius, ae_true, _state);
kdtreequeryresultsx(&s->tree, &s->calcbufx, _state);
kdtreequeryresultstags(&s->tree, &s->calcbuftags, _state);
for(i=0; i<=s->ny-1; i++)
{
for(j=0; j<=lx-1; j++)
{
tg = s->calcbuftags.ptr.p_int[j];
rcur = s->wr.ptr.pp_double[tg][0];
bf = ae_exp(-(ae_sqr(s->calcbufxcx.ptr.p_double[0]-s->calcbufx.ptr.pp_double[j][0], _state)+ae_sqr(s->calcbufxcx.ptr.p_double[1]-s->calcbufx.ptr.pp_double[j][1], _state)+ae_sqr(s->calcbufxcx.ptr.p_double[2]-s->calcbufx.ptr.pp_double[j][2], _state))/ae_sqr(rcur, _state), _state);
for(k=0; k<=s->nl-1; k++)
{
y->ptr.p_double[i] = y->ptr.p_double[i]+bf*s->wr.ptr.pp_double[tg][1+k*s->ny+i];
t = bf*bf;
bf = t*t;
}
}
}
}
/*************************************************************************
This function calculates values of the RBF model at the regular grid.
Grid have N0*N1 points, with Point[I,J] = (X0[I], X1[J])
This function returns 0.0 when:
* model is not initialized
* NX<>2
*NY<>1
INPUT PARAMETERS:
S - RBF model
X0 - array of grid nodes, first coordinates, array[N0]
N0 - grid size (number of nodes) in the first dimension
X1 - array of grid nodes, second coordinates, array[N1]
N1 - grid size (number of nodes) in the second dimension
OUTPUT PARAMETERS:
Y - function values, array[N0,N1]. Y is out-variable and
is reallocated by this function.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfgridcalc2(rbfmodel* s,
/* Real */ ae_vector* x0,
ae_int_t n0,
/* Real */ ae_vector* x1,
ae_int_t n1,
/* Real */ ae_matrix* y,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector cpx0;
ae_vector cpx1;
ae_vector p01;
ae_vector p11;
ae_vector p2;
double rlimit;
double xcnorm2;
ae_int_t hp01;
double hcpx0;
double xc0;
double xc1;
double omega;
double radius;
ae_int_t i;
ae_int_t j;
ae_int_t k;
ae_int_t d;
ae_int_t i00;
ae_int_t i01;
ae_int_t i10;
ae_int_t i11;
ae_frame_make(_state, &_frame_block);
ae_matrix_clear(y);
ae_vector_init(&cpx0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&cpx1, 0, DT_REAL, _state, ae_true);
ae_vector_init(&p01, 0, DT_INT, _state, ae_true);
ae_vector_init(&p11, 0, DT_INT, _state, ae_true);
ae_vector_init(&p2, 0, DT_INT, _state, ae_true);
ae_assert(n0>0, "RBFGridCalc2: invalid value for N0 (N0<=0)!", _state);
ae_assert(n1>0, "RBFGridCalc2: invalid value for N1 (N1<=0)!", _state);
ae_assert(x0->cnt>=n0, "RBFGridCalc2: Length(X0)<N0", _state);
ae_assert(x1->cnt>=n1, "RBFGridCalc2: Length(X1)<N1", _state);
ae_assert(isfinitevector(x0, n0, _state), "RBFGridCalc2: X0 contains infinite or NaN values!", _state);
ae_assert(isfinitevector(x1, n1, _state), "RBFGridCalc2: X1 contains infinite or NaN values!", _state);
ae_matrix_set_length(y, n0, n1, _state);
for(i=0; i<=n0-1; i++)
{
for(j=0; j<=n1-1; j++)
{
y->ptr.pp_double[i][j] = 0;
}
}
if( (s->ny!=1||s->nx!=2)||s->nc==0 )
{
ae_frame_leave(_state);
return;
}
/*
*create and sort arrays
*/
ae_vector_set_length(&cpx0, n0, _state);
for(i=0; i<=n0-1; i++)
{
cpx0.ptr.p_double[i] = x0->ptr.p_double[i];
}
tagsort(&cpx0, n0, &p01, &p2, _state);
ae_vector_set_length(&cpx1, n1, _state);
for(i=0; i<=n1-1; i++)
{
cpx1.ptr.p_double[i] = x1->ptr.p_double[i];
}
tagsort(&cpx1, n1, &p11, &p2, _state);
/*
*calculate function's value
*/
for(i=0; i<=s->nc-1; i++)
{
radius = s->wr.ptr.pp_double[i][0];
for(d=0; d<=s->nl-1; d++)
{
omega = s->wr.ptr.pp_double[i][1+d];
rlimit = radius*rbf_rbffarradius;
/*
*search lower and upper indexes
*/
i00 = lowerbound(&cpx0, n0, s->xc.ptr.pp_double[i][0]-rlimit, _state);
i01 = upperbound(&cpx0, n0, s->xc.ptr.pp_double[i][0]+rlimit, _state);
i10 = lowerbound(&cpx1, n1, s->xc.ptr.pp_double[i][1]-rlimit, _state);
i11 = upperbound(&cpx1, n1, s->xc.ptr.pp_double[i][1]+rlimit, _state);
xc0 = s->xc.ptr.pp_double[i][0];
xc1 = s->xc.ptr.pp_double[i][1];
for(j=i00; j<=i01-1; j++)
{
hcpx0 = cpx0.ptr.p_double[j];
hp01 = p01.ptr.p_int[j];
for(k=i10; k<=i11-1; k++)
{
xcnorm2 = ae_sqr(hcpx0-xc0, _state)+ae_sqr(cpx1.ptr.p_double[k]-xc1, _state);
if( ae_fp_less_eq(xcnorm2,rlimit*rlimit) )
{
y->ptr.pp_double[hp01][p11.ptr.p_int[k]] = y->ptr.pp_double[hp01][p11.ptr.p_int[k]]+ae_exp(-xcnorm2/ae_sqr(radius, _state), _state)*omega;
}
}
}
radius = 0.5*radius;
}
}
/*
*add linear term
*/
for(i=0; i<=n0-1; i++)
{
for(j=0; j<=n1-1; j++)
{
y->ptr.pp_double[i][j] = y->ptr.pp_double[i][j]+s->v.ptr.pp_double[0][0]*x0->ptr.p_double[i]+s->v.ptr.pp_double[0][1]*x1->ptr.p_double[j]+s->v.ptr.pp_double[0][rbf_mxnx];
}
}
ae_frame_leave(_state);
}
/*************************************************************************
This function "unpacks" RBF model by extracting its coefficients.
INPUT PARAMETERS:
S - RBF model
OUTPUT PARAMETERS:
NX - dimensionality of argument
NY - dimensionality of the target function
XWR - model information, array[NC,NX+NY+1].
One row of the array corresponds to one basis function:
* first NX columns - coordinates of the center
* next NY columns - weights, one per dimension of the
function being modelled
* last column - radius, same for all dimensions of
the function being modelled
NC - number of the centers
V - polynomial term , array[NY,NX+1]. One row per one
dimension of the function being modelled. First NX
elements are linear coefficients, V[NX] is equal to the
constant part.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfunpack(rbfmodel* s,
ae_int_t* nx,
ae_int_t* ny,
/* Real */ ae_matrix* xwr,
ae_int_t* nc,
/* Real */ ae_matrix* v,
ae_state *_state)
{
ae_int_t i;
ae_int_t j;
double rcur;
*nx = 0;
*ny = 0;
ae_matrix_clear(xwr);
*nc = 0;
ae_matrix_clear(v);
*nx = s->nx;
*ny = s->ny;
*nc = s->nc;
/*
* Fill V
*/
ae_matrix_set_length(v, s->ny, s->nx+1, _state);
for(i=0; i<=s->ny-1; i++)
{
ae_v_move(&v->ptr.pp_double[i][0], 1, &s->v.ptr.pp_double[i][0], 1, ae_v_len(0,s->nx-1));
v->ptr.pp_double[i][s->nx] = s->v.ptr.pp_double[i][rbf_mxnx];
}
/*
* Fill XWR and V
*/
if( *nc*s->nl>0 )
{
ae_matrix_set_length(xwr, s->nc*s->nl, s->nx+s->ny+1, _state);
for(i=0; i<=s->nc-1; i++)
{
rcur = s->wr.ptr.pp_double[i][0];
for(j=0; j<=s->nl-1; j++)
{
ae_v_move(&xwr->ptr.pp_double[i*s->nl+j][0], 1, &s->xc.ptr.pp_double[i][0], 1, ae_v_len(0,s->nx-1));
ae_v_move(&xwr->ptr.pp_double[i*s->nl+j][s->nx], 1, &s->wr.ptr.pp_double[i][1+j*s->ny], 1, ae_v_len(s->nx,s->nx+s->ny-1));
xwr->ptr.pp_double[i*s->nl+j][s->nx+s->ny] = rcur;
rcur = 0.5*rcur;
}
}
}
}
/*************************************************************************
Serializer: allocation
-- ALGLIB --
Copyright 02.02.2012 by Bochkanov Sergey
*************************************************************************/
void rbfalloc(ae_serializer* s, rbfmodel* model, ae_state *_state)
{
/*
* Header
*/
ae_serializer_alloc_entry(s);
ae_serializer_alloc_entry(s);
/*
* Data
*/
ae_serializer_alloc_entry(s);
ae_serializer_alloc_entry(s);
ae_serializer_alloc_entry(s);
ae_serializer_alloc_entry(s);
kdtreealloc(s, &model->tree, _state);
allocrealmatrix(s, &model->xc, -1, -1, _state);
allocrealmatrix(s, &model->wr, -1, -1, _state);
ae_serializer_alloc_entry(s);
allocrealmatrix(s, &model->v, -1, -1, _state);
}
/*************************************************************************
Serializer: serialization
-- ALGLIB --
Copyright 02.02.2012 by Bochkanov Sergey
*************************************************************************/
void rbfserialize(ae_serializer* s, rbfmodel* model, ae_state *_state)
{
/*
* Header
*/
ae_serializer_serialize_int(s, getrbfserializationcode(_state), _state);
ae_serializer_serialize_int(s, rbf_rbffirstversion, _state);
/*
* Data
*/
ae_serializer_serialize_int(s, model->nx, _state);
ae_serializer_serialize_int(s, model->ny, _state);
ae_serializer_serialize_int(s, model->nc, _state);
ae_serializer_serialize_int(s, model->nl, _state);
kdtreeserialize(s, &model->tree, _state);
serializerealmatrix(s, &model->xc, -1, -1, _state);
serializerealmatrix(s, &model->wr, -1, -1, _state);
ae_serializer_serialize_double(s, model->rmax, _state);
serializerealmatrix(s, &model->v, -1, -1, _state);
}
/*************************************************************************
Serializer: unserialization
-- ALGLIB --
Copyright 02.02.2012 by Bochkanov Sergey
*************************************************************************/
void rbfunserialize(ae_serializer* s, rbfmodel* model, ae_state *_state)
{
ae_int_t i0;
ae_int_t i1;
ae_int_t nx;
ae_int_t ny;
_rbfmodel_clear(model);
/*
* Header
*/
ae_serializer_unserialize_int(s, &i0, _state);
ae_assert(i0==getrbfserializationcode(_state), "RBFUnserialize: stream header corrupted", _state);
ae_serializer_unserialize_int(s, &i1, _state);
ae_assert(i1==rbf_rbffirstversion, "RBFUnserialize: stream header corrupted", _state);
/*
* Unserialize primary model parameters, initialize model.
*
* It is necessary to call RBFCreate() because some internal fields
* which are NOT unserialized will need initialization.
*/
ae_serializer_unserialize_int(s, &nx, _state);
ae_serializer_unserialize_int(s, &ny, _state);
rbfcreate(nx, ny, model, _state);
ae_serializer_unserialize_int(s, &model->nc, _state);
ae_serializer_unserialize_int(s, &model->nl, _state);
kdtreeunserialize(s, &model->tree, _state);
unserializerealmatrix(s, &model->xc, _state);
unserializerealmatrix(s, &model->wr, _state);
ae_serializer_unserialize_double(s, &model->rmax, _state);
unserializerealmatrix(s, &model->v, _state);
}
/*************************************************************************
This function changes centers allocation algorithm to one which allocates
centers exactly at the dataset points (one input point = one center). This
function won't have effect until next call to RBFBuildModel().
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call
NOTE: this function has some serialization-related subtleties. We
recommend you to study serialization examples from ALGLIB Reference
Manual if you want to perform serialization of your models.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
static void rbf_rbfgridpoints(rbfmodel* s, ae_state *_state)
{
s->gridtype = 2;
}
/*************************************************************************
This function changes radii calculation algorithm to one which makes
radius for I-th node equal to R[i]=DistNN[i]*Q, where:
* R[i] is a radius calculated by the algorithm
* DistNN[i] is distance from I-th center to its nearest neighbor center
* Q is a scale parameter, which should be within [0.75,1.50], with
recommended value equal to 1.0
* after performing radii calculation, radii are transformed in order to
avoid situation when single outlier has very large radius and influences
many points across entire dataset. Transformation has following form:
new_r[i] = min(r[i],Z*median(r[]))
where r[i] is I-th radius, median() is a median radius across entire
dataset, Z is user-specified value which controls amount of deviation
from median radius.
This function won't have effect until next call to RBFBuildModel().
The idea behind this algorithm is to choose radii corresponding to basis
functions is such way that I-th radius is approximately equal to distance
from I-th center to its nearest neighbor. In this case interactions with
distant points will be insignificant, and we will get well conditioned
basis.
Properties of this basis depend on the value of Q:
* Q<0.75 will give perfectly conditioned basis, but terrible smoothness
properties (RBF interpolant will have sharp peaks around function values)
* Q>1.5 will lead to badly conditioned systems and slow convergence of the
underlying linear solver (although smoothness will be very good)
* Q around 1.0 gives good balance between smoothness and condition number
INPUT PARAMETERS:
S - RBF model, initialized by RBFCreate() call
Q - radius coefficient, Q>0
Z - z-parameter, Z>0
Default value of Q is equal to 1.0
Default value of Z is equal to 5.0
NOTE: this function has some serialization-related subtleties. We
recommend you to study serialization examples from ALGLIB Reference
Manual if you want to perform serialization of your models.
-- ALGLIB --
Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
static void rbf_rbfradnn(rbfmodel* s,
double q,
double z,
ae_state *_state)
{
ae_assert(ae_isfinite(q, _state)&&ae_fp_greater(q,0), "RBFRadNN: Q<=0, infinite or NAN", _state);
ae_assert(ae_isfinite(z, _state)&&ae_fp_greater(z,0), "RBFRadNN: Z<=0, infinite or NAN", _state);
s->fixrad = ae_false;
s->radvalue = q;
s->radzvalue = z;
}
static ae_bool rbf_buildlinearmodel(/* Real */ ae_matrix* x,
/* Real */ ae_matrix* y,
ae_int_t n,
ae_int_t ny,
ae_int_t modeltype,
/* Real */ ae_matrix* v,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector tmpy;
ae_matrix a;
double scaling;
ae_vector shifting;
double mn;
double mx;
ae_vector c;
lsfitreport rep;
ae_int_t i;
ae_int_t j;
ae_int_t k;
ae_int_t info;
ae_bool result;
ae_frame_make(_state, &_frame_block);
ae_matrix_clear(v);
ae_vector_init(&tmpy, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&a, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&shifting, 0, DT_REAL, _state, ae_true);
ae_vector_init(&c, 0, DT_REAL, _state, ae_true);
_lsfitreport_init(&rep, _state, ae_true);
ae_assert(n>=0, "BuildLinearModel: N<0", _state);
ae_assert(ny>0, "BuildLinearModel: NY<=0", _state);
/*
* Handle degenerate case (N=0)
*/
result = ae_true;
ae_matrix_set_length(v, ny, rbf_mxnx+1, _state);
if( n==0 )
{
for(j=0; j<=rbf_mxnx; j++)
{
for(i=0; i<=ny-1; i++)
{
v->ptr.pp_double[i][j] = 0;
}
}
ae_frame_leave(_state);
return result;
}
/*
* Allocate temporaries
*/
ae_vector_set_length(&tmpy, n, _state);
/*
* General linear model.
*/
if( modeltype==1 )
{
/*
* Calculate scaling/shifting, transform variables, prepare LLS problem
*/
ae_matrix_set_length(&a, n, rbf_mxnx+1, _state);
ae_vector_set_length(&shifting, rbf_mxnx, _state);
scaling = 0;
for(i=0; i<=rbf_mxnx-1; i++)
{
mn = x->ptr.pp_double[0][i];
mx = mn;
for(j=1; j<=n-1; j++)
{
if( ae_fp_greater(mn,x->ptr.pp_double[j][i]) )
{
mn = x->ptr.pp_double[j][i];
}
if( ae_fp_less(mx,x->ptr.pp_double[j][i]) )
{
mx = x->ptr.pp_double[j][i];
}
}
scaling = ae_maxreal(scaling, mx-mn, _state);
shifting.ptr.p_double[i] = 0.5*(mx+mn);
}
if( ae_fp_eq(scaling,0) )
{
scaling = 1;
}
else
{
scaling = 0.5*scaling;
}
for(i=0; i<=n-1; i++)
{
for(j=0; j<=rbf_mxnx-1; j++)
{
a.ptr.pp_double[i][j] = (x->ptr.pp_double[i][j]-shifting.ptr.p_double[j])/scaling;
}
}
for(i=0; i<=n-1; i++)
{
a.ptr.pp_double[i][rbf_mxnx] = 1;
}
/*
* Solve linear system in transformed variables, make backward
*/
for(i=0; i<=ny-1; i++)
{
for(j=0; j<=n-1; j++)
{
tmpy.ptr.p_double[j] = y->ptr.pp_double[j][i];
}
lsfitlinear(&tmpy, &a, n, rbf_mxnx+1, &info, &c, &rep, _state);
if( info<=0 )
{
result = ae_false;
ae_frame_leave(_state);
return result;
}
for(j=0; j<=rbf_mxnx-1; j++)
{
v->ptr.pp_double[i][j] = c.ptr.p_double[j]/scaling;
}
v->ptr.pp_double[i][rbf_mxnx] = c.ptr.p_double[rbf_mxnx];
for(j=0; j<=rbf_mxnx-1; j++)
{
v->ptr.pp_double[i][rbf_mxnx] = v->ptr.pp_double[i][rbf_mxnx]-shifting.ptr.p_double[j]*v->ptr.pp_double[i][j];
}
for(j=0; j<=n-1; j++)
{
for(k=0; k<=rbf_mxnx-1; k++)
{
y->ptr.pp_double[j][i] = y->ptr.pp_double[j][i]-x->ptr.pp_double[j][k]*v->ptr.pp_double[i][k];
}
y->ptr.pp_double[j][i] = y->ptr.pp_double[j][i]-v->ptr.pp_double[i][rbf_mxnx];
}
}
ae_frame_leave(_state);
return result;
}
/*
* Constant model, very simple
*/
if( modeltype==2 )
{
for(i=0; i<=ny-1; i++)
{
for(j=0; j<=rbf_mxnx; j++)
{
v->ptr.pp_double[i][j] = 0;
}
for(j=0; j<=n-1; j++)
{
v->ptr.pp_double[i][rbf_mxnx] = v->ptr.pp_double[i][rbf_mxnx]+y->ptr.pp_double[j][i];
}
if( n>0 )
{
v->ptr.pp_double[i][rbf_mxnx] = v->ptr.pp_double[i][rbf_mxnx]/n;
}
for(j=0; j<=n-1; j++)
{
y->ptr.pp_double[j][i] = y->ptr.pp_double[j][i]-v->ptr.pp_double[i][rbf_mxnx];
}
}
ae_frame_leave(_state);
return result;
}
/*
* Zero model
*/
ae_assert(modeltype==3, "BuildLinearModel: unknown model type", _state);
for(i=0; i<=ny-1; i++)
{
for(j=0; j<=rbf_mxnx; j++)
{
v->ptr.pp_double[i][j] = 0;
}
}
ae_frame_leave(_state);
return result;
}
static void rbf_buildrbfmodellsqr(/* Real */ ae_matrix* x,
/* Real */ ae_matrix* y,
/* Real */ ae_matrix* xc,
/* Real */ ae_vector* r,
ae_int_t n,
ae_int_t nc,
ae_int_t ny,
kdtree* pointstree,
kdtree* centerstree,
double epsort,
double epserr,
ae_int_t maxits,
ae_int_t* gnnz,
ae_int_t* snnz,
/* Real */ ae_matrix* w,
ae_int_t* info,
ae_int_t* iterationscount,
ae_int_t* nmv,
ae_state *_state)
{
ae_frame _frame_block;
linlsqrstate state;
linlsqrreport lsqrrep;
sparsematrix spg;
sparsematrix sps;
ae_vector nearcenterscnt;
ae_vector nearpointscnt;
ae_vector skipnearpointscnt;
ae_vector farpointscnt;
ae_int_t maxnearcenterscnt;
ae_int_t maxnearpointscnt;
ae_int_t maxfarpointscnt;
ae_int_t sumnearcenterscnt;
ae_int_t sumnearpointscnt;
ae_int_t sumfarpointscnt;
double maxrad;
ae_vector pointstags;
ae_vector centerstags;
ae_matrix nearpoints;
ae_matrix nearcenters;
ae_matrix farpoints;
ae_int_t tmpi;
ae_int_t pointscnt;
ae_int_t centerscnt;
ae_vector xcx;
ae_vector tmpy;
ae_vector tc;
ae_vector g;
ae_vector c;
ae_int_t i;
ae_int_t j;
ae_int_t k;
ae_int_t sind;
ae_matrix a;
double vv;
double vx;
double vy;
double vz;
double vr;
double gnorm2;
ae_vector tmp0;
ae_vector tmp1;
ae_vector tmp2;
double fx;
ae_matrix xx;
ae_matrix cx;
double mrad;
ae_frame_make(_state, &_frame_block);
*gnnz = 0;
*snnz = 0;
ae_matrix_clear(w);
*info = 0;
*iterationscount = 0;
*nmv = 0;
_linlsqrstate_init(&state, _state, ae_true);
_linlsqrreport_init(&lsqrrep, _state, ae_true);
_sparsematrix_init(&spg, _state, ae_true);
_sparsematrix_init(&sps, _state, ae_true);
ae_vector_init(&nearcenterscnt, 0, DT_INT, _state, ae_true);
ae_vector_init(&nearpointscnt, 0, DT_INT, _state, ae_true);
ae_vector_init(&skipnearpointscnt, 0, DT_INT, _state, ae_true);
ae_vector_init(&farpointscnt, 0, DT_INT, _state, ae_true);
ae_vector_init(&pointstags, 0, DT_INT, _state, ae_true);
ae_vector_init(&centerstags, 0, DT_INT, _state, ae_true);
ae_matrix_init(&nearpoints, 0, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&nearcenters, 0, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&farpoints, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&xcx, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmpy, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tc, 0, DT_REAL, _state, ae_true);
ae_vector_init(&g, 0, DT_REAL, _state, ae_true);
ae_vector_init(&c, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&a, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmp0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmp1, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmp2, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&xx, 0, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&cx, 0, 0, DT_REAL, _state, ae_true);
/*
* Handle special cases: NC=0
*/
if( nc==0 )
{
*info = 1;
*iterationscount = 0;
*nmv = 0;
ae_frame_leave(_state);
return;
}
/*
* Prepare for general case, NC>0
*/
ae_vector_set_length(&xcx, rbf_mxnx, _state);
ae_vector_set_length(&pointstags, n, _state);
ae_vector_set_length(&centerstags, nc, _state);
*info = -1;
*iterationscount = 0;
*nmv = 0;
/*
* This block prepares quantities used to compute approximate cardinal basis functions (ACBFs):
* * NearCentersCnt[] - array[NC], whose elements store number of near centers used to build ACBF
* * NearPointsCnt[] - array[NC], number of near points used to build ACBF
* * FarPointsCnt[] - array[NC], number of far points (ones where ACBF is nonzero)
* * MaxNearCentersCnt - max(NearCentersCnt)
* * MaxNearPointsCnt - max(NearPointsCnt)
* * SumNearCentersCnt - sum(NearCentersCnt)
* * SumNearPointsCnt - sum(NearPointsCnt)
* * SumFarPointsCnt - sum(FarPointsCnt)
*/
ae_vector_set_length(&nearcenterscnt, nc, _state);
ae_vector_set_length(&nearpointscnt, nc, _state);
ae_vector_set_length(&skipnearpointscnt, nc, _state);
ae_vector_set_length(&farpointscnt, nc, _state);
maxnearcenterscnt = 0;
maxnearpointscnt = 0;
maxfarpointscnt = 0;
sumnearcenterscnt = 0;
sumnearpointscnt = 0;
sumfarpointscnt = 0;
for(i=0; i<=nc-1; i++)
{
for(j=0; j<=rbf_mxnx-1; j++)
{
xcx.ptr.p_double[j] = xc->ptr.pp_double[i][j];
}
/*
* Determine number of near centers and maximum radius of near centers
*/
nearcenterscnt.ptr.p_int[i] = kdtreequeryrnn(centerstree, &xcx, r->ptr.p_double[i]*rbf_rbfnearradius, ae_true, _state);
kdtreequeryresultstags(centerstree, &centerstags, _state);
maxrad = 0;
for(j=0; j<=nearcenterscnt.ptr.p_int[i]-1; j++)
{
maxrad = ae_maxreal(maxrad, ae_fabs(r->ptr.p_double[centerstags.ptr.p_int[j]], _state), _state);
}
/*
* Determine number of near points (ones which used to build ACBF)
* and skipped points (the most near points which are NOT used to build ACBF
* and are NOT included in the near points count
*/
skipnearpointscnt.ptr.p_int[i] = kdtreequeryrnn(pointstree, &xcx, 0.1*r->ptr.p_double[i], ae_true, _state);
nearpointscnt.ptr.p_int[i] = kdtreequeryrnn(pointstree, &xcx, (r->ptr.p_double[i]+maxrad)*rbf_rbfnearradius, ae_true, _state)-skipnearpointscnt.ptr.p_int[i];
ae_assert(nearpointscnt.ptr.p_int[i]>=0, "BuildRBFModelLSQR: internal error", _state);
/*
* Determine number of far points
*/
farpointscnt.ptr.p_int[i] = kdtreequeryrnn(pointstree, &xcx, ae_maxreal(r->ptr.p_double[i]*rbf_rbfnearradius+maxrad*rbf_rbffarradius, r->ptr.p_double[i]*rbf_rbffarradius, _state), ae_true, _state);
/*
* calculate sum and max, make some basic checks
*/
ae_assert(nearcenterscnt.ptr.p_int[i]>0, "BuildRBFModelLSQR: internal error", _state);
maxnearcenterscnt = ae_maxint(maxnearcenterscnt, nearcenterscnt.ptr.p_int[i], _state);
maxnearpointscnt = ae_maxint(maxnearpointscnt, nearpointscnt.ptr.p_int[i], _state);
maxfarpointscnt = ae_maxint(maxfarpointscnt, farpointscnt.ptr.p_int[i], _state);
sumnearcenterscnt = sumnearcenterscnt+nearcenterscnt.ptr.p_int[i];
sumnearpointscnt = sumnearpointscnt+nearpointscnt.ptr.p_int[i];
sumfarpointscnt = sumfarpointscnt+farpointscnt.ptr.p_int[i];
}
*snnz = sumnearcenterscnt;
*gnnz = sumfarpointscnt;
ae_assert(maxnearcenterscnt>0, "BuildRBFModelLSQR: internal error", _state);
/*
* Allocate temporaries.
*
* NOTE: we want to avoid allocation of zero-size arrays, so we
* use max(desired_size,1) instead of desired_size when performing
* memory allocation.
*/
ae_matrix_set_length(&a, maxnearpointscnt+maxnearcenterscnt, maxnearcenterscnt, _state);
ae_vector_set_length(&tmpy, maxnearpointscnt+maxnearcenterscnt, _state);
ae_vector_set_length(&g, maxnearcenterscnt, _state);
ae_vector_set_length(&c, maxnearcenterscnt, _state);
ae_matrix_set_length(&nearcenters, maxnearcenterscnt, rbf_mxnx, _state);
ae_matrix_set_length(&nearpoints, ae_maxint(maxnearpointscnt, 1, _state), rbf_mxnx, _state);
ae_matrix_set_length(&farpoints, ae_maxint(maxfarpointscnt, 1, _state), rbf_mxnx, _state);
/*
* fill matrix SpG
*/
sparsecreate(n, nc, *gnnz, &spg, _state);
sparsecreate(nc, nc, *snnz, &sps, _state);
for(i=0; i<=nc-1; i++)
{
centerscnt = nearcenterscnt.ptr.p_int[i];
/*
* main center
*/
for(j=0; j<=rbf_mxnx-1; j++)
{
xcx.ptr.p_double[j] = xc->ptr.pp_double[i][j];
}
/*
* center's tree
*/
tmpi = kdtreequeryknn(centerstree, &xcx, centerscnt, ae_true, _state);
ae_assert(tmpi==centerscnt, "BuildRBFModelLSQR: internal error", _state);
kdtreequeryresultsx(centerstree, &cx, _state);
kdtreequeryresultstags(centerstree, &centerstags, _state);
/*
* point's tree
*/
mrad = 0;
for(j=0; j<=centerscnt-1; j++)
{
mrad = ae_maxreal(mrad, r->ptr.p_double[centerstags.ptr.p_int[j]], _state);
}
/*
* we need to be sure that 'CTree' contains
* at least one side center
*/
sparseset(&sps, i, i, 1, _state);
c.ptr.p_double[0] = 1.0;
for(j=1; j<=centerscnt-1; j++)
{
c.ptr.p_double[j] = 0.0;
}
if( centerscnt>1&&nearpointscnt.ptr.p_int[i]>0 )
{
/*
* first KDTree request for points
*/
pointscnt = nearpointscnt.ptr.p_int[i];
tmpi = kdtreequeryknn(pointstree, &xcx, skipnearpointscnt.ptr.p_int[i]+nearpointscnt.ptr.p_int[i], ae_true, _state);
ae_assert(tmpi==skipnearpointscnt.ptr.p_int[i]+nearpointscnt.ptr.p_int[i], "BuildRBFModelLSQR: internal error", _state);
kdtreequeryresultsx(pointstree, &xx, _state);
sind = skipnearpointscnt.ptr.p_int[i];
for(j=0; j<=pointscnt-1; j++)
{
vx = xx.ptr.pp_double[sind+j][0];
vy = xx.ptr.pp_double[sind+j][1];
vz = xx.ptr.pp_double[sind+j][2];
for(k=0; k<=centerscnt-1; k++)
{
vr = 0.0;
vv = vx-cx.ptr.pp_double[k][0];
vr = vr+vv*vv;
vv = vy-cx.ptr.pp_double[k][1];
vr = vr+vv*vv;
vv = vz-cx.ptr.pp_double[k][2];
vr = vr+vv*vv;
vv = r->ptr.p_double[centerstags.ptr.p_int[k]];
a.ptr.pp_double[j][k] = ae_exp(-vr/(vv*vv), _state);
}
}
for(j=0; j<=centerscnt-1; j++)
{
g.ptr.p_double[j] = ae_exp(-(ae_sqr(xcx.ptr.p_double[0]-cx.ptr.pp_double[j][0], _state)+ae_sqr(xcx.ptr.p_double[1]-cx.ptr.pp_double[j][1], _state)+ae_sqr(xcx.ptr.p_double[2]-cx.ptr.pp_double[j][2], _state))/ae_sqr(r->ptr.p_double[centerstags.ptr.p_int[j]], _state), _state);
}
/*
* calculate the problem
*/
gnorm2 = ae_v_dotproduct(&g.ptr.p_double[0], 1, &g.ptr.p_double[0], 1, ae_v_len(0,centerscnt-1));
for(j=0; j<=pointscnt-1; j++)
{
vv = ae_v_dotproduct(&a.ptr.pp_double[j][0], 1, &g.ptr.p_double[0], 1, ae_v_len(0,centerscnt-1));
vv = vv/gnorm2;
tmpy.ptr.p_double[j] = -vv;
ae_v_subd(&a.ptr.pp_double[j][0], 1, &g.ptr.p_double[0], 1, ae_v_len(0,centerscnt-1), vv);
}
for(j=pointscnt; j<=pointscnt+centerscnt-1; j++)
{
for(k=0; k<=centerscnt-1; k++)
{
a.ptr.pp_double[j][k] = 0.0;
}
a.ptr.pp_double[j][j-pointscnt] = 1.0E-6;
tmpy.ptr.p_double[j] = 0.0;
}
fblssolvels(&a, &tmpy, pointscnt+centerscnt, centerscnt, &tmp0, &tmp1, &tmp2, _state);
ae_v_move(&c.ptr.p_double[0], 1, &tmpy.ptr.p_double[0], 1, ae_v_len(0,centerscnt-1));
vv = ae_v_dotproduct(&g.ptr.p_double[0], 1, &c.ptr.p_double[0], 1, ae_v_len(0,centerscnt-1));
vv = vv/gnorm2;
ae_v_subd(&c.ptr.p_double[0], 1, &g.ptr.p_double[0], 1, ae_v_len(0,centerscnt-1), vv);
vv = 1/gnorm2;
ae_v_addd(&c.ptr.p_double[0], 1, &g.ptr.p_double[0], 1, ae_v_len(0,centerscnt-1), vv);
for(j=0; j<=centerscnt-1; j++)
{
sparseset(&sps, i, centerstags.ptr.p_int[j], c.ptr.p_double[j], _state);
}
}
/*
* second KDTree request for points
*/
pointscnt = farpointscnt.ptr.p_int[i];
tmpi = kdtreequeryknn(pointstree, &xcx, pointscnt, ae_true, _state);
ae_assert(tmpi==pointscnt, "BuildRBFModelLSQR: internal error", _state);
kdtreequeryresultsx(pointstree, &xx, _state);
kdtreequeryresultstags(pointstree, &pointstags, _state);
/*
*fill SpG matrix
*/
for(j=0; j<=pointscnt-1; j++)
{
fx = 0;
vx = xx.ptr.pp_double[j][0];
vy = xx.ptr.pp_double[j][1];
vz = xx.ptr.pp_double[j][2];
for(k=0; k<=centerscnt-1; k++)
{
vr = 0.0;
vv = vx-cx.ptr.pp_double[k][0];
vr = vr+vv*vv;
vv = vy-cx.ptr.pp_double[k][1];
vr = vr+vv*vv;
vv = vz-cx.ptr.pp_double[k][2];
vr = vr+vv*vv;
vv = r->ptr.p_double[centerstags.ptr.p_int[k]];
vv = vv*vv;
fx = fx+c.ptr.p_double[k]*ae_exp(-vr/vv, _state);
}
sparseset(&spg, pointstags.ptr.p_int[j], i, fx, _state);
}
}
sparseconverttocrs(&spg, _state);
sparseconverttocrs(&sps, _state);
/*
* solve by LSQR method
*/
ae_vector_set_length(&tmpy, n, _state);
ae_vector_set_length(&tc, nc, _state);
ae_matrix_set_length(w, nc, ny, _state);
linlsqrcreate(n, nc, &state, _state);
linlsqrsetcond(&state, epsort, epserr, maxits, _state);
for(i=0; i<=ny-1; i++)
{
for(j=0; j<=n-1; j++)
{
tmpy.ptr.p_double[j] = y->ptr.pp_double[j][i];
}
linlsqrsolvesparse(&state, &spg, &tmpy, _state);
linlsqrresults(&state, &c, &lsqrrep, _state);
if( lsqrrep.terminationtype<=0 )
{
*info = -4;
ae_frame_leave(_state);
return;
}
sparsemtv(&sps, &c, &tc, _state);
for(j=0; j<=nc-1; j++)
{
w->ptr.pp_double[j][i] = tc.ptr.p_double[j];
}
*iterationscount = *iterationscount+lsqrrep.iterationscount;
*nmv = *nmv+lsqrrep.nmv;
}
*info = 1;
ae_frame_leave(_state);
}
static void rbf_buildrbfmlayersmodellsqr(/* Real */ ae_matrix* x,
/* Real */ ae_matrix* y,
/* Real */ ae_matrix* xc,
double rval,
/* Real */ ae_vector* r,
ae_int_t n,
ae_int_t* nc,
ae_int_t ny,
ae_int_t nlayers,
kdtree* centerstree,
double epsort,
double epserr,
ae_int_t maxits,
double lambdav,
ae_int_t* annz,
/* Real */ ae_matrix* w,
ae_int_t* info,
ae_int_t* iterationscount,
ae_int_t* nmv,
ae_state *_state)
{
ae_frame _frame_block;
linlsqrstate state;
linlsqrreport lsqrrep;
sparsematrix spa;
double anorm;
ae_vector omega;
ae_vector xx;
ae_vector tmpy;
ae_matrix cx;
double yval;
ae_int_t nec;
ae_vector centerstags;
ae_int_t layer;
ae_int_t i;
ae_int_t j;
ae_int_t k;
double v;
double rmaxbefore;
double rmaxafter;
ae_frame_make(_state, &_frame_block);
ae_matrix_clear(xc);
ae_vector_clear(r);
*nc = 0;
*annz = 0;
ae_matrix_clear(w);
*info = 0;
*iterationscount = 0;
*nmv = 0;
_linlsqrstate_init(&state, _state, ae_true);
_linlsqrreport_init(&lsqrrep, _state, ae_true);
_sparsematrix_init(&spa, _state, ae_true);
ae_vector_init(&omega, 0, DT_REAL, _state, ae_true);
ae_vector_init(&xx, 0, DT_REAL, _state, ae_true);
ae_vector_init(&tmpy, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&cx, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&centerstags, 0, DT_INT, _state, ae_true);
ae_assert(nlayers>=0, "BuildRBFMLayersModelLSQR: invalid argument(NLayers<0)", _state);
ae_assert(n>=0, "BuildRBFMLayersModelLSQR: invalid argument(N<0)", _state);
ae_assert(rbf_mxnx>0&&rbf_mxnx<=3, "BuildRBFMLayersModelLSQR: internal error(invalid global const MxNX: either MxNX<=0 or MxNX>3)", _state);
*annz = 0;
if( n==0||nlayers==0 )
{
*info = 1;
*iterationscount = 0;
*nmv = 0;
ae_frame_leave(_state);
return;
}
*nc = n*nlayers;
ae_vector_set_length(&xx, rbf_mxnx, _state);
ae_vector_set_length(&centerstags, n, _state);
ae_matrix_set_length(xc, *nc, rbf_mxnx, _state);
ae_vector_set_length(r, *nc, _state);
for(i=0; i<=*nc-1; i++)
{
for(j=0; j<=rbf_mxnx-1; j++)
{
xc->ptr.pp_double[i][j] = x->ptr.pp_double[i%n][j];
}
}
for(i=0; i<=*nc-1; i++)
{
r->ptr.p_double[i] = rval/ae_pow(2, i/n, _state);
}
for(i=0; i<=n-1; i++)
{
centerstags.ptr.p_int[i] = i;
}
kdtreebuildtagged(xc, &centerstags, n, rbf_mxnx, 0, 2, centerstree, _state);
ae_vector_set_length(&omega, n, _state);
ae_vector_set_length(&tmpy, n, _state);
ae_matrix_set_length(w, *nc, ny, _state);
*info = -1;
*iterationscount = 0;
*nmv = 0;
linlsqrcreate(n, n, &state, _state);
linlsqrsetcond(&state, epsort, epserr, maxits, _state);
linlsqrsetlambdai(&state, 1.0E-6, _state);
/*
* calculate number of non-zero elements for sparse matrix
*/
for(i=0; i<=n-1; i++)
{
for(j=0; j<=rbf_mxnx-1; j++)
{
xx.ptr.p_double[j] = x->ptr.pp_double[i][j];
}
*annz = *annz+kdtreequeryrnn(centerstree, &xx, r->ptr.p_double[0]*rbf_rbfmlradius, ae_true, _state);
}
for(layer=0; layer<=nlayers-1; layer++)
{
/*
* Fill sparse matrix, calculate norm(A)
*/
anorm = 0.0;
sparsecreate(n, n, *annz, &spa, _state);
for(i=0; i<=n-1; i++)
{
for(j=0; j<=rbf_mxnx-1; j++)
{
xx.ptr.p_double[j] = x->ptr.pp_double[i][j];
}
nec = kdtreequeryrnn(centerstree, &xx, r->ptr.p_double[layer*n]*rbf_rbfmlradius, ae_true, _state);
kdtreequeryresultsx(centerstree, &cx, _state);
kdtreequeryresultstags(centerstree, &centerstags, _state);
for(j=0; j<=nec-1; j++)
{
v = ae_exp(-(ae_sqr(xx.ptr.p_double[0]-cx.ptr.pp_double[j][0], _state)+ae_sqr(xx.ptr.p_double[1]-cx.ptr.pp_double[j][1], _state)+ae_sqr(xx.ptr.p_double[2]-cx.ptr.pp_double[j][2], _state))/ae_sqr(r->ptr.p_double[layer*n+centerstags.ptr.p_int[j]], _state), _state);
sparseset(&spa, i, centerstags.ptr.p_int[j], v, _state);
anorm = anorm+ae_sqr(v, _state);
}
}
anorm = ae_sqrt(anorm, _state);
sparseconverttocrs(&spa, _state);
/*
* Calculate maximum residual before adding new layer.
* This value is not used by algorithm, the only purpose is to make debugging easier.
*/
rmaxbefore = 0.0;
for(j=0; j<=n-1; j++)
{
for(i=0; i<=ny-1; i++)
{
rmaxbefore = ae_maxreal(rmaxbefore, ae_fabs(y->ptr.pp_double[j][i], _state), _state);
}
}
/*
* Process NY dimensions of the target function
*/
for(i=0; i<=ny-1; i++)
{
for(j=0; j<=n-1; j++)
{
tmpy.ptr.p_double[j] = y->ptr.pp_double[j][i];
}
/*
* calculate Omega for current layer
*/
linlsqrsetlambdai(&state, lambdav*anorm/n, _state);
linlsqrsolvesparse(&state, &spa, &tmpy, _state);
linlsqrresults(&state, &omega, &lsqrrep, _state);
if( lsqrrep.terminationtype<=0 )
{
*info = -4;
ae_frame_leave(_state);
return;
}
/*
* calculate error for current layer
*/
for(j=0; j<=n-1; j++)
{
yval = 0;
for(k=0; k<=rbf_mxnx-1; k++)
{
xx.ptr.p_double[k] = x->ptr.pp_double[j][k];
}
nec = kdtreequeryrnn(centerstree, &xx, r->ptr.p_double[layer*n]*rbf_rbffarradius, ae_true, _state);
kdtreequeryresultsx(centerstree, &cx, _state);
kdtreequeryresultstags(centerstree, &centerstags, _state);
for(k=0; k<=nec-1; k++)
{
yval = yval+omega.ptr.p_double[centerstags.ptr.p_int[k]]*ae_exp(-(ae_sqr(xx.ptr.p_double[0]-cx.ptr.pp_double[k][0], _state)+ae_sqr(xx.ptr.p_double[1]-cx.ptr.pp_double[k][1], _state)+ae_sqr(xx.ptr.p_double[2]-cx.ptr.pp_double[k][2], _state))/ae_sqr(r->ptr.p_double[layer*n+centerstags.ptr.p_int[k]], _state), _state);
}
y->ptr.pp_double[j][i] = y->ptr.pp_double[j][i]-yval;
}
/*
* write Omega in out parameter W
*/
for(j=0; j<=n-1; j++)
{
w->ptr.pp_double[layer*n+j][i] = omega.ptr.p_double[j];
}
*iterationscount = *iterationscount+lsqrrep.iterationscount;
*nmv = *nmv+lsqrrep.nmv;
}
/*
* Calculate maximum residual before adding new layer.
* This value is not used by algorithm, the only purpose is to make debugging easier.
*/
rmaxafter = 0.0;
for(j=0; j<=n-1; j++)
{
for(i=0; i<=ny-1; i++)
{
rmaxafter = ae_maxreal(rmaxafter, ae_fabs(y->ptr.pp_double[j][i], _state), _state);
}
}
}
*info = 1;
ae_frame_leave(_state);
}
ae_bool _rbfmodel_init(void* _p, ae_state *_state, ae_bool make_automatic)
{
rbfmodel *p = (rbfmodel*)_p;
ae_touch_ptr((void*)p);
if( !_kdtree_init(&p->tree, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init(&p->xc, 0, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init(&p->wr, 0, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init(&p->v, 0, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init(&p->x, 0, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init(&p->y, 0, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->calcbufxcx, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init(&p->calcbufx, 0, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->calcbuftags, 0, DT_INT, _state, make_automatic) )
return ae_false;
return ae_true;
}
ae_bool _rbfmodel_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic)
{
rbfmodel *dst = (rbfmodel*)_dst;
rbfmodel *src = (rbfmodel*)_src;
dst->ny = src->ny;
dst->nx = src->nx;
dst->nc = src->nc;
dst->nl = src->nl;
if( !_kdtree_init_copy(&dst->tree, &src->tree, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init_copy(&dst->xc, &src->xc, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init_copy(&dst->wr, &src->wr, _state, make_automatic) )
return ae_false;
dst->rmax = src->rmax;
if( !ae_matrix_init_copy(&dst->v, &src->v, _state, make_automatic) )
return ae_false;
dst->gridtype = src->gridtype;
dst->fixrad = src->fixrad;
dst->lambdav = src->lambdav;
dst->radvalue = src->radvalue;
dst->radzvalue = src->radzvalue;
dst->nlayers = src->nlayers;
dst->aterm = src->aterm;
dst->algorithmtype = src->algorithmtype;
dst->epsort = src->epsort;
dst->epserr = src->epserr;
dst->maxits = src->maxits;
dst->h = src->h;
dst->n = src->n;
if( !ae_matrix_init_copy(&dst->x, &src->x, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init_copy(&dst->y, &src->y, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->calcbufxcx, &src->calcbufxcx, _state, make_automatic) )
return ae_false;
if( !ae_matrix_init_copy(&dst->calcbufx, &src->calcbufx, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->calcbuftags, &src->calcbuftags, _state, make_automatic) )
return ae_false;
return ae_true;
}
void _rbfmodel_clear(void* _p)
{
rbfmodel *p = (rbfmodel*)_p;
ae_touch_ptr((void*)p);
_kdtree_clear(&p->tree);
ae_matrix_clear(&p->xc);
ae_matrix_clear(&p->wr);
ae_matrix_clear(&p->v);
ae_matrix_clear(&p->x);
ae_matrix_clear(&p->y);
ae_vector_clear(&p->calcbufxcx);
ae_matrix_clear(&p->calcbufx);
ae_vector_clear(&p->calcbuftags);
}
void _rbfmodel_destroy(void* _p)
{
rbfmodel *p = (rbfmodel*)_p;
ae_touch_ptr((void*)p);
_kdtree_destroy(&p->tree);
ae_matrix_destroy(&p->xc);
ae_matrix_destroy(&p->wr);
ae_matrix_destroy(&p->v);
ae_matrix_destroy(&p->x);
ae_matrix_destroy(&p->y);
ae_vector_destroy(&p->calcbufxcx);
ae_matrix_destroy(&p->calcbufx);
ae_vector_destroy(&p->calcbuftags);
}
ae_bool _rbfreport_init(void* _p, ae_state *_state, ae_bool make_automatic)
{
rbfreport *p = (rbfreport*)_p;
ae_touch_ptr((void*)p);
return ae_true;
}
ae_bool _rbfreport_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic)
{
rbfreport *dst = (rbfreport*)_dst;
rbfreport *src = (rbfreport*)_src;
dst->arows = src->arows;
dst->acols = src->acols;
dst->annz = src->annz;
dst->iterationscount = src->iterationscount;
dst->nmv = src->nmv;
dst->terminationtype = src->terminationtype;
return ae_true;
}
void _rbfreport_clear(void* _p)
{
rbfreport *p = (rbfreport*)_p;
ae_touch_ptr((void*)p);
}
void _rbfreport_destroy(void* _p)
{
rbfreport *p = (rbfreport*)_p;
ae_touch_ptr((void*)p);
}
/*************************************************************************
This subroutine calculates the value of the bilinear or bicubic spline at
the given point X.
Input parameters:
C - coefficients table.
Built by BuildBilinearSpline or BuildBicubicSpline.
X, Y- point
Result:
S(x,y)
-- ALGLIB PROJECT --
Copyright 05.07.2007 by Bochkanov Sergey
*************************************************************************/
double spline2dcalc(spline2dinterpolant* c,
double x,
double y,
ae_state *_state)
{
double v;
double vx;
double vy;
double vxy;
double result;
ae_assert(c->stype==-1||c->stype==-3, "Spline2DCalc: incorrect C (incorrect parameter C.SType)", _state);
ae_assert(ae_isfinite(x, _state)&&ae_isfinite(y, _state), "Spline2DCalc: X or Y contains NaN or Infinite value", _state);
if( c->d!=1 )
{
result = 0;
return result;
}
spline2ddiff(c, x, y, &v, &vx, &vy, &vxy, _state);
result = v;
return result;
}
/*************************************************************************
This subroutine calculates the value of the bilinear or bicubic spline at
the given point X and its derivatives.
Input parameters:
C - spline interpolant.
X, Y- point
Output parameters:
F - S(x,y)
FX - dS(x,y)/dX
FY - dS(x,y)/dY
FXY - d2S(x,y)/dXdY
-- ALGLIB PROJECT --
Copyright 05.07.2007 by Bochkanov Sergey
*************************************************************************/
void spline2ddiff(spline2dinterpolant* c,
double x,
double y,
double* f,
double* fx,
double* fy,
double* fxy,
ae_state *_state)
{
double t;
double dt;
double u;
double du;
ae_int_t ix;
ae_int_t iy;
ae_int_t l;
ae_int_t r;
ae_int_t h;
ae_int_t s1;
ae_int_t s2;
ae_int_t s3;
ae_int_t s4;
ae_int_t sfx;
ae_int_t sfy;
ae_int_t sfxy;
double y1;
double y2;
double y3;
double y4;
double v;
double t0;
double t1;
double t2;
double t3;
double u0;
double u1;
double u2;
double u3;
*f = 0;
*fx = 0;
*fy = 0;
*fxy = 0;
ae_assert(c->stype==-1||c->stype==-3, "Spline2DDiff: incorrect C (incorrect parameter C.SType)", _state);
ae_assert(ae_isfinite(x, _state)&&ae_isfinite(y, _state), "Spline2DDiff: X or Y contains NaN or Infinite value", _state);
/*
* Prepare F, dF/dX, dF/dY, d2F/dXdY
*/
*f = 0;
*fx = 0;
*fy = 0;
*fxy = 0;
if( c->d!=1 )
{
return;
}
/*
* Binary search in the [ x[0], ..., x[n-2] ] (x[n-1] is not included)
*/
l = 0;
r = c->n-1;
while(l!=r-1)
{
h = (l+r)/2;
if( ae_fp_greater_eq(c->x.ptr.p_double[h],x) )
{
r = h;
}
else
{
l = h;
}
}
t = (x-c->x.ptr.p_double[l])/(c->x.ptr.p_double[l+1]-c->x.ptr.p_double[l]);
dt = 1.0/(c->x.ptr.p_double[l+1]-c->x.ptr.p_double[l]);
ix = l;
/*
* Binary search in the [ y[0], ..., y[m-2] ] (y[m-1] is not included)
*/
l = 0;
r = c->m-1;
while(l!=r-1)
{
h = (l+r)/2;
if( ae_fp_greater_eq(c->y.ptr.p_double[h],y) )
{
r = h;
}
else
{
l = h;
}
}
u = (y-c->y.ptr.p_double[l])/(c->y.ptr.p_double[l+1]-c->y.ptr.p_double[l]);
du = 1.0/(c->y.ptr.p_double[l+1]-c->y.ptr.p_double[l]);
iy = l;
/*
* Bilinear interpolation
*/
if( c->stype==-1 )
{
y1 = c->f.ptr.p_double[c->n*iy+ix];
y2 = c->f.ptr.p_double[c->n*iy+(ix+1)];
y3 = c->f.ptr.p_double[c->n*(iy+1)+(ix+1)];
y4 = c->f.ptr.p_double[c->n*(iy+1)+ix];
*f = (1-t)*(1-u)*y1+t*(1-u)*y2+t*u*y3+(1-t)*u*y4;
*fx = (-(1-u)*y1+(1-u)*y2+u*y3-u*y4)*dt;
*fy = (-(1-t)*y1-t*y2+t*y3+(1-t)*y4)*du;
*fxy = (y1-y2+y3-y4)*du*dt;
return;
}
/*
* Bicubic interpolation
*/
if( c->stype==-3 )
{
/*
* Prepare info
*/
t0 = 1;
t1 = t;
t2 = ae_sqr(t, _state);
t3 = t*t2;
u0 = 1;
u1 = u;
u2 = ae_sqr(u, _state);
u3 = u*u2;
sfx = c->n*c->m;
sfy = 2*c->n*c->m;
sfxy = 3*c->n*c->m;
s1 = c->n*iy+ix;
s2 = c->n*iy+(ix+1);
s3 = c->n*(iy+1)+(ix+1);
s4 = c->n*(iy+1)+ix;
/*
* Calculate
*/
v = c->f.ptr.p_double[s1];
*f = *f+v*t0*u0;
v = c->f.ptr.p_double[sfy+s1]/du;
*f = *f+v*t0*u1;
*fy = *fy+v*t0*u0*du;
v = -3*c->f.ptr.p_double[s1]+3*c->f.ptr.p_double[s4]-2*c->f.ptr.p_double[sfy+s1]/du-c->f.ptr.p_double[sfy+s4]/du;
*f = *f+v*t0*u2;
*fy = *fy+2*v*t0*u1*du;
v = 2*c->f.ptr.p_double[s1]-2*c->f.ptr.p_double[s4]+c->f.ptr.p_double[sfy+s1]/du+c->f.ptr.p_double[sfy+s4]/du;
*f = *f+v*t0*u3;
*fy = *fy+3*v*t0*u2*du;
v = c->f.ptr.p_double[sfx+s1]/dt;
*f = *f+v*t1*u0;
*fx = *fx+v*t0*u0*dt;
v = c->f.ptr.p_double[sfxy+s1]/(dt*du);
*f = *f+v*t1*u1;
*fx = *fx+v*t0*u1*dt;
*fy = *fy+v*t1*u0*du;
*fxy = *fxy+v*t0*u0*dt*du;
v = -3*c->f.ptr.p_double[sfx+s1]/dt+3*c->f.ptr.p_double[sfx+s4]/dt-2*c->f.ptr.p_double[sfxy+s1]/(dt*du)-c->f.ptr.p_double[sfxy+s4]/(dt*du);
*f = *f+v*t1*u2;
*fx = *fx+v*t0*u2*dt;
*fy = *fy+2*v*t1*u1*du;
*fxy = *fxy+2*v*t0*u1*dt*du;
v = 2*c->f.ptr.p_double[sfx+s1]/dt-2*c->f.ptr.p_double[sfx+s4]/dt+c->f.ptr.p_double[sfxy+s1]/(dt*du)+c->f.ptr.p_double[sfxy+s4]/(dt*du);
*f = *f+v*t1*u3;
*fx = *fx+v*t0*u3*dt;
*fy = *fy+3*v*t1*u2*du;
*fxy = *fxy+3*v*t0*u2*dt*du;
v = -3*c->f.ptr.p_double[s1]+3*c->f.ptr.p_double[s2]-2*c->f.ptr.p_double[sfx+s1]/dt-c->f.ptr.p_double[sfx+s2]/dt;
*f = *f+v*t2*u0;
*fx = *fx+2*v*t1*u0*dt;
v = -3*c->f.ptr.p_double[sfy+s1]/du+3*c->f.ptr.p_double[sfy+s2]/du-2*c->f.ptr.p_double[sfxy+s1]/(dt*du)-c->f.ptr.p_double[sfxy+s2]/(dt*du);
*f = *f+v*t2*u1;
*fx = *fx+2*v*t1*u1*dt;
*fy = *fy+v*t2*u0*du;
*fxy = *fxy+2*v*t1*u0*dt*du;
v = 9*c->f.ptr.p_double[s1]-9*c->f.ptr.p_double[s2]+9*c->f.ptr.p_double[s3]-9*c->f.ptr.p_double[s4]+6*c->f.ptr.p_double[sfx+s1]/dt+3*c->f.ptr.p_double[sfx+s2]/dt-3*c->f.ptr.p_double[sfx+s3]/dt-6*c->f.ptr.p_double[sfx+s4]/dt+6*c->f.ptr.p_double[sfy+s1]/du-6*c->f.ptr.p_double[sfy+s2]/du-3*c->f.ptr.p_double[sfy+s3]/du+3*c->f.ptr.p_double[sfy+s4]/du+4*c->f.ptr.p_double[sfxy+s1]/(dt*du)+2*c->f.ptr.p_double[sfxy+s2]/(dt*du)+c->f.ptr.p_double[sfxy+s3]/(dt*du)+2*c->f.ptr.p_double[sfxy+s4]/(dt*du);
*f = *f+v*t2*u2;
*fx = *fx+2*v*t1*u2*dt;
*fy = *fy+2*v*t2*u1*du;
*fxy = *fxy+4*v*t1*u1*dt*du;
v = -6*c->f.ptr.p_double[s1]+6*c->f.ptr.p_double[s2]-6*c->f.ptr.p_double[s3]+6*c->f.ptr.p_double[s4]-4*c->f.ptr.p_double[sfx+s1]/dt-2*c->f.ptr.p_double[sfx+s2]/dt+2*c->f.ptr.p_double[sfx+s3]/dt+4*c->f.ptr.p_double[sfx+s4]/dt-3*c->f.ptr.p_double[sfy+s1]/du+3*c->f.ptr.p_double[sfy+s2]/du+3*c->f.ptr.p_double[sfy+s3]/du-3*c->f.ptr.p_double[sfy+s4]/du-2*c->f.ptr.p_double[sfxy+s1]/(dt*du)-c->f.ptr.p_double[sfxy+s2]/(dt*du)-c->f.ptr.p_double[sfxy+s3]/(dt*du)-2*c->f.ptr.p_double[sfxy+s4]/(dt*du);
*f = *f+v*t2*u3;
*fx = *fx+2*v*t1*u3*dt;
*fy = *fy+3*v*t2*u2*du;
*fxy = *fxy+6*v*t1*u2*dt*du;
v = 2*c->f.ptr.p_double[s1]-2*c->f.ptr.p_double[s2]+c->f.ptr.p_double[sfx+s1]/dt+c->f.ptr.p_double[sfx+s2]/dt;
*f = *f+v*t3*u0;
*fx = *fx+3*v*t2*u0*dt;
v = 2*c->f.ptr.p_double[sfy+s1]/du-2*c->f.ptr.p_double[sfy+s2]/du+c->f.ptr.p_double[sfxy+s1]/(dt*du)+c->f.ptr.p_double[sfxy+s2]/(dt*du);
*f = *f+v*t3*u1;
*fx = *fx+3*v*t2*u1*dt;
*fy = *fy+v*t3*u0*du;
*fxy = *fxy+3*v*t2*u0*dt*du;
v = -6*c->f.ptr.p_double[s1]+6*c->f.ptr.p_double[s2]-6*c->f.ptr.p_double[s3]+6*c->f.ptr.p_double[s4]-3*c->f.ptr.p_double[sfx+s1]/dt-3*c->f.ptr.p_double[sfx+s2]/dt+3*c->f.ptr.p_double[sfx+s3]/dt+3*c->f.ptr.p_double[sfx+s4]/dt-4*c->f.ptr.p_double[sfy+s1]/du+4*c->f.ptr.p_double[sfy+s2]/du+2*c->f.ptr.p_double[sfy+s3]/du-2*c->f.ptr.p_double[sfy+s4]/du-2*c->f.ptr.p_double[sfxy+s1]/(dt*du)-2*c->f.ptr.p_double[sfxy+s2]/(dt*du)-c->f.ptr.p_double[sfxy+s3]/(dt*du)-c->f.ptr.p_double[sfxy+s4]/(dt*du);
*f = *f+v*t3*u2;
*fx = *fx+3*v*t2*u2*dt;
*fy = *fy+2*v*t3*u1*du;
*fxy = *fxy+6*v*t2*u1*dt*du;
v = 4*c->f.ptr.p_double[s1]-4*c->f.ptr.p_double[s2]+4*c->f.ptr.p_double[s3]-4*c->f.ptr.p_double[s4]+2*c->f.ptr.p_double[sfx+s1]/dt+2*c->f.ptr.p_double[sfx+s2]/dt-2*c->f.ptr.p_double[sfx+s3]/dt-2*c->f.ptr.p_double[sfx+s4]/dt+2*c->f.ptr.p_double[sfy+s1]/du-2*c->f.ptr.p_double[sfy+s2]/du-2*c->f.ptr.p_double[sfy+s3]/du+2*c->f.ptr.p_double[sfy+s4]/du+c->f.ptr.p_double[sfxy+s1]/(dt*du)+c->f.ptr.p_double[sfxy+s2]/(dt*du)+c->f.ptr.p_double[sfxy+s3]/(dt*du)+c->f.ptr.p_double[sfxy+s4]/(dt*du);
*f = *f+v*t3*u3;
*fx = *fx+3*v*t2*u3*dt;
*fy = *fy+3*v*t3*u2*du;
*fxy = *fxy+9*v*t2*u2*dt*du;
return;
}
}
/*************************************************************************
This subroutine performs linear transformation of the spline argument.
Input parameters:
C - spline interpolant
AX, BX - transformation coefficients: x = A*t + B
AY, BY - transformation coefficients: y = A*u + B
Result:
C - transformed spline
-- ALGLIB PROJECT --
Copyright 30.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline2dlintransxy(spline2dinterpolant* c,
double ax,
double bx,
double ay,
double by,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector x;
ae_vector y;
ae_vector f;
ae_vector v;
ae_int_t i;
ae_int_t j;
ae_int_t k;
ae_frame_make(_state, &_frame_block);
ae_vector_init(&x, 0, DT_REAL, _state, ae_true);
ae_vector_init(&y, 0, DT_REAL, _state, ae_true);
ae_vector_init(&f, 0, DT_REAL, _state, ae_true);
ae_vector_init(&v, 0, DT_REAL, _state, ae_true);
ae_assert(c->stype==-3||c->stype==-1, "Spline2DLinTransXY: incorrect C (incorrect parameter C.SType)", _state);
ae_assert(ae_isfinite(ax, _state), "Spline2DLinTransXY: AX is infinite or NaN", _state);
ae_assert(ae_isfinite(bx, _state), "Spline2DLinTransXY: BX is infinite or NaN", _state);
ae_assert(ae_isfinite(ay, _state), "Spline2DLinTransXY: AY is infinite or NaN", _state);
ae_assert(ae_isfinite(by, _state), "Spline2DLinTransXY: BY is infinite or NaN", _state);
ae_vector_set_length(&x, c->n, _state);
ae_vector_set_length(&y, c->m, _state);
ae_vector_set_length(&f, c->m*c->n*c->d, _state);
for(j=0; j<=c->n-1; j++)
{
x.ptr.p_double[j] = c->x.ptr.p_double[j];
}
for(i=0; i<=c->m-1; i++)
{
y.ptr.p_double[i] = c->y.ptr.p_double[i];
}
for(i=0; i<=c->m-1; i++)
{
for(j=0; j<=c->n-1; j++)
{
for(k=0; k<=c->d-1; k++)
{
f.ptr.p_double[c->d*(i*c->n+j)+k] = c->f.ptr.p_double[c->d*(i*c->n+j)+k];
}
}
}
/*
* Handle different combinations of AX/AY
*/
if( ae_fp_eq(ax,0)&&ae_fp_neq(ay,0) )
{
for(i=0; i<=c->m-1; i++)
{
spline2dcalcvbuf(c, bx, y.ptr.p_double[i], &v, _state);
y.ptr.p_double[i] = (y.ptr.p_double[i]-by)/ay;
for(j=0; j<=c->n-1; j++)
{
for(k=0; k<=c->d-1; k++)
{
f.ptr.p_double[c->d*(i*c->n+j)+k] = v.ptr.p_double[k];
}
}
}
}
if( ae_fp_neq(ax,0)&&ae_fp_eq(ay,0) )
{
for(j=0; j<=c->n-1; j++)
{
spline2dcalcvbuf(c, x.ptr.p_double[j], by, &v, _state);
x.ptr.p_double[j] = (x.ptr.p_double[j]-bx)/ax;
for(i=0; i<=c->m-1; i++)
{
for(k=0; k<=c->d-1; k++)
{
f.ptr.p_double[c->d*(i*c->n+j)+k] = v.ptr.p_double[k];
}
}
}
}
if( ae_fp_neq(ax,0)&&ae_fp_neq(ay,0) )
{
for(j=0; j<=c->n-1; j++)
{
x.ptr.p_double[j] = (x.ptr.p_double[j]-bx)/ax;
}
for(i=0; i<=c->m-1; i++)
{
y.ptr.p_double[i] = (y.ptr.p_double[i]-by)/ay;
}
}
if( ae_fp_eq(ax,0)&&ae_fp_eq(ay,0) )
{
spline2dcalcvbuf(c, bx, by, &v, _state);
for(i=0; i<=c->m-1; i++)
{
for(j=0; j<=c->n-1; j++)
{
for(k=0; k<=c->d-1; k++)
{
f.ptr.p_double[c->d*(i*c->n+j)+k] = v.ptr.p_double[k];
}
}
}
}
/*
* Rebuild spline
*/
if( c->stype==-3 )
{
spline2dbuildbicubicv(&x, c->n, &y, c->m, &f, c->d, c, _state);
}
if( c->stype==-1 )
{
spline2dbuildbilinearv(&x, c->n, &y, c->m, &f, c->d, c, _state);
}
ae_frame_leave(_state);
}
/*************************************************************************
This subroutine performs linear transformation of the spline.
Input parameters:
C - spline interpolant.
A, B- transformation coefficients: S2(x,y) = A*S(x,y) + B
Output parameters:
C - transformed spline
-- ALGLIB PROJECT --
Copyright 30.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline2dlintransf(spline2dinterpolant* c,
double a,
double b,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector x;
ae_vector y;
ae_vector f;
ae_int_t i;
ae_int_t j;
ae_frame_make(_state, &_frame_block);
ae_vector_init(&x, 0, DT_REAL, _state, ae_true);
ae_vector_init(&y, 0, DT_REAL, _state, ae_true);
ae_vector_init(&f, 0, DT_REAL, _state, ae_true);
ae_assert(c->stype==-3||c->stype==-1, "Spline2DLinTransF: incorrect C (incorrect parameter C.SType)", _state);
ae_vector_set_length(&x, c->n, _state);
ae_vector_set_length(&y, c->m, _state);
ae_vector_set_length(&f, c->m*c->n*c->d, _state);
for(j=0; j<=c->n-1; j++)
{
x.ptr.p_double[j] = c->x.ptr.p_double[j];
}
for(i=0; i<=c->m-1; i++)
{
y.ptr.p_double[i] = c->y.ptr.p_double[i];
}
for(i=0; i<=c->m*c->n*c->d-1; i++)
{
f.ptr.p_double[i] = a*c->f.ptr.p_double[i]+b;
}
if( c->stype==-3 )
{
spline2dbuildbicubicv(&x, c->n, &y, c->m, &f, c->d, c, _state);
}
if( c->stype==-1 )
{
spline2dbuildbilinearv(&x, c->n, &y, c->m, &f, c->d, c, _state);
}
ae_frame_leave(_state);
}
/*************************************************************************
This subroutine makes the copy of the spline model.
Input parameters:
C - spline interpolant
Output parameters:
CC - spline copy
-- ALGLIB PROJECT --
Copyright 29.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline2dcopy(spline2dinterpolant* c,
spline2dinterpolant* cc,
ae_state *_state)
{
ae_int_t tblsize;
_spline2dinterpolant_clear(cc);
ae_assert(c->k==1||c->k==3, "Spline2DCopy: incorrect C (incorrect parameter C.K)", _state);
cc->k = c->k;
cc->n = c->n;
cc->m = c->m;
cc->d = c->d;
cc->stype = c->stype;
tblsize = -1;
if( c->stype==-3 )
{
tblsize = 4*c->n*c->m*c->d;
}
if( c->stype==-1 )
{
tblsize = c->n*c->m*c->d;
}
ae_assert(tblsize>0, "Spline2DCopy: internal error", _state);
ae_vector_set_length(&cc->x, cc->n, _state);
ae_vector_set_length(&cc->y, cc->m, _state);
ae_vector_set_length(&cc->f, tblsize, _state);
ae_v_move(&cc->x.ptr.p_double[0], 1, &c->x.ptr.p_double[0], 1, ae_v_len(0,cc->n-1));
ae_v_move(&cc->y.ptr.p_double[0], 1, &c->y.ptr.p_double[0], 1, ae_v_len(0,cc->m-1));
ae_v_move(&cc->f.ptr.p_double[0], 1, &c->f.ptr.p_double[0], 1, ae_v_len(0,tblsize-1));
}
/*************************************************************************
Bicubic spline resampling
Input parameters:
A - function values at the old grid,
array[0..OldHeight-1, 0..OldWidth-1]
OldHeight - old grid height, OldHeight>1
OldWidth - old grid width, OldWidth>1
NewHeight - new grid height, NewHeight>1
NewWidth - new grid width, NewWidth>1
Output parameters:
B - function values at the new grid,
array[0..NewHeight-1, 0..NewWidth-1]
-- ALGLIB routine --
15 May, 2007
Copyright by Bochkanov Sergey
*************************************************************************/
void spline2dresamplebicubic(/* Real */ ae_matrix* a,
ae_int_t oldheight,
ae_int_t oldwidth,
/* Real */ ae_matrix* b,
ae_int_t newheight,
ae_int_t newwidth,
ae_state *_state)
{
ae_frame _frame_block;
ae_matrix buf;
ae_vector x;
ae_vector y;
spline1dinterpolant c;
ae_int_t mw;
ae_int_t mh;
ae_int_t i;
ae_int_t j;
ae_frame_make(_state, &_frame_block);
ae_matrix_clear(b);
ae_matrix_init(&buf, 0, 0, DT_REAL, _state, ae_true);
ae_vector_init(&x, 0, DT_REAL, _state, ae_true);
ae_vector_init(&y, 0, DT_REAL, _state, ae_true);
_spline1dinterpolant_init(&c, _state, ae_true);
ae_assert(oldwidth>1&&oldheight>1, "Spline2DResampleBicubic: width/height less than 1", _state);
ae_assert(newwidth>1&&newheight>1, "Spline2DResampleBicubic: width/height less than 1", _state);
/*
* Prepare
*/
mw = ae_maxint(oldwidth, newwidth, _state);
mh = ae_maxint(oldheight, newheight, _state);
ae_matrix_set_length(b, newheight, newwidth, _state);
ae_matrix_set_length(&buf, oldheight, newwidth, _state);
ae_vector_set_length(&x, ae_maxint(mw, mh, _state), _state);
ae_vector_set_length(&y, ae_maxint(mw, mh, _state), _state);
/*
* Horizontal interpolation
*/
for(i=0; i<=oldheight-1; i++)
{
/*
* Fill X, Y
*/
for(j=0; j<=oldwidth-1; j++)
{
x.ptr.p_double[j] = (double)j/(double)(oldwidth-1);
y.ptr.p_double[j] = a->ptr.pp_double[i][j];
}
/*
* Interpolate and place result into temporary matrix
*/
spline1dbuildcubic(&x, &y, oldwidth, 0, 0.0, 0, 0.0, &c, _state);
for(j=0; j<=newwidth-1; j++)
{
buf.ptr.pp_double[i][j] = spline1dcalc(&c, (double)j/(double)(newwidth-1), _state);
}
}
/*
* Vertical interpolation
*/
for(j=0; j<=newwidth-1; j++)
{
/*
* Fill X, Y
*/
for(i=0; i<=oldheight-1; i++)
{
x.ptr.p_double[i] = (double)i/(double)(oldheight-1);
y.ptr.p_double[i] = buf.ptr.pp_double[i][j];
}
/*
* Interpolate and place result into B
*/
spline1dbuildcubic(&x, &y, oldheight, 0, 0.0, 0, 0.0, &c, _state);
for(i=0; i<=newheight-1; i++)
{
b->ptr.pp_double[i][j] = spline1dcalc(&c, (double)i/(double)(newheight-1), _state);
}
}
ae_frame_leave(_state);
}
/*************************************************************************
Bilinear spline resampling
Input parameters:
A - function values at the old grid,
array[0..OldHeight-1, 0..OldWidth-1]
OldHeight - old grid height, OldHeight>1
OldWidth - old grid width, OldWidth>1
NewHeight - new grid height, NewHeight>1
NewWidth - new grid width, NewWidth>1
Output parameters:
B - function values at the new grid,
array[0..NewHeight-1, 0..NewWidth-1]
-- ALGLIB routine --
09.07.2007
Copyright by Bochkanov Sergey
*************************************************************************/
void spline2dresamplebilinear(/* Real */ ae_matrix* a,
ae_int_t oldheight,
ae_int_t oldwidth,
/* Real */ ae_matrix* b,
ae_int_t newheight,
ae_int_t newwidth,
ae_state *_state)
{
ae_int_t l;
ae_int_t c;
double t;
double u;
ae_int_t i;
ae_int_t j;
ae_matrix_clear(b);
ae_assert(oldwidth>1&&oldheight>1, "Spline2DResampleBilinear: width/height less than 1", _state);
ae_assert(newwidth>1&&newheight>1, "Spline2DResampleBilinear: width/height less than 1", _state);
ae_matrix_set_length(b, newheight, newwidth, _state);
for(i=0; i<=newheight-1; i++)
{
for(j=0; j<=newwidth-1; j++)
{
l = i*(oldheight-1)/(newheight-1);
if( l==oldheight-1 )
{
l = oldheight-2;
}
u = (double)i/(double)(newheight-1)*(oldheight-1)-l;
c = j*(oldwidth-1)/(newwidth-1);
if( c==oldwidth-1 )
{
c = oldwidth-2;
}
t = (double)(j*(oldwidth-1))/(double)(newwidth-1)-c;
b->ptr.pp_double[i][j] = (1-t)*(1-u)*a->ptr.pp_double[l][c]+t*(1-u)*a->ptr.pp_double[l][c+1]+t*u*a->ptr.pp_double[l+1][c+1]+(1-t)*u*a->ptr.pp_double[l+1][c];
}
}
}
/*************************************************************************
This subroutine builds bilinear vector-valued spline.
Input parameters:
X - spline abscissas, array[0..N-1]
Y - spline ordinates, array[0..M-1]
F - function values, array[0..M*N*D-1]:
* first D elements store D values at (X[0],Y[0])
* next D elements store D values at (X[1],Y[0])
* general form - D function values at (X[i],Y[j]) are stored
at F[D*(J*N+I)...D*(J*N+I)+D-1].
M,N - grid size, M>=2, N>=2
D - vector dimension, D>=1
Output parameters:
C - spline interpolant
-- ALGLIB PROJECT --
Copyright 16.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildbilinearv(/* Real */ ae_vector* x,
ae_int_t n,
/* Real */ ae_vector* y,
ae_int_t m,
/* Real */ ae_vector* f,
ae_int_t d,
spline2dinterpolant* c,
ae_state *_state)
{
double t;
ae_int_t i;
ae_int_t j;
ae_int_t k;
ae_int_t i0;
_spline2dinterpolant_clear(c);
ae_assert(n>=2, "Spline2DBuildBilinearV: N is less then 2", _state);
ae_assert(m>=2, "Spline2DBuildBilinearV: M is less then 2", _state);
ae_assert(d>=1, "Spline2DBuildBilinearV: invalid argument D (D<1)", _state);
ae_assert(x->cnt>=n&&y->cnt>=m, "Spline2DBuildBilinearV: length of X or Y is too short (Length(X/Y)<N/M)", _state);
ae_assert(isfinitevector(x, n, _state)&&isfinitevector(y, m, _state), "Spline2DBuildBilinearV: X or Y contains NaN or Infinite value", _state);
k = n*m*d;
ae_assert(f->cnt>=k, "Spline2DBuildBilinearV: length of F is too short (Length(F)<N*M*D)", _state);
ae_assert(isfinitevector(f, k, _state), "Spline2DBuildBilinearV: F contains NaN or Infinite value", _state);
/*
* Fill interpolant
*/
c->k = 1;
c->n = n;
c->m = m;
c->d = d;
c->stype = -1;
ae_vector_set_length(&c->x, c->n, _state);
ae_vector_set_length(&c->y, c->m, _state);
ae_vector_set_length(&c->f, k, _state);
for(i=0; i<=c->n-1; i++)
{
c->x.ptr.p_double[i] = x->ptr.p_double[i];
}
for(i=0; i<=c->m-1; i++)
{
c->y.ptr.p_double[i] = y->ptr.p_double[i];
}
for(i=0; i<=k-1; i++)
{
c->f.ptr.p_double[i] = f->ptr.p_double[i];
}
/*
* Sort points
*/
for(j=0; j<=c->n-1; j++)
{
k = j;
for(i=j+1; i<=c->n-1; i++)
{
if( ae_fp_less(c->x.ptr.p_double[i],c->x.ptr.p_double[k]) )
{
k = i;
}
}
if( k!=j )
{
for(i=0; i<=c->m-1; i++)
{
for(i0=0; i0<=c->d-1; i0++)
{
t = c->f.ptr.p_double[c->d*(i*c->n+j)+i0];
c->f.ptr.p_double[c->d*(i*c->n+j)+i0] = c->f.ptr.p_double[c->d*(i*c->n+k)+i0];
c->f.ptr.p_double[c->d*(i*c->n+k)+i0] = t;
}
}
t = c->x.ptr.p_double[j];
c->x.ptr.p_double[j] = c->x.ptr.p_double[k];
c->x.ptr.p_double[k] = t;
}
}
for(i=0; i<=c->m-1; i++)
{
k = i;
for(j=i+1; j<=c->m-1; j++)
{
if( ae_fp_less(c->y.ptr.p_double[j],c->y.ptr.p_double[k]) )
{
k = j;
}
}
if( k!=i )
{
for(j=0; j<=c->n-1; j++)
{
for(i0=0; i0<=c->d-1; i0++)
{
t = c->f.ptr.p_double[c->d*(i*c->n+j)+i0];
c->f.ptr.p_double[c->d*(i*c->n+j)+i0] = c->f.ptr.p_double[c->d*(k*c->n+j)+i0];
c->f.ptr.p_double[c->d*(k*c->n+j)+i0] = t;
}
}
t = c->y.ptr.p_double[i];
c->y.ptr.p_double[i] = c->y.ptr.p_double[k];
c->y.ptr.p_double[k] = t;
}
}
}
/*************************************************************************
This subroutine builds bicubic vector-valued spline.
Input parameters:
X - spline abscissas, array[0..N-1]
Y - spline ordinates, array[0..M-1]
F - function values, array[0..M*N*D-1]:
* first D elements store D values at (X[0],Y[0])
* next D elements store D values at (X[1],Y[0])
* general form - D function values at (X[i],Y[j]) are stored
at F[D*(J*N+I)...D*(J*N+I)+D-1].
M,N - grid size, M>=2, N>=2
D - vector dimension, D>=1
Output parameters:
C - spline interpolant
-- ALGLIB PROJECT --
Copyright 16.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildbicubicv(/* Real */ ae_vector* x,
ae_int_t n,
/* Real */ ae_vector* y,
ae_int_t m,
/* Real */ ae_vector* f,
ae_int_t d,
spline2dinterpolant* c,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector _f;
ae_matrix tf;
ae_matrix dx;
ae_matrix dy;
ae_matrix dxy;
double t;
ae_int_t i;
ae_int_t j;
ae_int_t k;
ae_int_t di;
ae_frame_make(_state, &_frame_block);
ae_vector_init_copy(&_f, f, _state, ae_true);
f = &_f;
_spline2dinterpolant_clear(c);
ae_matrix_init(&tf, 0, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&dx, 0, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&dy, 0, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&dxy, 0, 0, DT_REAL, _state, ae_true);
ae_assert(n>=2, "Spline2DBuildBicubicV: N is less than 2", _state);
ae_assert(m>=2, "Spline2DBuildBicubicV: M is less than 2", _state);
ae_assert(d>=1, "Spline2DBuildBicubicV: invalid argument D (D<1)", _state);
ae_assert(x->cnt>=n&&y->cnt>=m, "Spline2DBuildBicubicV: length of X or Y is too short (Length(X/Y)<N/M)", _state);
ae_assert(isfinitevector(x, n, _state)&&isfinitevector(y, m, _state), "Spline2DBuildBicubicV: X or Y contains NaN or Infinite value", _state);
k = n*m*d;
ae_assert(f->cnt>=k, "Spline2DBuildBicubicV: length of F is too short (Length(F)<N*M*D)", _state);
ae_assert(isfinitevector(f, k, _state), "Spline2DBuildBicubicV: F contains NaN or Infinite value", _state);
/*
* Fill interpolant:
* F[0]...F[N*M*D-1]:
* f(i,j) table. f(0,0), f(0, 1), f(0,2) and so on...
* F[N*M*D]...F[2*N*M*D-1]:
* df(i,j)/dx table.
* F[2*N*M*D]...F[3*N*M*D-1]:
* df(i,j)/dy table.
* F[3*N*M*D]...F[4*N*M*D-1]:
* d2f(i,j)/dxdy table.
*/
c->k = 3;
c->d = d;
c->n = n;
c->m = m;
c->stype = -3;
k = 4*k;
ae_vector_set_length(&c->x, c->n, _state);
ae_vector_set_length(&c->y, c->m, _state);
ae_vector_set_length(&c->f, k, _state);
ae_matrix_set_length(&tf, c->m, c->n, _state);
for(i=0; i<=c->n-1; i++)
{
c->x.ptr.p_double[i] = x->ptr.p_double[i];
}
for(i=0; i<=c->m-1; i++)
{
c->y.ptr.p_double[i] = y->ptr.p_double[i];
}
/*
* Sort points
*/
for(j=0; j<=c->n-1; j++)
{
k = j;
for(i=j+1; i<=c->n-1; i++)
{
if( ae_fp_less(c->x.ptr.p_double[i],c->x.ptr.p_double[k]) )
{
k = i;
}
}
if( k!=j )
{
for(i=0; i<=c->m-1; i++)
{
for(di=0; di<=c->d-1; di++)
{
t = f->ptr.p_double[c->d*(i*c->n+j)+di];
f->ptr.p_double[c->d*(i*c->n+j)+di] = f->ptr.p_double[c->d*(i*c->n+k)+di];
f->ptr.p_double[c->d*(i*c->n+k)+di] = t;
}
}
t = c->x.ptr.p_double[j];
c->x.ptr.p_double[j] = c->x.ptr.p_double[k];
c->x.ptr.p_double[k] = t;
}
}
for(i=0; i<=c->m-1; i++)
{
k = i;
for(j=i+1; j<=c->m-1; j++)
{
if( ae_fp_less(c->y.ptr.p_double[j],c->y.ptr.p_double[k]) )
{
k = j;
}
}
if( k!=i )
{
for(j=0; j<=c->n-1; j++)
{
for(di=0; di<=c->d-1; di++)
{
t = f->ptr.p_double[c->d*(i*c->n+j)+di];
f->ptr.p_double[c->d*(i*c->n+j)+di] = f->ptr.p_double[c->d*(k*c->n+j)+di];
f->ptr.p_double[c->d*(k*c->n+j)+di] = t;
}
}
t = c->y.ptr.p_double[i];
c->y.ptr.p_double[i] = c->y.ptr.p_double[k];
c->y.ptr.p_double[k] = t;
}
}
for(di=0; di<=c->d-1; di++)
{
for(i=0; i<=c->m-1; i++)
{
for(j=0; j<=c->n-1; j++)
{
tf.ptr.pp_double[i][j] = f->ptr.p_double[c->d*(i*c->n+j)+di];
}
}
spline2d_bicubiccalcderivatives(&tf, &c->x, &c->y, c->m, c->n, &dx, &dy, &dxy, _state);
for(i=0; i<=c->m-1; i++)
{
for(j=0; j<=c->n-1; j++)
{
k = c->d*(i*c->n+j)+di;
c->f.ptr.p_double[k] = tf.ptr.pp_double[i][j];
c->f.ptr.p_double[c->n*c->m*c->d+k] = dx.ptr.pp_double[i][j];
c->f.ptr.p_double[2*c->n*c->m*c->d+k] = dy.ptr.pp_double[i][j];
c->f.ptr.p_double[3*c->n*c->m*c->d+k] = dxy.ptr.pp_double[i][j];
}
}
}
ae_frame_leave(_state);
}
/*************************************************************************
This subroutine calculates bilinear or bicubic vector-valued spline at the
given point (X,Y).
INPUT PARAMETERS:
C - spline interpolant.
X, Y- point
F - output buffer, possibly preallocated array. In case array size
is large enough to store result, it is not reallocated. Array
which is too short will be reallocated
OUTPUT PARAMETERS:
F - array[D] (or larger) which stores function values
-- ALGLIB PROJECT --
Copyright 16.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline2dcalcvbuf(spline2dinterpolant* c,
double x,
double y,
/* Real */ ae_vector* f,
ae_state *_state)
{
double t;
double dt;
double u;
double du;
ae_int_t ix;
ae_int_t iy;
ae_int_t l;
ae_int_t r;
ae_int_t h;
ae_int_t s1;
ae_int_t s2;
ae_int_t s3;
ae_int_t s4;
ae_int_t sfx;
ae_int_t sfy;
ae_int_t sfxy;
double y1;
double y2;
double y3;
double y4;
double v;
double t0;
double t1;
double t2;
double t3;
double u0;
double u1;
double u2;
double u3;
ae_int_t i;
ae_assert(c->stype==-1||c->stype==-3, "Spline2DCalcVBuf: incorrect C (incorrect parameter C.SType)", _state);
ae_assert(ae_isfinite(x, _state)&&ae_isfinite(y, _state), "Spline2DCalcVBuf: either X=NaN/Infinite or Y=NaN/Infinite", _state);
rvectorsetlengthatleast(f, c->d, _state);
/*
* Binary search in the [ x[0], ..., x[n-2] ] (x[n-1] is not included)
*/
l = 0;
r = c->n-1;
while(l!=r-1)
{
h = (l+r)/2;
if( ae_fp_greater_eq(c->x.ptr.p_double[h],x) )
{
r = h;
}
else
{
l = h;
}
}
t = (x-c->x.ptr.p_double[l])/(c->x.ptr.p_double[l+1]-c->x.ptr.p_double[l]);
dt = 1.0/(c->x.ptr.p_double[l+1]-c->x.ptr.p_double[l]);
ix = l;
/*
* Binary search in the [ y[0], ..., y[m-2] ] (y[m-1] is not included)
*/
l = 0;
r = c->m-1;
while(l!=r-1)
{
h = (l+r)/2;
if( ae_fp_greater_eq(c->y.ptr.p_double[h],y) )
{
r = h;
}
else
{
l = h;
}
}
u = (y-c->y.ptr.p_double[l])/(c->y.ptr.p_double[l+1]-c->y.ptr.p_double[l]);
du = 1.0/(c->y.ptr.p_double[l+1]-c->y.ptr.p_double[l]);
iy = l;
/*
* Bilinear interpolation
*/
if( c->stype==-1 )
{
for(i=0; i<=c->d-1; i++)
{
y1 = c->f.ptr.p_double[c->d*(c->n*iy+ix)+i];
y2 = c->f.ptr.p_double[c->d*(c->n*iy+(ix+1))+i];
y3 = c->f.ptr.p_double[c->d*(c->n*(iy+1)+(ix+1))+i];
y4 = c->f.ptr.p_double[c->d*(c->n*(iy+1)+ix)+i];
f->ptr.p_double[i] = (1-t)*(1-u)*y1+t*(1-u)*y2+t*u*y3+(1-t)*u*y4;
}
return;
}
/*
* Bicubic interpolation
*/
if( c->stype==-3 )
{
/*
* Prepare info
*/
t0 = 1;
t1 = t;
t2 = ae_sqr(t, _state);
t3 = t*t2;
u0 = 1;
u1 = u;
u2 = ae_sqr(u, _state);
u3 = u*u2;
sfx = c->n*c->m*c->d;
sfy = 2*c->n*c->m*c->d;
sfxy = 3*c->n*c->m*c->d;
for(i=0; i<=c->d-1; i++)
{
/*
* Prepare F, dF/dX, dF/dY, d2F/dXdY
*/
f->ptr.p_double[i] = 0;
s1 = c->d*(c->n*iy+ix)+i;
s2 = c->d*(c->n*iy+(ix+1))+i;
s3 = c->d*(c->n*(iy+1)+(ix+1))+i;
s4 = c->d*(c->n*(iy+1)+ix)+i;
/*
* Calculate
*/
v = c->f.ptr.p_double[s1];
f->ptr.p_double[i] = f->ptr.p_double[i]+v*t0*u0;
v = c->f.ptr.p_double[sfy+s1]/du;
f->ptr.p_double[i] = f->ptr.p_double[i]+v*t0*u1;
v = -3*c->f.ptr.p_double[s1]+3*c->f.ptr.p_double[s4]-2*c->f.ptr.p_double[sfy+s1]/du-c->f.ptr.p_double[sfy+s4]/du;
f->ptr.p_double[i] = f->ptr.p_double[i]+v*t0*u2;
v = 2*c->f.ptr.p_double[s1]-2*c->f.ptr.p_double[s4]+c->f.ptr.p_double[sfy+s1]/du+c->f.ptr.p_double[sfy+s4]/du;
f->ptr.p_double[i] = f->ptr.p_double[i]+v*t0*u3;
v = c->f.ptr.p_double[sfx+s1]/dt;
f->ptr.p_double[i] = f->ptr.p_double[i]+v*t1*u0;
v = c->f.ptr.p_double[sfxy+s1]/(dt*du);
f->ptr.p_double[i] = f->ptr.p_double[i]+v*t1*u1;
v = -3*c->f.ptr.p_double[sfx+s1]/dt+3*c->f.ptr.p_double[sfx+s4]/dt-2*c->f.ptr.p_double[sfxy+s1]/(dt*du)-c->f.ptr.p_double[sfxy+s4]/(dt*du);
f->ptr.p_double[i] = f->ptr.p_double[i]+v*t1*u2;
v = 2*c->f.ptr.p_double[sfx+s1]/dt-2*c->f.ptr.p_double[sfx+s4]/dt+c->f.ptr.p_double[sfxy+s1]/(dt*du)+c->f.ptr.p_double[sfxy+s4]/(dt*du);
f->ptr.p_double[i] = f->ptr.p_double[i]+v*t1*u3;
v = -3*c->f.ptr.p_double[s1]+3*c->f.ptr.p_double[s2]-2*c->f.ptr.p_double[sfx+s1]/dt-c->f.ptr.p_double[sfx+s2]/dt;
f->ptr.p_double[i] = f->ptr.p_double[i]+v*t2*u0;
v = -3*c->f.ptr.p_double[sfy+s1]/du+3*c->f.ptr.p_double[sfy+s2]/du-2*c->f.ptr.p_double[sfxy+s1]/(dt*du)-c->f.ptr.p_double[sfxy+s2]/(dt*du);
f->ptr.p_double[i] = f->ptr.p_double[i]+v*t2*u1;
v = 9*c->f.ptr.p_double[s1]-9*c->f.ptr.p_double[s2]+9*c->f.ptr.p_double[s3]-9*c->f.ptr.p_double[s4]+6*c->f.ptr.p_double[sfx+s1]/dt+3*c->f.ptr.p_double[sfx+s2]/dt-3*c->f.ptr.p_double[sfx+s3]/dt-6*c->f.ptr.p_double[sfx+s4]/dt+6*c->f.ptr.p_double[sfy+s1]/du-6*c->f.ptr.p_double[sfy+s2]/du-3*c->f.ptr.p_double[sfy+s3]/du+3*c->f.ptr.p_double[sfy+s4]/du+4*c->f.ptr.p_double[sfxy+s1]/(dt*du)+2*c->f.ptr.p_double[sfxy+s2]/(dt*du)+c->f.ptr.p_double[sfxy+s3]/(dt*du)+2*c->f.ptr.p_double[sfxy+s4]/(dt*du);
f->ptr.p_double[i] = f->ptr.p_double[i]+v*t2*u2;
v = -6*c->f.ptr.p_double[s1]+6*c->f.ptr.p_double[s2]-6*c->f.ptr.p_double[s3]+6*c->f.ptr.p_double[s4]-4*c->f.ptr.p_double[sfx+s1]/dt-2*c->f.ptr.p_double[sfx+s2]/dt+2*c->f.ptr.p_double[sfx+s3]/dt+4*c->f.ptr.p_double[sfx+s4]/dt-3*c->f.ptr.p_double[sfy+s1]/du+3*c->f.ptr.p_double[sfy+s2]/du+3*c->f.ptr.p_double[sfy+s3]/du-3*c->f.ptr.p_double[sfy+s4]/du-2*c->f.ptr.p_double[sfxy+s1]/(dt*du)-c->f.ptr.p_double[sfxy+s2]/(dt*du)-c->f.ptr.p_double[sfxy+s3]/(dt*du)-2*c->f.ptr.p_double[sfxy+s4]/(dt*du);
f->ptr.p_double[i] = f->ptr.p_double[i]+v*t2*u3;
v = 2*c->f.ptr.p_double[s1]-2*c->f.ptr.p_double[s2]+c->f.ptr.p_double[sfx+s1]/dt+c->f.ptr.p_double[sfx+s2]/dt;
f->ptr.p_double[i] = f->ptr.p_double[i]+v*t3*u0;
v = 2*c->f.ptr.p_double[sfy+s1]/du-2*c->f.ptr.p_double[sfy+s2]/du+c->f.ptr.p_double[sfxy+s1]/(dt*du)+c->f.ptr.p_double[sfxy+s2]/(dt*du);
f->ptr.p_double[i] = f->ptr.p_double[i]+v*t3*u1;
v = -6*c->f.ptr.p_double[s1]+6*c->f.ptr.p_double[s2]-6*c->f.ptr.p_double[s3]+6*c->f.ptr.p_double[s4]-3*c->f.ptr.p_double[sfx+s1]/dt-3*c->f.ptr.p_double[sfx+s2]/dt+3*c->f.ptr.p_double[sfx+s3]/dt+3*c->f.ptr.p_double[sfx+s4]/dt-4*c->f.ptr.p_double[sfy+s1]/du+4*c->f.ptr.p_double[sfy+s2]/du+2*c->f.ptr.p_double[sfy+s3]/du-2*c->f.ptr.p_double[sfy+s4]/du-2*c->f.ptr.p_double[sfxy+s1]/(dt*du)-2*c->f.ptr.p_double[sfxy+s2]/(dt*du)-c->f.ptr.p_double[sfxy+s3]/(dt*du)-c->f.ptr.p_double[sfxy+s4]/(dt*du);
f->ptr.p_double[i] = f->ptr.p_double[i]+v*t3*u2;
v = 4*c->f.ptr.p_double[s1]-4*c->f.ptr.p_double[s2]+4*c->f.ptr.p_double[s3]-4*c->f.ptr.p_double[s4]+2*c->f.ptr.p_double[sfx+s1]/dt+2*c->f.ptr.p_double[sfx+s2]/dt-2*c->f.ptr.p_double[sfx+s3]/dt-2*c->f.ptr.p_double[sfx+s4]/dt+2*c->f.ptr.p_double[sfy+s1]/du-2*c->f.ptr.p_double[sfy+s2]/du-2*c->f.ptr.p_double[sfy+s3]/du+2*c->f.ptr.p_double[sfy+s4]/du+c->f.ptr.p_double[sfxy+s1]/(dt*du)+c->f.ptr.p_double[sfxy+s2]/(dt*du)+c->f.ptr.p_double[sfxy+s3]/(dt*du)+c->f.ptr.p_double[sfxy+s4]/(dt*du);
f->ptr.p_double[i] = f->ptr.p_double[i]+v*t3*u3;
}
return;
}
}
/*************************************************************************
This subroutine calculates bilinear or bicubic vector-valued spline at the
given point (X,Y).
INPUT PARAMETERS:
C - spline interpolant.
X, Y- point
OUTPUT PARAMETERS:
F - array[D] which stores function values. F is out-parameter and
it is reallocated after call to this function. In case you
want to reuse previously allocated F, you may use
Spline2DCalcVBuf(), which reallocates F only when it is too
small.
-- ALGLIB PROJECT --
Copyright 16.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline2dcalcv(spline2dinterpolant* c,
double x,
double y,
/* Real */ ae_vector* f,
ae_state *_state)
{
ae_vector_clear(f);
ae_assert(c->stype==-1||c->stype==-3, "Spline2DCalcV: incorrect C (incorrect parameter C.SType)", _state);
ae_assert(ae_isfinite(x, _state)&&ae_isfinite(y, _state), "Spline2DCalcV: either X=NaN/Infinite or Y=NaN/Infinite", _state);
ae_vector_set_length(f, c->d, _state);
spline2dcalcvbuf(c, x, y, f, _state);
}
/*************************************************************************
This subroutine unpacks two-dimensional spline into the coefficients table
Input parameters:
C - spline interpolant.
Result:
M, N- grid size (x-axis and y-axis)
D - number of components
Tbl - coefficients table, unpacked format,
D - components: [0..(N-1)*(M-1)*D-1, 0..19].
For T=0..D-1 (component index), I = 0...N-2 (x index),
J=0..M-2 (y index):
K := T + I*D + J*D*(N-1)
K-th row stores decomposition for T-th component of the
vector-valued function
Tbl[K,0] = X[i]
Tbl[K,1] = X[i+1]
Tbl[K,2] = Y[j]
Tbl[K,3] = Y[j+1]
Tbl[K,4] = C00
Tbl[K,5] = C01
Tbl[K,6] = C02
Tbl[K,7] = C03
Tbl[K,8] = C10
Tbl[K,9] = C11
...
Tbl[K,19] = C33
On each grid square spline is equals to:
S(x) = SUM(c[i,j]*(t^i)*(u^j), i=0..3, j=0..3)
t = x-x[j]
u = y-y[i]
-- ALGLIB PROJECT --
Copyright 16.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline2dunpackv(spline2dinterpolant* c,
ae_int_t* m,
ae_int_t* n,
ae_int_t* d,
/* Real */ ae_matrix* tbl,
ae_state *_state)
{
ae_int_t k;
ae_int_t p;
ae_int_t ci;
ae_int_t cj;
ae_int_t s1;
ae_int_t s2;
ae_int_t s3;
ae_int_t s4;
ae_int_t sfx;
ae_int_t sfy;
ae_int_t sfxy;
double y1;
double y2;
double y3;
double y4;
double dt;
double du;
ae_int_t i;
ae_int_t j;
ae_int_t k0;
*m = 0;
*n = 0;
*d = 0;
ae_matrix_clear(tbl);
ae_assert(c->stype==-3||c->stype==-1, "Spline2DUnpackV: incorrect C (incorrect parameter C.SType)", _state);
*n = c->n;
*m = c->m;
*d = c->d;
ae_matrix_set_length(tbl, (*n-1)*(*m-1)*(*d), 20, _state);
sfx = *n*(*m)*(*d);
sfy = 2*(*n)*(*m)*(*d);
sfxy = 3*(*n)*(*m)*(*d);
for(i=0; i<=*m-2; i++)
{
for(j=0; j<=*n-2; j++)
{
for(k=0; k<=*d-1; k++)
{
p = *d*(i*(*n-1)+j)+k;
tbl->ptr.pp_double[p][0] = c->x.ptr.p_double[j];
tbl->ptr.pp_double[p][1] = c->x.ptr.p_double[j+1];
tbl->ptr.pp_double[p][2] = c->y.ptr.p_double[i];
tbl->ptr.pp_double[p][3] = c->y.ptr.p_double[i+1];
dt = 1/(tbl->ptr.pp_double[p][1]-tbl->ptr.pp_double[p][0]);
du = 1/(tbl->ptr.pp_double[p][3]-tbl->ptr.pp_double[p][2]);
/*
* Bilinear interpolation
*/
if( c->stype==-1 )
{
for(k0=4; k0<=19; k0++)
{
tbl->ptr.pp_double[p][k0] = 0;
}
y1 = c->f.ptr.p_double[*d*(*n*i+j)+k];
y2 = c->f.ptr.p_double[*d*(*n*i+(j+1))+k];
y3 = c->f.ptr.p_double[*d*(*n*(i+1)+(j+1))+k];
y4 = c->f.ptr.p_double[*d*(*n*(i+1)+j)+k];
tbl->ptr.pp_double[p][4] = y1;
tbl->ptr.pp_double[p][4+1*4+0] = y2-y1;
tbl->ptr.pp_double[p][4+0*4+1] = y4-y1;
tbl->ptr.pp_double[p][4+1*4+1] = y3-y2-y4+y1;
}
/*
* Bicubic interpolation
*/
if( c->stype==-3 )
{
s1 = *d*(*n*i+j)+k;
s2 = *d*(*n*i+(j+1))+k;
s3 = *d*(*n*(i+1)+(j+1))+k;
s4 = *d*(*n*(i+1)+j)+k;
tbl->ptr.pp_double[p][4+0*4+0] = c->f.ptr.p_double[s1];
tbl->ptr.pp_double[p][4+0*4+1] = c->f.ptr.p_double[sfy+s1]/du;
tbl->ptr.pp_double[p][4+0*4+2] = -3*c->f.ptr.p_double[s1]+3*c->f.ptr.p_double[s4]-2*c->f.ptr.p_double[sfy+s1]/du-c->f.ptr.p_double[sfy+s4]/du;
tbl->ptr.pp_double[p][4+0*4+3] = 2*c->f.ptr.p_double[s1]-2*c->f.ptr.p_double[s4]+c->f.ptr.p_double[sfy+s1]/du+c->f.ptr.p_double[sfy+s4]/du;
tbl->ptr.pp_double[p][4+1*4+0] = c->f.ptr.p_double[sfx+s1]/dt;
tbl->ptr.pp_double[p][4+1*4+1] = c->f.ptr.p_double[sfxy+s1]/(dt*du);
tbl->ptr.pp_double[p][4+1*4+2] = -3*c->f.ptr.p_double[sfx+s1]/dt+3*c->f.ptr.p_double[sfx+s4]/dt-2*c->f.ptr.p_double[sfxy+s1]/(dt*du)-c->f.ptr.p_double[sfxy+s4]/(dt*du);
tbl->ptr.pp_double[p][4+1*4+3] = 2*c->f.ptr.p_double[sfx+s1]/dt-2*c->f.ptr.p_double[sfx+s4]/dt+c->f.ptr.p_double[sfxy+s1]/(dt*du)+c->f.ptr.p_double[sfxy+s4]/(dt*du);
tbl->ptr.pp_double[p][4+2*4+0] = -3*c->f.ptr.p_double[s1]+3*c->f.ptr.p_double[s2]-2*c->f.ptr.p_double[sfx+s1]/dt-c->f.ptr.p_double[sfx+s2]/dt;
tbl->ptr.pp_double[p][4+2*4+1] = -3*c->f.ptr.p_double[sfy+s1]/du+3*c->f.ptr.p_double[sfy+s2]/du-2*c->f.ptr.p_double[sfxy+s1]/(dt*du)-c->f.ptr.p_double[sfxy+s2]/(dt*du);
tbl->ptr.pp_double[p][4+2*4+2] = 9*c->f.ptr.p_double[s1]-9*c->f.ptr.p_double[s2]+9*c->f.ptr.p_double[s3]-9*c->f.ptr.p_double[s4]+6*c->f.ptr.p_double[sfx+s1]/dt+3*c->f.ptr.p_double[sfx+s2]/dt-3*c->f.ptr.p_double[sfx+s3]/dt-6*c->f.ptr.p_double[sfx+s4]/dt+6*c->f.ptr.p_double[sfy+s1]/du-6*c->f.ptr.p_double[sfy+s2]/du-3*c->f.ptr.p_double[sfy+s3]/du+3*c->f.ptr.p_double[sfy+s4]/du+4*c->f.ptr.p_double[sfxy+s1]/(dt*du)+2*c->f.ptr.p_double[sfxy+s2]/(dt*du)+c->f.ptr.p_double[sfxy+s3]/(dt*du)+2*c->f.ptr.p_double[sfxy+s4]/(dt*du);
tbl->ptr.pp_double[p][4+2*4+3] = -6*c->f.ptr.p_double[s1]+6*c->f.ptr.p_double[s2]-6*c->f.ptr.p_double[s3]+6*c->f.ptr.p_double[s4]-4*c->f.ptr.p_double[sfx+s1]/dt-2*c->f.ptr.p_double[sfx+s2]/dt+2*c->f.ptr.p_double[sfx+s3]/dt+4*c->f.ptr.p_double[sfx+s4]/dt-3*c->f.ptr.p_double[sfy+s1]/du+3*c->f.ptr.p_double[sfy+s2]/du+3*c->f.ptr.p_double[sfy+s3]/du-3*c->f.ptr.p_double[sfy+s4]/du-2*c->f.ptr.p_double[sfxy+s1]/(dt*du)-c->f.ptr.p_double[sfxy+s2]/(dt*du)-c->f.ptr.p_double[sfxy+s3]/(dt*du)-2*c->f.ptr.p_double[sfxy+s4]/(dt*du);
tbl->ptr.pp_double[p][4+3*4+0] = 2*c->f.ptr.p_double[s1]-2*c->f.ptr.p_double[s2]+c->f.ptr.p_double[sfx+s1]/dt+c->f.ptr.p_double[sfx+s2]/dt;
tbl->ptr.pp_double[p][4+3*4+1] = 2*c->f.ptr.p_double[sfy+s1]/du-2*c->f.ptr.p_double[sfy+s2]/du+c->f.ptr.p_double[sfxy+s1]/(dt*du)+c->f.ptr.p_double[sfxy+s2]/(dt*du);
tbl->ptr.pp_double[p][4+3*4+2] = -6*c->f.ptr.p_double[s1]+6*c->f.ptr.p_double[s2]-6*c->f.ptr.p_double[s3]+6*c->f.ptr.p_double[s4]-3*c->f.ptr.p_double[sfx+s1]/dt-3*c->f.ptr.p_double[sfx+s2]/dt+3*c->f.ptr.p_double[sfx+s3]/dt+3*c->f.ptr.p_double[sfx+s4]/dt-4*c->f.ptr.p_double[sfy+s1]/du+4*c->f.ptr.p_double[sfy+s2]/du+2*c->f.ptr.p_double[sfy+s3]/du-2*c->f.ptr.p_double[sfy+s4]/du-2*c->f.ptr.p_double[sfxy+s1]/(dt*du)-2*c->f.ptr.p_double[sfxy+s2]/(dt*du)-c->f.ptr.p_double[sfxy+s3]/(dt*du)-c->f.ptr.p_double[sfxy+s4]/(dt*du);
tbl->ptr.pp_double[p][4+3*4+3] = 4*c->f.ptr.p_double[s1]-4*c->f.ptr.p_double[s2]+4*c->f.ptr.p_double[s3]-4*c->f.ptr.p_double[s4]+2*c->f.ptr.p_double[sfx+s1]/dt+2*c->f.ptr.p_double[sfx+s2]/dt-2*c->f.ptr.p_double[sfx+s3]/dt-2*c->f.ptr.p_double[sfx+s4]/dt+2*c->f.ptr.p_double[sfy+s1]/du-2*c->f.ptr.p_double[sfy+s2]/du-2*c->f.ptr.p_double[sfy+s3]/du+2*c->f.ptr.p_double[sfy+s4]/du+c->f.ptr.p_double[sfxy+s1]/(dt*du)+c->f.ptr.p_double[sfxy+s2]/(dt*du)+c->f.ptr.p_double[sfxy+s3]/(dt*du)+c->f.ptr.p_double[sfxy+s4]/(dt*du);
}
/*
* Rescale Cij
*/
for(ci=0; ci<=3; ci++)
{
for(cj=0; cj<=3; cj++)
{
tbl->ptr.pp_double[p][4+ci*4+cj] = tbl->ptr.pp_double[p][4+ci*4+cj]*ae_pow(dt, ci, _state)*ae_pow(du, cj, _state);
}
}
}
}
}
}
/*************************************************************************
This subroutine was deprecated in ALGLIB 3.6.0
We recommend you to switch to Spline2DBuildBilinearV(), which is more
flexible and accepts its arguments in more convenient order.
-- ALGLIB PROJECT --
Copyright 05.07.2007 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildbilinear(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Real */ ae_matrix* f,
ae_int_t m,
ae_int_t n,
spline2dinterpolant* c,
ae_state *_state)
{
double t;
ae_int_t i;
ae_int_t j;
ae_int_t k;
_spline2dinterpolant_clear(c);
ae_assert(n>=2, "Spline2DBuildBilinear: N<2", _state);
ae_assert(m>=2, "Spline2DBuildBilinear: M<2", _state);
ae_assert(x->cnt>=n&&y->cnt>=m, "Spline2DBuildBilinear: length of X or Y is too short (Length(X/Y)<N/M)", _state);
ae_assert(isfinitevector(x, n, _state)&&isfinitevector(y, m, _state), "Spline2DBuildBilinear: X or Y contains NaN or Infinite value", _state);
ae_assert(f->rows>=m&&f->cols>=n, "Spline2DBuildBilinear: size of F is too small (rows(F)<M or cols(F)<N)", _state);
ae_assert(apservisfinitematrix(f, m, n, _state), "Spline2DBuildBilinear: F contains NaN or Infinite value", _state);
/*
* Fill interpolant
*/
c->k = 1;
c->n = n;
c->m = m;
c->d = 1;
c->stype = -1;
ae_vector_set_length(&c->x, c->n, _state);
ae_vector_set_length(&c->y, c->m, _state);
ae_vector_set_length(&c->f, c->n*c->m, _state);
for(i=0; i<=c->n-1; i++)
{
c->x.ptr.p_double[i] = x->ptr.p_double[i];
}
for(i=0; i<=c->m-1; i++)
{
c->y.ptr.p_double[i] = y->ptr.p_double[i];
}
for(i=0; i<=c->m-1; i++)
{
for(j=0; j<=c->n-1; j++)
{
c->f.ptr.p_double[i*c->n+j] = f->ptr.pp_double[i][j];
}
}
/*
* Sort points
*/
for(j=0; j<=c->n-1; j++)
{
k = j;
for(i=j+1; i<=c->n-1; i++)
{
if( ae_fp_less(c->x.ptr.p_double[i],c->x.ptr.p_double[k]) )
{
k = i;
}
}
if( k!=j )
{
for(i=0; i<=c->m-1; i++)
{
t = c->f.ptr.p_double[i*c->n+j];
c->f.ptr.p_double[i*c->n+j] = c->f.ptr.p_double[i*c->n+k];
c->f.ptr.p_double[i*c->n+k] = t;
}
t = c->x.ptr.p_double[j];
c->x.ptr.p_double[j] = c->x.ptr.p_double[k];
c->x.ptr.p_double[k] = t;
}
}
for(i=0; i<=c->m-1; i++)
{
k = i;
for(j=i+1; j<=c->m-1; j++)
{
if( ae_fp_less(c->y.ptr.p_double[j],c->y.ptr.p_double[k]) )
{
k = j;
}
}
if( k!=i )
{
for(j=0; j<=c->n-1; j++)
{
t = c->f.ptr.p_double[i*c->n+j];
c->f.ptr.p_double[i*c->n+j] = c->f.ptr.p_double[k*c->n+j];
c->f.ptr.p_double[k*c->n+j] = t;
}
t = c->y.ptr.p_double[i];
c->y.ptr.p_double[i] = c->y.ptr.p_double[k];
c->y.ptr.p_double[k] = t;
}
}
}
/*************************************************************************
This subroutine was deprecated in ALGLIB 3.6.0
We recommend you to switch to Spline2DBuildBicubicV(), which is more
flexible and accepts its arguments in more convenient order.
-- ALGLIB PROJECT --
Copyright 05.07.2007 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildbicubic(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
/* Real */ ae_matrix* f,
ae_int_t m,
ae_int_t n,
spline2dinterpolant* c,
ae_state *_state)
{
ae_frame _frame_block;
ae_matrix _f;
ae_int_t sfx;
ae_int_t sfy;
ae_int_t sfxy;
ae_matrix dx;
ae_matrix dy;
ae_matrix dxy;
double t;
ae_int_t i;
ae_int_t j;
ae_int_t k;
ae_frame_make(_state, &_frame_block);
ae_matrix_init_copy(&_f, f, _state, ae_true);
f = &_f;
_spline2dinterpolant_clear(c);
ae_matrix_init(&dx, 0, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&dy, 0, 0, DT_REAL, _state, ae_true);
ae_matrix_init(&dxy, 0, 0, DT_REAL, _state, ae_true);
ae_assert(n>=2, "Spline2DBuildBicubicSpline: N<2", _state);
ae_assert(m>=2, "Spline2DBuildBicubicSpline: M<2", _state);
ae_assert(x->cnt>=n&&y->cnt>=m, "Spline2DBuildBicubic: length of X or Y is too short (Length(X/Y)<N/M)", _state);
ae_assert(isfinitevector(x, n, _state)&&isfinitevector(y, m, _state), "Spline2DBuildBicubic: X or Y contains NaN or Infinite value", _state);
ae_assert(f->rows>=m&&f->cols>=n, "Spline2DBuildBicubic: size of F is too small (rows(F)<M or cols(F)<N)", _state);
ae_assert(apservisfinitematrix(f, m, n, _state), "Spline2DBuildBicubic: F contains NaN or Infinite value", _state);
/*
* Fill interpolant:
* F[0]...F[N*M-1]:
* f(i,j) table. f(0,0), f(0, 1), f(0,2) and so on...
* F[N*M]...F[2*N*M-1]:
* df(i,j)/dx table.
* F[2*N*M]...F[3*N*M-1]:
* df(i,j)/dy table.
* F[3*N*M]...F[4*N*M-1]:
* d2f(i,j)/dxdy table.
*/
c->k = 3;
c->d = 1;
c->n = n;
c->m = m;
c->stype = -3;
sfx = c->n*c->m;
sfy = 2*c->n*c->m;
sfxy = 3*c->n*c->m;
ae_vector_set_length(&c->x, c->n, _state);
ae_vector_set_length(&c->y, c->m, _state);
ae_vector_set_length(&c->f, 4*c->n*c->m, _state);
for(i=0; i<=c->n-1; i++)
{
c->x.ptr.p_double[i] = x->ptr.p_double[i];
}
for(i=0; i<=c->m-1; i++)
{
c->y.ptr.p_double[i] = y->ptr.p_double[i];
}
/*
* Sort points
*/
for(j=0; j<=c->n-1; j++)
{
k = j;
for(i=j+1; i<=c->n-1; i++)
{
if( ae_fp_less(c->x.ptr.p_double[i],c->x.ptr.p_double[k]) )
{
k = i;
}
}
if( k!=j )
{
for(i=0; i<=c->m-1; i++)
{
t = f->ptr.pp_double[i][j];
f->ptr.pp_double[i][j] = f->ptr.pp_double[i][k];
f->ptr.pp_double[i][k] = t;
}
t = c->x.ptr.p_double[j];
c->x.ptr.p_double[j] = c->x.ptr.p_double[k];
c->x.ptr.p_double[k] = t;
}
}
for(i=0; i<=c->m-1; i++)
{
k = i;
for(j=i+1; j<=c->m-1; j++)
{
if( ae_fp_less(c->y.ptr.p_double[j],c->y.ptr.p_double[k]) )
{
k = j;
}
}
if( k!=i )
{
for(j=0; j<=c->n-1; j++)
{
t = f->ptr.pp_double[i][j];
f->ptr.pp_double[i][j] = f->ptr.pp_double[k][j];
f->ptr.pp_double[k][j] = t;
}
t = c->y.ptr.p_double[i];
c->y.ptr.p_double[i] = c->y.ptr.p_double[k];
c->y.ptr.p_double[k] = t;
}
}
spline2d_bicubiccalcderivatives(f, &c->x, &c->y, c->m, c->n, &dx, &dy, &dxy, _state);
for(i=0; i<=c->m-1; i++)
{
for(j=0; j<=c->n-1; j++)
{
k = i*c->n+j;
c->f.ptr.p_double[k] = f->ptr.pp_double[i][j];
c->f.ptr.p_double[sfx+k] = dx.ptr.pp_double[i][j];
c->f.ptr.p_double[sfy+k] = dy.ptr.pp_double[i][j];
c->f.ptr.p_double[sfxy+k] = dxy.ptr.pp_double[i][j];
}
}
ae_frame_leave(_state);
}
/*************************************************************************
This subroutine was deprecated in ALGLIB 3.6.0
We recommend you to switch to Spline2DUnpackV(), which is more flexible
and accepts its arguments in more convenient order.
-- ALGLIB PROJECT --
Copyright 29.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline2dunpack(spline2dinterpolant* c,
ae_int_t* m,
ae_int_t* n,
/* Real */ ae_matrix* tbl,
ae_state *_state)
{
ae_int_t k;
ae_int_t p;
ae_int_t ci;
ae_int_t cj;
ae_int_t s1;
ae_int_t s2;
ae_int_t s3;
ae_int_t s4;
ae_int_t sfx;
ae_int_t sfy;
ae_int_t sfxy;
double y1;
double y2;
double y3;
double y4;
double dt;
double du;
ae_int_t i;
ae_int_t j;
*m = 0;
*n = 0;
ae_matrix_clear(tbl);
ae_assert(c->stype==-3||c->stype==-1, "Spline2DUnpack: incorrect C (incorrect parameter C.SType)", _state);
if( c->d!=1 )
{
*n = 0;
*m = 0;
return;
}
*n = c->n;
*m = c->m;
ae_matrix_set_length(tbl, (*n-1)*(*m-1), 20, _state);
sfx = *n*(*m);
sfy = 2*(*n)*(*m);
sfxy = 3*(*n)*(*m);
/*
* Fill
*/
for(i=0; i<=*m-2; i++)
{
for(j=0; j<=*n-2; j++)
{
p = i*(*n-1)+j;
tbl->ptr.pp_double[p][0] = c->x.ptr.p_double[j];
tbl->ptr.pp_double[p][1] = c->x.ptr.p_double[j+1];
tbl->ptr.pp_double[p][2] = c->y.ptr.p_double[i];
tbl->ptr.pp_double[p][3] = c->y.ptr.p_double[i+1];
dt = 1/(tbl->ptr.pp_double[p][1]-tbl->ptr.pp_double[p][0]);
du = 1/(tbl->ptr.pp_double[p][3]-tbl->ptr.pp_double[p][2]);
/*
* Bilinear interpolation
*/
if( c->stype==-1 )
{
for(k=4; k<=19; k++)
{
tbl->ptr.pp_double[p][k] = 0;
}
y1 = c->f.ptr.p_double[*n*i+j];
y2 = c->f.ptr.p_double[*n*i+(j+1)];
y3 = c->f.ptr.p_double[*n*(i+1)+(j+1)];
y4 = c->f.ptr.p_double[*n*(i+1)+j];
tbl->ptr.pp_double[p][4] = y1;
tbl->ptr.pp_double[p][4+1*4+0] = y2-y1;
tbl->ptr.pp_double[p][4+0*4+1] = y4-y1;
tbl->ptr.pp_double[p][4+1*4+1] = y3-y2-y4+y1;
}
/*
* Bicubic interpolation
*/
if( c->stype==-3 )
{
s1 = *n*i+j;
s2 = *n*i+(j+1);
s3 = *n*(i+1)+(j+1);
s4 = *n*(i+1)+j;
tbl->ptr.pp_double[p][4+0*4+0] = c->f.ptr.p_double[s1];
tbl->ptr.pp_double[p][4+0*4+1] = c->f.ptr.p_double[sfy+s1]/du;
tbl->ptr.pp_double[p][4+0*4+2] = -3*c->f.ptr.p_double[s1]+3*c->f.ptr.p_double[s4]-2*c->f.ptr.p_double[sfy+s1]/du-c->f.ptr.p_double[sfy+s4]/du;
tbl->ptr.pp_double[p][4+0*4+3] = 2*c->f.ptr.p_double[s1]-2*c->f.ptr.p_double[s4]+c->f.ptr.p_double[sfy+s1]/du+c->f.ptr.p_double[sfy+s4]/du;
tbl->ptr.pp_double[p][4+1*4+0] = c->f.ptr.p_double[sfx+s1]/dt;
tbl->ptr.pp_double[p][4+1*4+1] = c->f.ptr.p_double[sfxy+s1]/(dt*du);
tbl->ptr.pp_double[p][4+1*4+2] = -3*c->f.ptr.p_double[sfx+s1]/dt+3*c->f.ptr.p_double[sfx+s4]/dt-2*c->f.ptr.p_double[sfxy+s1]/(dt*du)-c->f.ptr.p_double[sfxy+s4]/(dt*du);
tbl->ptr.pp_double[p][4+1*4+3] = 2*c->f.ptr.p_double[sfx+s1]/dt-2*c->f.ptr.p_double[sfx+s4]/dt+c->f.ptr.p_double[sfxy+s1]/(dt*du)+c->f.ptr.p_double[sfxy+s4]/(dt*du);
tbl->ptr.pp_double[p][4+2*4+0] = -3*c->f.ptr.p_double[s1]+3*c->f.ptr.p_double[s2]-2*c->f.ptr.p_double[sfx+s1]/dt-c->f.ptr.p_double[sfx+s2]/dt;
tbl->ptr.pp_double[p][4+2*4+1] = -3*c->f.ptr.p_double[sfy+s1]/du+3*c->f.ptr.p_double[sfy+s2]/du-2*c->f.ptr.p_double[sfxy+s1]/(dt*du)-c->f.ptr.p_double[sfxy+s2]/(dt*du);
tbl->ptr.pp_double[p][4+2*4+2] = 9*c->f.ptr.p_double[s1]-9*c->f.ptr.p_double[s2]+9*c->f.ptr.p_double[s3]-9*c->f.ptr.p_double[s4]+6*c->f.ptr.p_double[sfx+s1]/dt+3*c->f.ptr.p_double[sfx+s2]/dt-3*c->f.ptr.p_double[sfx+s3]/dt-6*c->f.ptr.p_double[sfx+s4]/dt+6*c->f.ptr.p_double[sfy+s1]/du-6*c->f.ptr.p_double[sfy+s2]/du-3*c->f.ptr.p_double[sfy+s3]/du+3*c->f.ptr.p_double[sfy+s4]/du+4*c->f.ptr.p_double[sfxy+s1]/(dt*du)+2*c->f.ptr.p_double[sfxy+s2]/(dt*du)+c->f.ptr.p_double[sfxy+s3]/(dt*du)+2*c->f.ptr.p_double[sfxy+s4]/(dt*du);
tbl->ptr.pp_double[p][4+2*4+3] = -6*c->f.ptr.p_double[s1]+6*c->f.ptr.p_double[s2]-6*c->f.ptr.p_double[s3]+6*c->f.ptr.p_double[s4]-4*c->f.ptr.p_double[sfx+s1]/dt-2*c->f.ptr.p_double[sfx+s2]/dt+2*c->f.ptr.p_double[sfx+s3]/dt+4*c->f.ptr.p_double[sfx+s4]/dt-3*c->f.ptr.p_double[sfy+s1]/du+3*c->f.ptr.p_double[sfy+s2]/du+3*c->f.ptr.p_double[sfy+s3]/du-3*c->f.ptr.p_double[sfy+s4]/du-2*c->f.ptr.p_double[sfxy+s1]/(dt*du)-c->f.ptr.p_double[sfxy+s2]/(dt*du)-c->f.ptr.p_double[sfxy+s3]/(dt*du)-2*c->f.ptr.p_double[sfxy+s4]/(dt*du);
tbl->ptr.pp_double[p][4+3*4+0] = 2*c->f.ptr.p_double[s1]-2*c->f.ptr.p_double[s2]+c->f.ptr.p_double[sfx+s1]/dt+c->f.ptr.p_double[sfx+s2]/dt;
tbl->ptr.pp_double[p][4+3*4+1] = 2*c->f.ptr.p_double[sfy+s1]/du-2*c->f.ptr.p_double[sfy+s2]/du+c->f.ptr.p_double[sfxy+s1]/(dt*du)+c->f.ptr.p_double[sfxy+s2]/(dt*du);
tbl->ptr.pp_double[p][4+3*4+2] = -6*c->f.ptr.p_double[s1]+6*c->f.ptr.p_double[s2]-6*c->f.ptr.p_double[s3]+6*c->f.ptr.p_double[s4]-3*c->f.ptr.p_double[sfx+s1]/dt-3*c->f.ptr.p_double[sfx+s2]/dt+3*c->f.ptr.p_double[sfx+s3]/dt+3*c->f.ptr.p_double[sfx+s4]/dt-4*c->f.ptr.p_double[sfy+s1]/du+4*c->f.ptr.p_double[sfy+s2]/du+2*c->f.ptr.p_double[sfy+s3]/du-2*c->f.ptr.p_double[sfy+s4]/du-2*c->f.ptr.p_double[sfxy+s1]/(dt*du)-2*c->f.ptr.p_double[sfxy+s2]/(dt*du)-c->f.ptr.p_double[sfxy+s3]/(dt*du)-c->f.ptr.p_double[sfxy+s4]/(dt*du);
tbl->ptr.pp_double[p][4+3*4+3] = 4*c->f.ptr.p_double[s1]-4*c->f.ptr.p_double[s2]+4*c->f.ptr.p_double[s3]-4*c->f.ptr.p_double[s4]+2*c->f.ptr.p_double[sfx+s1]/dt+2*c->f.ptr.p_double[sfx+s2]/dt-2*c->f.ptr.p_double[sfx+s3]/dt-2*c->f.ptr.p_double[sfx+s4]/dt+2*c->f.ptr.p_double[sfy+s1]/du-2*c->f.ptr.p_double[sfy+s2]/du-2*c->f.ptr.p_double[sfy+s3]/du+2*c->f.ptr.p_double[sfy+s4]/du+c->f.ptr.p_double[sfxy+s1]/(dt*du)+c->f.ptr.p_double[sfxy+s2]/(dt*du)+c->f.ptr.p_double[sfxy+s3]/(dt*du)+c->f.ptr.p_double[sfxy+s4]/(dt*du);
}
/*
* Rescale Cij
*/
for(ci=0; ci<=3; ci++)
{
for(cj=0; cj<=3; cj++)
{
tbl->ptr.pp_double[p][4+ci*4+cj] = tbl->ptr.pp_double[p][4+ci*4+cj]*ae_pow(dt, ci, _state)*ae_pow(du, cj, _state);
}
}
}
}
}
/*************************************************************************
Internal subroutine.
Calculation of the first derivatives and the cross-derivative.
*************************************************************************/
static void spline2d_bicubiccalcderivatives(/* Real */ ae_matrix* a,
/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t m,
ae_int_t n,
/* Real */ ae_matrix* dx,
/* Real */ ae_matrix* dy,
/* Real */ ae_matrix* dxy,
ae_state *_state)
{
ae_frame _frame_block;
ae_int_t i;
ae_int_t j;
ae_vector xt;
ae_vector ft;
double s;
double ds;
double d2s;
spline1dinterpolant c;
ae_frame_make(_state, &_frame_block);
ae_matrix_clear(dx);
ae_matrix_clear(dy);
ae_matrix_clear(dxy);
ae_vector_init(&xt, 0, DT_REAL, _state, ae_true);
ae_vector_init(&ft, 0, DT_REAL, _state, ae_true);
_spline1dinterpolant_init(&c, _state, ae_true);
ae_matrix_set_length(dx, m, n, _state);
ae_matrix_set_length(dy, m, n, _state);
ae_matrix_set_length(dxy, m, n, _state);
/*
* dF/dX
*/
ae_vector_set_length(&xt, n, _state);
ae_vector_set_length(&ft, n, _state);
for(i=0; i<=m-1; i++)
{
for(j=0; j<=n-1; j++)
{
xt.ptr.p_double[j] = x->ptr.p_double[j];
ft.ptr.p_double[j] = a->ptr.pp_double[i][j];
}
spline1dbuildcubic(&xt, &ft, n, 0, 0.0, 0, 0.0, &c, _state);
for(j=0; j<=n-1; j++)
{
spline1ddiff(&c, x->ptr.p_double[j], &s, &ds, &d2s, _state);
dx->ptr.pp_double[i][j] = ds;
}
}
/*
* dF/dY
*/
ae_vector_set_length(&xt, m, _state);
ae_vector_set_length(&ft, m, _state);
for(j=0; j<=n-1; j++)
{
for(i=0; i<=m-1; i++)
{
xt.ptr.p_double[i] = y->ptr.p_double[i];
ft.ptr.p_double[i] = a->ptr.pp_double[i][j];
}
spline1dbuildcubic(&xt, &ft, m, 0, 0.0, 0, 0.0, &c, _state);
for(i=0; i<=m-1; i++)
{
spline1ddiff(&c, y->ptr.p_double[i], &s, &ds, &d2s, _state);
dy->ptr.pp_double[i][j] = ds;
}
}
/*
* d2F/dXdY
*/
ae_vector_set_length(&xt, n, _state);
ae_vector_set_length(&ft, n, _state);
for(i=0; i<=m-1; i++)
{
for(j=0; j<=n-1; j++)
{
xt.ptr.p_double[j] = x->ptr.p_double[j];
ft.ptr.p_double[j] = dy->ptr.pp_double[i][j];
}
spline1dbuildcubic(&xt, &ft, n, 0, 0.0, 0, 0.0, &c, _state);
for(j=0; j<=n-1; j++)
{
spline1ddiff(&c, x->ptr.p_double[j], &s, &ds, &d2s, _state);
dxy->ptr.pp_double[i][j] = ds;
}
}
ae_frame_leave(_state);
}
ae_bool _spline2dinterpolant_init(void* _p, ae_state *_state, ae_bool make_automatic)
{
spline2dinterpolant *p = (spline2dinterpolant*)_p;
ae_touch_ptr((void*)p);
if( !ae_vector_init(&p->x, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->y, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->f, 0, DT_REAL, _state, make_automatic) )
return ae_false;
return ae_true;
}
ae_bool _spline2dinterpolant_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic)
{
spline2dinterpolant *dst = (spline2dinterpolant*)_dst;
spline2dinterpolant *src = (spline2dinterpolant*)_src;
dst->k = src->k;
dst->stype = src->stype;
dst->n = src->n;
dst->m = src->m;
dst->d = src->d;
if( !ae_vector_init_copy(&dst->x, &src->x, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->y, &src->y, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->f, &src->f, _state, make_automatic) )
return ae_false;
return ae_true;
}
void _spline2dinterpolant_clear(void* _p)
{
spline2dinterpolant *p = (spline2dinterpolant*)_p;
ae_touch_ptr((void*)p);
ae_vector_clear(&p->x);
ae_vector_clear(&p->y);
ae_vector_clear(&p->f);
}
void _spline2dinterpolant_destroy(void* _p)
{
spline2dinterpolant *p = (spline2dinterpolant*)_p;
ae_touch_ptr((void*)p);
ae_vector_destroy(&p->x);
ae_vector_destroy(&p->y);
ae_vector_destroy(&p->f);
}
/*************************************************************************
This subroutine calculates the value of the trilinear or tricubic spline at
the given point (X,Y,Z).
INPUT PARAMETERS:
C - coefficients table.
Built by BuildBilinearSpline or BuildBicubicSpline.
X, Y,
Z - point
Result:
S(x,y,z)
-- ALGLIB PROJECT --
Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
double spline3dcalc(spline3dinterpolant* c,
double x,
double y,
double z,
ae_state *_state)
{
double v;
double vx;
double vy;
double vxy;
double result;
ae_assert(c->stype==-1||c->stype==-3, "Spline3DCalc: incorrect C (incorrect parameter C.SType)", _state);
ae_assert((ae_isfinite(x, _state)&&ae_isfinite(y, _state))&&ae_isfinite(z, _state), "Spline3DCalc: X=NaN/Infinite, Y=NaN/Infinite or Z=NaN/Infinite", _state);
if( c->d!=1 )
{
result = 0;
return result;
}
spline3d_spline3ddiff(c, x, y, z, &v, &vx, &vy, &vxy, _state);
result = v;
return result;
}
/*************************************************************************
This subroutine performs linear transformation of the spline argument.
INPUT PARAMETERS:
C - spline interpolant
AX, BX - transformation coefficients: x = A*u + B
AY, BY - transformation coefficients: y = A*v + B
AZ, BZ - transformation coefficients: z = A*w + B
OUTPUT PARAMETERS:
C - transformed spline
-- ALGLIB PROJECT --
Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dlintransxyz(spline3dinterpolant* c,
double ax,
double bx,
double ay,
double by,
double az,
double bz,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector x;
ae_vector y;
ae_vector z;
ae_vector f;
ae_vector v;
ae_int_t i;
ae_int_t j;
ae_int_t k;
ae_int_t di;
ae_frame_make(_state, &_frame_block);
ae_vector_init(&x, 0, DT_REAL, _state, ae_true);
ae_vector_init(&y, 0, DT_REAL, _state, ae_true);
ae_vector_init(&z, 0, DT_REAL, _state, ae_true);
ae_vector_init(&f, 0, DT_REAL, _state, ae_true);
ae_vector_init(&v, 0, DT_REAL, _state, ae_true);
ae_assert(c->stype==-3||c->stype==-1, "Spline3DLinTransXYZ: incorrect C (incorrect parameter C.SType)", _state);
ae_vector_set_length(&x, c->n, _state);
ae_vector_set_length(&y, c->m, _state);
ae_vector_set_length(&z, c->l, _state);
ae_vector_set_length(&f, c->m*c->n*c->l*c->d, _state);
for(j=0; j<=c->n-1; j++)
{
x.ptr.p_double[j] = c->x.ptr.p_double[j];
}
for(i=0; i<=c->m-1; i++)
{
y.ptr.p_double[i] = c->y.ptr.p_double[i];
}
for(i=0; i<=c->l-1; i++)
{
z.ptr.p_double[i] = c->z.ptr.p_double[i];
}
/*
* Handle different combinations of zero/nonzero AX/AY/AZ
*/
if( (ae_fp_neq(ax,0)&&ae_fp_neq(ay,0))&&ae_fp_neq(az,0) )
{
ae_v_move(&f.ptr.p_double[0], 1, &c->f.ptr.p_double[0], 1, ae_v_len(0,c->m*c->n*c->l*c->d-1));
}
if( (ae_fp_eq(ax,0)&&ae_fp_neq(ay,0))&&ae_fp_neq(az,0) )
{
for(i=0; i<=c->m-1; i++)
{
for(j=0; j<=c->l-1; j++)
{
spline3dcalcv(c, bx, y.ptr.p_double[i], z.ptr.p_double[j], &v, _state);
for(k=0; k<=c->n-1; k++)
{
for(di=0; di<=c->d-1; di++)
{
f.ptr.p_double[c->d*(c->n*(c->m*j+i)+k)+di] = v.ptr.p_double[di];
}
}
}
}
ax = 1;
bx = 0;
}
if( (ae_fp_neq(ax,0)&&ae_fp_eq(ay,0))&&ae_fp_neq(az,0) )
{
for(i=0; i<=c->n-1; i++)
{
for(j=0; j<=c->l-1; j++)
{
spline3dcalcv(c, x.ptr.p_double[i], by, z.ptr.p_double[j], &v, _state);
for(k=0; k<=c->m-1; k++)
{
for(di=0; di<=c->d-1; di++)
{
f.ptr.p_double[c->d*(c->n*(c->m*j+k)+i)+di] = v.ptr.p_double[di];
}
}
}
}
ay = 1;
by = 0;
}
if( (ae_fp_neq(ax,0)&&ae_fp_neq(ay,0))&&ae_fp_eq(az,0) )
{
for(i=0; i<=c->n-1; i++)
{
for(j=0; j<=c->m-1; j++)
{
spline3dcalcv(c, x.ptr.p_double[i], y.ptr.p_double[j], bz, &v, _state);
for(k=0; k<=c->l-1; k++)
{
for(di=0; di<=c->d-1; di++)
{
f.ptr.p_double[c->d*(c->n*(c->m*k+j)+i)+di] = v.ptr.p_double[di];
}
}
}
}
az = 1;
bz = 0;
}
if( (ae_fp_eq(ax,0)&&ae_fp_eq(ay,0))&&ae_fp_neq(az,0) )
{
for(i=0; i<=c->l-1; i++)
{
spline3dcalcv(c, bx, by, z.ptr.p_double[i], &v, _state);
for(k=0; k<=c->m-1; k++)
{
for(j=0; j<=c->n-1; j++)
{
for(di=0; di<=c->d-1; di++)
{
f.ptr.p_double[c->d*(c->n*(c->m*i+k)+j)+di] = v.ptr.p_double[di];
}
}
}
}
ax = 1;
bx = 0;
ay = 1;
by = 0;
}
if( (ae_fp_eq(ax,0)&&ae_fp_neq(ay,0))&&ae_fp_eq(az,0) )
{
for(i=0; i<=c->m-1; i++)
{
spline3dcalcv(c, bx, y.ptr.p_double[i], bz, &v, _state);
for(k=0; k<=c->l-1; k++)
{
for(j=0; j<=c->n-1; j++)
{
for(di=0; di<=c->d-1; di++)
{
f.ptr.p_double[c->d*(c->n*(c->m*k+i)+j)+di] = v.ptr.p_double[di];
}
}
}
}
ax = 1;
bx = 0;
az = 1;
bz = 0;
}
if( (ae_fp_neq(ax,0)&&ae_fp_eq(ay,0))&&ae_fp_eq(az,0) )
{
for(i=0; i<=c->n-1; i++)
{
spline3dcalcv(c, x.ptr.p_double[i], by, bz, &v, _state);
for(k=0; k<=c->l-1; k++)
{
for(j=0; j<=c->m-1; j++)
{
for(di=0; di<=c->d-1; di++)
{
f.ptr.p_double[c->d*(c->n*(c->m*k+j)+i)+di] = v.ptr.p_double[di];
}
}
}
}
ay = 1;
by = 0;
az = 1;
bz = 0;
}
if( (ae_fp_eq(ax,0)&&ae_fp_eq(ay,0))&&ae_fp_eq(az,0) )
{
spline3dcalcv(c, bx, by, bz, &v, _state);
for(k=0; k<=c->l-1; k++)
{
for(j=0; j<=c->m-1; j++)
{
for(i=0; i<=c->n-1; i++)
{
for(di=0; di<=c->d-1; di++)
{
f.ptr.p_double[c->d*(c->n*(c->m*k+j)+i)+di] = v.ptr.p_double[di];
}
}
}
}
ax = 1;
bx = 0;
ay = 1;
by = 0;
az = 1;
bz = 0;
}
/*
* General case: AX<>0, AY<>0, AZ<>0
* Unpack, scale and pack again.
*/
for(i=0; i<=c->n-1; i++)
{
x.ptr.p_double[i] = (x.ptr.p_double[i]-bx)/ax;
}
for(i=0; i<=c->m-1; i++)
{
y.ptr.p_double[i] = (y.ptr.p_double[i]-by)/ay;
}
for(i=0; i<=c->l-1; i++)
{
z.ptr.p_double[i] = (z.ptr.p_double[i]-bz)/az;
}
if( c->stype==-1 )
{
spline3dbuildtrilinearv(&x, c->n, &y, c->m, &z, c->l, &f, c->d, c, _state);
}
ae_frame_leave(_state);
}
/*************************************************************************
This subroutine performs linear transformation of the spline.
INPUT PARAMETERS:
C - spline interpolant.
A, B- transformation coefficients: S2(x,y) = A*S(x,y,z) + B
OUTPUT PARAMETERS:
C - transformed spline
-- ALGLIB PROJECT --
Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dlintransf(spline3dinterpolant* c,
double a,
double b,
ae_state *_state)
{
ae_frame _frame_block;
ae_vector x;
ae_vector y;
ae_vector z;
ae_vector f;
ae_int_t i;
ae_int_t j;
ae_frame_make(_state, &_frame_block);
ae_vector_init(&x, 0, DT_REAL, _state, ae_true);
ae_vector_init(&y, 0, DT_REAL, _state, ae_true);
ae_vector_init(&z, 0, DT_REAL, _state, ae_true);
ae_vector_init(&f, 0, DT_REAL, _state, ae_true);
ae_assert(c->stype==-3||c->stype==-1, "Spline3DLinTransF: incorrect C (incorrect parameter C.SType)", _state);
ae_vector_set_length(&x, c->n, _state);
ae_vector_set_length(&y, c->m, _state);
ae_vector_set_length(&z, c->l, _state);
ae_vector_set_length(&f, c->m*c->n*c->l*c->d, _state);
for(j=0; j<=c->n-1; j++)
{
x.ptr.p_double[j] = c->x.ptr.p_double[j];
}
for(i=0; i<=c->m-1; i++)
{
y.ptr.p_double[i] = c->y.ptr.p_double[i];
}
for(i=0; i<=c->l-1; i++)
{
z.ptr.p_double[i] = c->z.ptr.p_double[i];
}
for(i=0; i<=c->m*c->n*c->l*c->d-1; i++)
{
f.ptr.p_double[i] = a*c->f.ptr.p_double[i]+b;
}
if( c->stype==-1 )
{
spline3dbuildtrilinearv(&x, c->n, &y, c->m, &z, c->l, &f, c->d, c, _state);
}
ae_frame_leave(_state);
}
/*************************************************************************
This subroutine makes the copy of the spline model.
INPUT PARAMETERS:
C - spline interpolant
OUTPUT PARAMETERS:
CC - spline copy
-- ALGLIB PROJECT --
Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dcopy(spline3dinterpolant* c,
spline3dinterpolant* cc,
ae_state *_state)
{
ae_int_t tblsize;
_spline3dinterpolant_clear(cc);
ae_assert(c->k==1||c->k==3, "Spline3DCopy: incorrect C (incorrect parameter C.K)", _state);
cc->k = c->k;
cc->n = c->n;
cc->m = c->m;
cc->l = c->l;
cc->d = c->d;
tblsize = c->n*c->m*c->l*c->d;
cc->stype = c->stype;
ae_vector_set_length(&cc->x, cc->n, _state);
ae_vector_set_length(&cc->y, cc->m, _state);
ae_vector_set_length(&cc->z, cc->l, _state);
ae_vector_set_length(&cc->f, tblsize, _state);
ae_v_move(&cc->x.ptr.p_double[0], 1, &c->x.ptr.p_double[0], 1, ae_v_len(0,cc->n-1));
ae_v_move(&cc->y.ptr.p_double[0], 1, &c->y.ptr.p_double[0], 1, ae_v_len(0,cc->m-1));
ae_v_move(&cc->z.ptr.p_double[0], 1, &c->z.ptr.p_double[0], 1, ae_v_len(0,cc->l-1));
ae_v_move(&cc->f.ptr.p_double[0], 1, &c->f.ptr.p_double[0], 1, ae_v_len(0,tblsize-1));
}
/*************************************************************************
Trilinear spline resampling
INPUT PARAMETERS:
A - array[0..OldXCount*OldYCount*OldZCount-1], function
values at the old grid, :
A[0] x=0,y=0,z=0
A[1] x=1,y=0,z=0
A[..] ...
A[..] x=oldxcount-1,y=0,z=0
A[..] x=0,y=1,z=0
A[..] ...
...
OldZCount - old Z-count, OldZCount>1
OldYCount - old Y-count, OldYCount>1
OldXCount - old X-count, OldXCount>1
NewZCount - new Z-count, NewZCount>1
NewYCount - new Y-count, NewYCount>1
NewXCount - new X-count, NewXCount>1
OUTPUT PARAMETERS:
B - array[0..NewXCount*NewYCount*NewZCount-1], function
values at the new grid:
B[0] x=0,y=0,z=0
B[1] x=1,y=0,z=0
B[..] ...
B[..] x=newxcount-1,y=0,z=0
B[..] x=0,y=1,z=0
B[..] ...
...
-- ALGLIB routine --
26.04.2012
Copyright by Bochkanov Sergey
*************************************************************************/
void spline3dresampletrilinear(/* Real */ ae_vector* a,
ae_int_t oldzcount,
ae_int_t oldycount,
ae_int_t oldxcount,
ae_int_t newzcount,
ae_int_t newycount,
ae_int_t newxcount,
/* Real */ ae_vector* b,
ae_state *_state)
{
double xd;
double yd;
double zd;
double c0;
double c1;
double c2;
double c3;
ae_int_t ix;
ae_int_t iy;
ae_int_t iz;
ae_int_t i;
ae_int_t j;
ae_int_t k;
ae_vector_clear(b);
ae_assert((oldycount>1&&oldzcount>1)&&oldxcount>1, "Spline3DResampleTrilinear: length/width/height less than 1", _state);
ae_assert((newycount>1&&newzcount>1)&&newxcount>1, "Spline3DResampleTrilinear: length/width/height less than 1", _state);
ae_assert(a->cnt>=oldycount*oldzcount*oldxcount, "Spline3DResampleTrilinear: length/width/height less than 1", _state);
ae_vector_set_length(b, newxcount*newycount*newzcount, _state);
for(i=0; i<=newxcount-1; i++)
{
for(j=0; j<=newycount-1; j++)
{
for(k=0; k<=newzcount-1; k++)
{
ix = i*(oldxcount-1)/(newxcount-1);
if( ix==oldxcount-1 )
{
ix = oldxcount-2;
}
xd = (double)(i*(oldxcount-1))/(double)(newxcount-1)-ix;
iy = j*(oldycount-1)/(newycount-1);
if( iy==oldycount-1 )
{
iy = oldycount-2;
}
yd = (double)(j*(oldycount-1))/(double)(newycount-1)-iy;
iz = k*(oldzcount-1)/(newzcount-1);
if( iz==oldzcount-1 )
{
iz = oldzcount-2;
}
zd = (double)(k*(oldzcount-1))/(double)(newzcount-1)-iz;
c0 = a->ptr.p_double[oldxcount*(oldycount*iz+iy)+ix]*(1-xd)+a->ptr.p_double[oldxcount*(oldycount*iz+iy)+(ix+1)]*xd;
c1 = a->ptr.p_double[oldxcount*(oldycount*iz+(iy+1))+ix]*(1-xd)+a->ptr.p_double[oldxcount*(oldycount*iz+(iy+1))+(ix+1)]*xd;
c2 = a->ptr.p_double[oldxcount*(oldycount*(iz+1)+iy)+ix]*(1-xd)+a->ptr.p_double[oldxcount*(oldycount*(iz+1)+iy)+(ix+1)]*xd;
c3 = a->ptr.p_double[oldxcount*(oldycount*(iz+1)+(iy+1))+ix]*(1-xd)+a->ptr.p_double[oldxcount*(oldycount*(iz+1)+(iy+1))+(ix+1)]*xd;
c0 = c0*(1-yd)+c1*yd;
c1 = c2*(1-yd)+c3*yd;
b->ptr.p_double[newxcount*(newycount*k+j)+i] = c0*(1-zd)+c1*zd;
}
}
}
}
/*************************************************************************
This subroutine builds trilinear vector-valued spline.
INPUT PARAMETERS:
X - spline abscissas, array[0..N-1]
Y - spline ordinates, array[0..M-1]
Z - spline applicates, array[0..L-1]
F - function values, array[0..M*N*L*D-1]:
* first D elements store D values at (X[0],Y[0],Z[0])
* next D elements store D values at (X[1],Y[0],Z[0])
* next D elements store D values at (X[2],Y[0],Z[0])
* ...
* next D elements store D values at (X[0],Y[1],Z[0])
* next D elements store D values at (X[1],Y[1],Z[0])
* next D elements store D values at (X[2],Y[1],Z[0])
* ...
* next D elements store D values at (X[0],Y[0],Z[1])
* next D elements store D values at (X[1],Y[0],Z[1])
* next D elements store D values at (X[2],Y[0],Z[1])
* ...
* general form - D function values at (X[i],Y[j]) are stored
at F[D*(N*(M*K+J)+I)...D*(N*(M*K+J)+I)+D-1].
M,N,
L - grid size, M>=2, N>=2, L>=2
D - vector dimension, D>=1
OUTPUT PARAMETERS:
C - spline interpolant
-- ALGLIB PROJECT --
Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dbuildtrilinearv(/* Real */ ae_vector* x,
ae_int_t n,
/* Real */ ae_vector* y,
ae_int_t m,
/* Real */ ae_vector* z,
ae_int_t l,
/* Real */ ae_vector* f,
ae_int_t d,
spline3dinterpolant* c,
ae_state *_state)
{
double t;
ae_int_t tblsize;
ae_int_t i;
ae_int_t j;
ae_int_t k;
ae_int_t i0;
ae_int_t j0;
_spline3dinterpolant_clear(c);
ae_assert(m>=2, "Spline3DBuildTrilinearV: M<2", _state);
ae_assert(n>=2, "Spline3DBuildTrilinearV: N<2", _state);
ae_assert(l>=2, "Spline3DBuildTrilinearV: L<2", _state);
ae_assert(d>=1, "Spline3DBuildTrilinearV: D<1", _state);
ae_assert((x->cnt>=n&&y->cnt>=m)&&z->cnt>=l, "Spline3DBuildTrilinearV: length of X, Y or Z is too short (Length(X/Y/Z)<N/M/L)", _state);
ae_assert((isfinitevector(x, n, _state)&&isfinitevector(y, m, _state))&&isfinitevector(z, l, _state), "Spline3DBuildTrilinearV: X, Y or Z contains NaN or Infinite value", _state);
tblsize = n*m*l*d;
ae_assert(f->cnt>=tblsize, "Spline3DBuildTrilinearV: length of F is too short (Length(F)<N*M*L*D)", _state);
ae_assert(isfinitevector(f, tblsize, _state), "Spline3DBuildTrilinearV: F contains NaN or Infinite value", _state);
/*
* Fill interpolant
*/
c->k = 1;
c->n = n;
c->m = m;
c->l = l;
c->d = d;
c->stype = -1;
ae_vector_set_length(&c->x, c->n, _state);
ae_vector_set_length(&c->y, c->m, _state);
ae_vector_set_length(&c->z, c->l, _state);
ae_vector_set_length(&c->f, tblsize, _state);
for(i=0; i<=c->n-1; i++)
{
c->x.ptr.p_double[i] = x->ptr.p_double[i];
}
for(i=0; i<=c->m-1; i++)
{
c->y.ptr.p_double[i] = y->ptr.p_double[i];
}
for(i=0; i<=c->l-1; i++)
{
c->z.ptr.p_double[i] = z->ptr.p_double[i];
}
for(i=0; i<=tblsize-1; i++)
{
c->f.ptr.p_double[i] = f->ptr.p_double[i];
}
/*
* Sort points:
* * sort x;
* * sort y;
* * sort z.
*/
for(j=0; j<=c->n-1; j++)
{
k = j;
for(i=j+1; i<=c->n-1; i++)
{
if( ae_fp_less(c->x.ptr.p_double[i],c->x.ptr.p_double[k]) )
{
k = i;
}
}
if( k!=j )
{
for(i=0; i<=c->m-1; i++)
{
for(j0=0; j0<=c->l-1; j0++)
{
for(i0=0; i0<=c->d-1; i0++)
{
t = c->f.ptr.p_double[c->d*(c->n*(c->m*j0+i)+j)+i0];
c->f.ptr.p_double[c->d*(c->n*(c->m*j0+i)+j)+i0] = c->f.ptr.p_double[c->d*(c->n*(c->m*j0+i)+k)+i0];
c->f.ptr.p_double[c->d*(c->n*(c->m*j0+i)+k)+i0] = t;
}
}
}
t = c->x.ptr.p_double[j];
c->x.ptr.p_double[j] = c->x.ptr.p_double[k];
c->x.ptr.p_double[k] = t;
}
}
for(i=0; i<=c->m-1; i++)
{
k = i;
for(j=i+1; j<=c->m-1; j++)
{
if( ae_fp_less(c->y.ptr.p_double[j],c->y.ptr.p_double[k]) )
{
k = j;
}
}
if( k!=i )
{
for(j=0; j<=c->n-1; j++)
{
for(j0=0; j0<=c->l-1; j0++)
{
for(i0=0; i0<=c->d-1; i0++)
{
t = c->f.ptr.p_double[c->d*(c->n*(c->m*j0+i)+j)+i0];
c->f.ptr.p_double[c->d*(c->n*(c->m*j0+i)+j)+i0] = c->f.ptr.p_double[c->d*(c->n*(c->m*j0+k)+j)+i0];
c->f.ptr.p_double[c->d*(c->n*(c->m*j0+k)+j)+i0] = t;
}
}
}
t = c->y.ptr.p_double[i];
c->y.ptr.p_double[i] = c->y.ptr.p_double[k];
c->y.ptr.p_double[k] = t;
}
}
for(k=0; k<=c->l-1; k++)
{
i = k;
for(j=i+1; j<=c->l-1; j++)
{
if( ae_fp_less(c->z.ptr.p_double[j],c->z.ptr.p_double[i]) )
{
i = j;
}
}
if( i!=k )
{
for(j=0; j<=c->m-1; j++)
{
for(j0=0; j0<=c->n-1; j0++)
{
for(i0=0; i0<=c->d-1; i0++)
{
t = c->f.ptr.p_double[c->d*(c->n*(c->m*k+j)+j0)+i0];
c->f.ptr.p_double[c->d*(c->n*(c->m*k+j)+j0)+i0] = c->f.ptr.p_double[c->d*(c->n*(c->m*i+j)+j0)+i0];
c->f.ptr.p_double[c->d*(c->n*(c->m*i+j)+j0)+i0] = t;
}
}
}
t = c->z.ptr.p_double[k];
c->z.ptr.p_double[k] = c->z.ptr.p_double[i];
c->z.ptr.p_double[i] = t;
}
}
}
/*************************************************************************
This subroutine calculates bilinear or bicubic vector-valued spline at the
given point (X,Y,Z).
INPUT PARAMETERS:
C - spline interpolant.
X, Y,
Z - point
F - output buffer, possibly preallocated array. In case array size
is large enough to store result, it is not reallocated. Array
which is too short will be reallocated
OUTPUT PARAMETERS:
F - array[D] (or larger) which stores function values
-- ALGLIB PROJECT --
Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dcalcvbuf(spline3dinterpolant* c,
double x,
double y,
double z,
/* Real */ ae_vector* f,
ae_state *_state)
{
double xd;
double yd;
double zd;
double c0;
double c1;
double c2;
double c3;
ae_int_t ix;
ae_int_t iy;
ae_int_t iz;
ae_int_t l;
ae_int_t r;
ae_int_t h;
ae_int_t i;
ae_assert(c->stype==-1||c->stype==-3, "Spline3DCalcVBuf: incorrect C (incorrect parameter C.SType)", _state);
ae_assert((ae_isfinite(x, _state)&&ae_isfinite(y, _state))&&ae_isfinite(z, _state), "Spline3DCalcVBuf: X, Y or Z contains NaN/Infinite", _state);
rvectorsetlengthatleast(f, c->d, _state);
/*
* Binary search in the [ x[0], ..., x[n-2] ] (x[n-1] is not included)
*/
l = 0;
r = c->n-1;
while(l!=r-1)
{
h = (l+r)/2;
if( ae_fp_greater_eq(c->x.ptr.p_double[h],x) )
{
r = h;
}
else
{
l = h;
}
}
ix = l;
/*
* Binary search in the [ y[0], ..., y[n-2] ] (y[n-1] is not included)
*/
l = 0;
r = c->m-1;
while(l!=r-1)
{
h = (l+r)/2;
if( ae_fp_greater_eq(c->y.ptr.p_double[h],y) )
{
r = h;
}
else
{
l = h;
}
}
iy = l;
/*
* Binary search in the [ z[0], ..., z[n-2] ] (z[n-1] is not included)
*/
l = 0;
r = c->l-1;
while(l!=r-1)
{
h = (l+r)/2;
if( ae_fp_greater_eq(c->z.ptr.p_double[h],z) )
{
r = h;
}
else
{
l = h;
}
}
iz = l;
xd = (x-c->x.ptr.p_double[ix])/(c->x.ptr.p_double[ix+1]-c->x.ptr.p_double[ix]);
yd = (y-c->y.ptr.p_double[iy])/(c->y.ptr.p_double[iy+1]-c->y.ptr.p_double[iy]);
zd = (z-c->z.ptr.p_double[iz])/(c->z.ptr.p_double[iz+1]-c->z.ptr.p_double[iz]);
for(i=0; i<=c->d-1; i++)
{
/*
* Trilinear interpolation
*/
if( c->stype==-1 )
{
c0 = c->f.ptr.p_double[c->d*(c->n*(c->m*iz+iy)+ix)+i]*(1-xd)+c->f.ptr.p_double[c->d*(c->n*(c->m*iz+iy)+(ix+1))+i]*xd;
c1 = c->f.ptr.p_double[c->d*(c->n*(c->m*iz+(iy+1))+ix)+i]*(1-xd)+c->f.ptr.p_double[c->d*(c->n*(c->m*iz+(iy+1))+(ix+1))+i]*xd;
c2 = c->f.ptr.p_double[c->d*(c->n*(c->m*(iz+1)+iy)+ix)+i]*(1-xd)+c->f.ptr.p_double[c->d*(c->n*(c->m*(iz+1)+iy)+(ix+1))+i]*xd;
c3 = c->f.ptr.p_double[c->d*(c->n*(c->m*(iz+1)+(iy+1))+ix)+i]*(1-xd)+c->f.ptr.p_double[c->d*(c->n*(c->m*(iz+1)+(iy+1))+(ix+1))+i]*xd;
c0 = c0*(1-yd)+c1*yd;
c1 = c2*(1-yd)+c3*yd;
f->ptr.p_double[i] = c0*(1-zd)+c1*zd;
}
}
}
/*************************************************************************
This subroutine calculates trilinear or tricubic vector-valued spline at the
given point (X,Y,Z).
INPUT PARAMETERS:
C - spline interpolant.
X, Y,
Z - point
OUTPUT PARAMETERS:
F - array[D] which stores function values. F is out-parameter and
it is reallocated after call to this function. In case you
want to reuse previously allocated F, you may use
Spline2DCalcVBuf(), which reallocates F only when it is too
small.
-- ALGLIB PROJECT --
Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dcalcv(spline3dinterpolant* c,
double x,
double y,
double z,
/* Real */ ae_vector* f,
ae_state *_state)
{
ae_vector_clear(f);
ae_assert(c->stype==-1||c->stype==-3, "Spline3DCalcV: incorrect C (incorrect parameter C.SType)", _state);
ae_assert((ae_isfinite(x, _state)&&ae_isfinite(y, _state))&&ae_isfinite(z, _state), "Spline3DCalcV: X=NaN/Infinite, Y=NaN/Infinite or Z=NaN/Infinite", _state);
ae_vector_set_length(f, c->d, _state);
spline3dcalcvbuf(c, x, y, z, f, _state);
}
/*************************************************************************
This subroutine unpacks tri-dimensional spline into the coefficients table
INPUT PARAMETERS:
C - spline interpolant.
Result:
N - grid size (X)
M - grid size (Y)
L - grid size (Z)
D - number of components
SType- spline type. Currently, only one spline type is supported:
trilinear spline, as indicated by SType=1.
Tbl - spline coefficients: [0..(N-1)*(M-1)*(L-1)*D-1, 0..13].
For T=0..D-1 (component index), I = 0...N-2 (x index),
J=0..M-2 (y index), K=0..L-2 (z index):
Q := T + I*D + J*D*(N-1) + K*D*(N-1)*(M-1),
Q-th row stores decomposition for T-th component of the
vector-valued function
Tbl[Q,0] = X[i]
Tbl[Q,1] = X[i+1]
Tbl[Q,2] = Y[j]
Tbl[Q,3] = Y[j+1]
Tbl[Q,4] = Z[k]
Tbl[Q,5] = Z[k+1]
Tbl[Q,6] = C000
Tbl[Q,7] = C100
Tbl[Q,8] = C010
Tbl[Q,9] = C110
Tbl[Q,10]= C001
Tbl[Q,11]= C101
Tbl[Q,12]= C011
Tbl[Q,13]= C111
On each grid square spline is equals to:
S(x) = SUM(c[i,j,k]*(x^i)*(y^j)*(z^k), i=0..1, j=0..1, k=0..1)
t = x-x[j]
u = y-y[i]
v = z-z[k]
NOTE: format of Tbl is given for SType=1. Future versions of
ALGLIB can use different formats for different values of
SType.
-- ALGLIB PROJECT --
Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dunpackv(spline3dinterpolant* c,
ae_int_t* n,
ae_int_t* m,
ae_int_t* l,
ae_int_t* d,
ae_int_t* stype,
/* Real */ ae_matrix* tbl,
ae_state *_state)
{
ae_int_t p;
ae_int_t ci;
ae_int_t cj;
ae_int_t ck;
double du;
double dv;
double dw;
ae_int_t i;
ae_int_t j;
ae_int_t k;
ae_int_t di;
ae_int_t i0;
*n = 0;
*m = 0;
*l = 0;
*d = 0;
*stype = 0;
ae_matrix_clear(tbl);
ae_assert(c->stype==-1, "Spline3DUnpackV: incorrect C (incorrect parameter C.SType)", _state);
*n = c->n;
*m = c->m;
*l = c->l;
*d = c->d;
*stype = ae_iabs(c->stype, _state);
ae_matrix_set_length(tbl, (*n-1)*(*m-1)*(*l-1)*(*d), 14, _state);
/*
* Fill
*/
for(i=0; i<=*n-2; i++)
{
for(j=0; j<=*m-2; j++)
{
for(k=0; k<=*l-2; k++)
{
for(di=0; di<=*d-1; di++)
{
p = *d*((*n-1)*((*m-1)*k+j)+i)+di;
tbl->ptr.pp_double[p][0] = c->x.ptr.p_double[i];
tbl->ptr.pp_double[p][1] = c->x.ptr.p_double[i+1];
tbl->ptr.pp_double[p][2] = c->y.ptr.p_double[j];
tbl->ptr.pp_double[p][3] = c->y.ptr.p_double[j+1];
tbl->ptr.pp_double[p][4] = c->z.ptr.p_double[k];
tbl->ptr.pp_double[p][5] = c->z.ptr.p_double[k+1];
du = 1/(tbl->ptr.pp_double[p][1]-tbl->ptr.pp_double[p][0]);
dv = 1/(tbl->ptr.pp_double[p][3]-tbl->ptr.pp_double[p][2]);
dw = 1/(tbl->ptr.pp_double[p][5]-tbl->ptr.pp_double[p][4]);
/*
* Trilinear interpolation
*/
if( c->stype==-1 )
{
for(i0=6; i0<=13; i0++)
{
tbl->ptr.pp_double[p][i0] = 0;
}
tbl->ptr.pp_double[p][6+2*(2*0+0)+0] = c->f.ptr.p_double[*d*(*n*(*m*k+j)+i)+di];
tbl->ptr.pp_double[p][6+2*(2*0+0)+1] = c->f.ptr.p_double[*d*(*n*(*m*k+j)+(i+1))+di]-c->f.ptr.p_double[*d*(*n*(*m*k+j)+i)+di];
tbl->ptr.pp_double[p][6+2*(2*0+1)+0] = c->f.ptr.p_double[*d*(*n*(*m*k+(j+1))+i)+di]-c->f.ptr.p_double[*d*(*n*(*m*k+j)+i)+di];
tbl->ptr.pp_double[p][6+2*(2*0+1)+1] = c->f.ptr.p_double[*d*(*n*(*m*k+(j+1))+(i+1))+di]-c->f.ptr.p_double[*d*(*n*(*m*k+(j+1))+i)+di]-c->f.ptr.p_double[*d*(*n*(*m*k+j)+(i+1))+di]+c->f.ptr.p_double[*d*(*n*(*m*k+j)+i)+di];
tbl->ptr.pp_double[p][6+2*(2*1+0)+0] = c->f.ptr.p_double[*d*(*n*(*m*(k+1)+j)+i)+di]-c->f.ptr.p_double[*d*(*n*(*m*k+j)+i)+di];
tbl->ptr.pp_double[p][6+2*(2*1+0)+1] = c->f.ptr.p_double[*d*(*n*(*m*(k+1)+j)+(i+1))+di]-c->f.ptr.p_double[*d*(*n*(*m*(k+1)+j)+i)+di]-c->f.ptr.p_double[*d*(*n*(*m*k+j)+(i+1))+di]+c->f.ptr.p_double[*d*(*n*(*m*k+j)+i)+di];
tbl->ptr.pp_double[p][6+2*(2*1+1)+0] = c->f.ptr.p_double[*d*(*n*(*m*(k+1)+(j+1))+i)+di]-c->f.ptr.p_double[*d*(*n*(*m*(k+1)+j)+i)+di]-c->f.ptr.p_double[*d*(*n*(*m*k+(j+1))+i)+di]+c->f.ptr.p_double[*d*(*n*(*m*k+j)+i)+di];
tbl->ptr.pp_double[p][6+2*(2*1+1)+1] = c->f.ptr.p_double[*d*(*n*(*m*(k+1)+(j+1))+(i+1))+di]-c->f.ptr.p_double[*d*(*n*(*m*(k+1)+(j+1))+i)+di]-c->f.ptr.p_double[*d*(*n*(*m*(k+1)+j)+(i+1))+di]+c->f.ptr.p_double[*d*(*n*(*m*(k+1)+j)+i)+di]-c->f.ptr.p_double[*d*(*n*(*m*k+(j+1))+(i+1))+di]+c->f.ptr.p_double[*d*(*n*(*m*k+(j+1))+i)+di]+c->f.ptr.p_double[*d*(*n*(*m*k+j)+(i+1))+di]-c->f.ptr.p_double[*d*(*n*(*m*k+j)+i)+di];
}
/*
* Rescale Cij
*/
for(ci=0; ci<=1; ci++)
{
for(cj=0; cj<=1; cj++)
{
for(ck=0; ck<=1; ck++)
{
tbl->ptr.pp_double[p][6+2*(2*ck+cj)+ci] = tbl->ptr.pp_double[p][6+2*(2*ck+cj)+ci]*ae_pow(du, ci, _state)*ae_pow(dv, cj, _state)*ae_pow(dw, ck, _state);
}
}
}
}
}
}
}
}
/*************************************************************************
This subroutine calculates the value of the trilinear(or tricubic;possible
will be later) spline at the given point X(and its derivatives; possible
will be later).
INPUT PARAMETERS:
C - spline interpolant.
X, Y, Z - point
OUTPUT PARAMETERS:
F - S(x,y,z)
FX - dS(x,y,z)/dX
FY - dS(x,y,z)/dY
FXY - d2S(x,y,z)/dXdY
-- ALGLIB PROJECT --
Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
static void spline3d_spline3ddiff(spline3dinterpolant* c,
double x,
double y,
double z,
double* f,
double* fx,
double* fy,
double* fxy,
ae_state *_state)
{
double xd;
double yd;
double zd;
double c0;
double c1;
double c2;
double c3;
ae_int_t ix;
ae_int_t iy;
ae_int_t iz;
ae_int_t l;
ae_int_t r;
ae_int_t h;
*f = 0;
*fx = 0;
*fy = 0;
*fxy = 0;
ae_assert(c->stype==-1||c->stype==-3, "Spline3DDiff: incorrect C (incorrect parameter C.SType)", _state);
ae_assert(ae_isfinite(x, _state)&&ae_isfinite(y, _state), "Spline3DDiff: X or Y contains NaN or Infinite value", _state);
/*
* Prepare F, dF/dX, dF/dY, d2F/dXdY
*/
*f = 0;
*fx = 0;
*fy = 0;
*fxy = 0;
if( c->d!=1 )
{
return;
}
/*
* Binary search in the [ x[0], ..., x[n-2] ] (x[n-1] is not included)
*/
l = 0;
r = c->n-1;
while(l!=r-1)
{
h = (l+r)/2;
if( ae_fp_greater_eq(c->x.ptr.p_double[h],x) )
{
r = h;
}
else
{
l = h;
}
}
ix = l;
/*
* Binary search in the [ y[0], ..., y[n-2] ] (y[n-1] is not included)
*/
l = 0;
r = c->m-1;
while(l!=r-1)
{
h = (l+r)/2;
if( ae_fp_greater_eq(c->y.ptr.p_double[h],y) )
{
r = h;
}
else
{
l = h;
}
}
iy = l;
/*
* Binary search in the [ z[0], ..., z[n-2] ] (z[n-1] is not included)
*/
l = 0;
r = c->l-1;
while(l!=r-1)
{
h = (l+r)/2;
if( ae_fp_greater_eq(c->z.ptr.p_double[h],z) )
{
r = h;
}
else
{
l = h;
}
}
iz = l;
xd = (x-c->x.ptr.p_double[ix])/(c->x.ptr.p_double[ix+1]-c->x.ptr.p_double[ix]);
yd = (y-c->y.ptr.p_double[iy])/(c->y.ptr.p_double[iy+1]-c->y.ptr.p_double[iy]);
zd = (z-c->z.ptr.p_double[iz])/(c->z.ptr.p_double[iz+1]-c->z.ptr.p_double[iz]);
/*
* Trilinear interpolation
*/
if( c->stype==-1 )
{
c0 = c->f.ptr.p_double[c->n*(c->m*iz+iy)+ix]*(1-xd)+c->f.ptr.p_double[c->n*(c->m*iz+iy)+(ix+1)]*xd;
c1 = c->f.ptr.p_double[c->n*(c->m*iz+(iy+1))+ix]*(1-xd)+c->f.ptr.p_double[c->n*(c->m*iz+(iy+1))+(ix+1)]*xd;
c2 = c->f.ptr.p_double[c->n*(c->m*(iz+1)+iy)+ix]*(1-xd)+c->f.ptr.p_double[c->n*(c->m*(iz+1)+iy)+(ix+1)]*xd;
c3 = c->f.ptr.p_double[c->n*(c->m*(iz+1)+(iy+1))+ix]*(1-xd)+c->f.ptr.p_double[c->n*(c->m*(iz+1)+(iy+1))+(ix+1)]*xd;
c0 = c0*(1-yd)+c1*yd;
c1 = c2*(1-yd)+c3*yd;
*f = c0*(1-zd)+c1*zd;
}
}
ae_bool _spline3dinterpolant_init(void* _p, ae_state *_state, ae_bool make_automatic)
{
spline3dinterpolant *p = (spline3dinterpolant*)_p;
ae_touch_ptr((void*)p);
if( !ae_vector_init(&p->x, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->y, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->z, 0, DT_REAL, _state, make_automatic) )
return ae_false;
if( !ae_vector_init(&p->f, 0, DT_REAL, _state, make_automatic) )
return ae_false;
return ae_true;
}
ae_bool _spline3dinterpolant_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic)
{
spline3dinterpolant *dst = (spline3dinterpolant*)_dst;
spline3dinterpolant *src = (spline3dinterpolant*)_src;
dst->k = src->k;
dst->stype = src->stype;
dst->n = src->n;
dst->m = src->m;
dst->l = src->l;
dst->d = src->d;
if( !ae_vector_init_copy(&dst->x, &src->x, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->y, &src->y, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->z, &src->z, _state, make_automatic) )
return ae_false;
if( !ae_vector_init_copy(&dst->f, &src->f, _state, make_automatic) )
return ae_false;
return ae_true;
}
void _spline3dinterpolant_clear(void* _p)
{
spline3dinterpolant *p = (spline3dinterpolant*)_p;
ae_touch_ptr((void*)p);
ae_vector_clear(&p->x);
ae_vector_clear(&p->y);
ae_vector_clear(&p->z);
ae_vector_clear(&p->f);
}
void _spline3dinterpolant_destroy(void* _p)
{
spline3dinterpolant *p = (spline3dinterpolant*)_p;
ae_touch_ptr((void*)p);
ae_vector_destroy(&p->x);
ae_vector_destroy(&p->y);
ae_vector_destroy(&p->z);
ae_vector_destroy(&p->f);
}
}
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