File: interpolation.h

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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 >>>
*************************************************************************/
#ifndef _interpolation_pkg_h
#define _interpolation_pkg_h
#include "ap.h"
#include "alglibinternal.h"
#include "alglibmisc.h"
#include "linalg.h"
#include "solvers.h"
#include "optimization.h"
#include "specialfunctions.h"
#include "integration.h"

/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS COMPUTATIONAL CORE DECLARATIONS (DATATYPES)
//
/////////////////////////////////////////////////////////////////////////
namespace alglib_impl
{
typedef struct
{
    ae_int_t n;
    ae_int_t nx;
    ae_int_t d;
    double r;
    ae_int_t nw;
    kdtree tree;
    ae_int_t modeltype;
    ae_matrix q;
    ae_vector xbuf;
    ae_vector tbuf;
    ae_vector rbuf;
    ae_matrix xybuf;
    ae_int_t debugsolverfailures;
    double debugworstrcond;
    double debugbestrcond;
} idwinterpolant;
typedef struct
{
    ae_int_t n;
    double sy;
    ae_vector x;
    ae_vector y;
    ae_vector w;
} barycentricinterpolant;
typedef struct
{
    ae_bool periodic;
    ae_int_t n;
    ae_int_t k;
    ae_int_t continuity;
    ae_vector x;
    ae_vector c;
} spline1dinterpolant;
typedef struct
{
    double taskrcond;
    double rmserror;
    double avgerror;
    double avgrelerror;
    double maxerror;
} polynomialfitreport;
typedef struct
{
    double taskrcond;
    ae_int_t dbest;
    double rmserror;
    double avgerror;
    double avgrelerror;
    double maxerror;
} barycentricfitreport;
typedef struct
{
    double taskrcond;
    double rmserror;
    double avgerror;
    double avgrelerror;
    double maxerror;
} spline1dfitreport;
typedef struct
{
    double taskrcond;
    ae_int_t iterationscount;
    ae_int_t varidx;
    double rmserror;
    double avgerror;
    double avgrelerror;
    double maxerror;
    double wrmserror;
    ae_matrix covpar;
    ae_vector errpar;
    ae_vector errcurve;
    ae_vector noise;
    double r2;
} lsfitreport;
typedef struct
{
    ae_int_t optalgo;
    ae_int_t m;
    ae_int_t k;
    double epsf;
    double epsx;
    ae_int_t maxits;
    double stpmax;
    ae_bool xrep;
    ae_vector s;
    ae_vector bndl;
    ae_vector bndu;
    ae_matrix taskx;
    ae_vector tasky;
    ae_int_t npoints;
    ae_vector taskw;
    ae_int_t nweights;
    ae_int_t wkind;
    ae_int_t wits;
    double diffstep;
    double teststep;
    ae_bool xupdated;
    ae_bool needf;
    ae_bool needfg;
    ae_bool needfgh;
    ae_int_t pointindex;
    ae_vector x;
    ae_vector c;
    double f;
    ae_vector g;
    ae_matrix h;
    ae_vector wcur;
    ae_vector tmp;
    ae_vector tmpf;
    ae_matrix tmpjac;
    ae_matrix tmpjacw;
    double tmpnoise;
    matinvreport invrep;
    ae_int_t repiterationscount;
    ae_int_t repterminationtype;
    ae_int_t repvaridx;
    double reprmserror;
    double repavgerror;
    double repavgrelerror;
    double repmaxerror;
    double repwrmserror;
    lsfitreport rep;
    minlmstate optstate;
    minlmreport optrep;
    ae_int_t prevnpt;
    ae_int_t prevalgo;
    rcommstate rstate;
} lsfitstate;
typedef struct
{
    ae_int_t n;
    ae_bool periodic;
    ae_vector p;
    spline1dinterpolant x;
    spline1dinterpolant y;
} pspline2interpolant;
typedef struct
{
    ae_int_t n;
    ae_bool periodic;
    ae_vector p;
    spline1dinterpolant x;
    spline1dinterpolant y;
    spline1dinterpolant z;
} pspline3interpolant;
typedef struct
{
    ae_int_t ny;
    ae_int_t nx;
    ae_int_t nc;
    ae_int_t nl;
    kdtree tree;
    ae_matrix xc;
    ae_matrix wr;
    double rmax;
    ae_matrix v;
    ae_int_t gridtype;
    ae_bool fixrad;
    double lambdav;
    double radvalue;
    double radzvalue;
    ae_int_t nlayers;
    ae_int_t aterm;
    ae_int_t algorithmtype;
    double epsort;
    double epserr;
    ae_int_t maxits;
    double h;
    ae_int_t n;
    ae_matrix x;
    ae_matrix y;
    ae_vector calcbufxcx;
    ae_matrix calcbufx;
    ae_vector calcbuftags;
} rbfmodel;
typedef struct
{
    ae_int_t arows;
    ae_int_t acols;
    ae_int_t annz;
    ae_int_t iterationscount;
    ae_int_t nmv;
    ae_int_t terminationtype;
} rbfreport;
typedef struct
{
    ae_int_t k;
    ae_int_t stype;
    ae_int_t n;
    ae_int_t m;
    ae_int_t d;
    ae_vector x;
    ae_vector y;
    ae_vector f;
} spline2dinterpolant;
typedef struct
{
    ae_int_t k;
    ae_int_t stype;
    ae_int_t n;
    ae_int_t m;
    ae_int_t l;
    ae_int_t d;
    ae_vector x;
    ae_vector y;
    ae_vector z;
    ae_vector f;
} spline3dinterpolant;

}

/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS C++ INTERFACE
//
/////////////////////////////////////////////////////////////////////////
namespace alglib
{

/*************************************************************************
IDW interpolant.
*************************************************************************/
class _idwinterpolant_owner
{
public:
    _idwinterpolant_owner();
    _idwinterpolant_owner(const _idwinterpolant_owner &rhs);
    _idwinterpolant_owner& operator=(const _idwinterpolant_owner &rhs);
    virtual ~_idwinterpolant_owner();
    alglib_impl::idwinterpolant* c_ptr();
    alglib_impl::idwinterpolant* c_ptr() const;
protected:
    alglib_impl::idwinterpolant *p_struct;
};
class idwinterpolant : public _idwinterpolant_owner
{
public:
    idwinterpolant();
    idwinterpolant(const idwinterpolant &rhs);
    idwinterpolant& operator=(const idwinterpolant &rhs);
    virtual ~idwinterpolant();

};

/*************************************************************************
Barycentric interpolant.
*************************************************************************/
class _barycentricinterpolant_owner
{
public:
    _barycentricinterpolant_owner();
    _barycentricinterpolant_owner(const _barycentricinterpolant_owner &rhs);
    _barycentricinterpolant_owner& operator=(const _barycentricinterpolant_owner &rhs);
    virtual ~_barycentricinterpolant_owner();
    alglib_impl::barycentricinterpolant* c_ptr();
    alglib_impl::barycentricinterpolant* c_ptr() const;
protected:
    alglib_impl::barycentricinterpolant *p_struct;
};
class barycentricinterpolant : public _barycentricinterpolant_owner
{
public:
    barycentricinterpolant();
    barycentricinterpolant(const barycentricinterpolant &rhs);
    barycentricinterpolant& operator=(const barycentricinterpolant &rhs);
    virtual ~barycentricinterpolant();

};



/*************************************************************************
1-dimensional spline interpolant
*************************************************************************/
class _spline1dinterpolant_owner
{
public:
    _spline1dinterpolant_owner();
    _spline1dinterpolant_owner(const _spline1dinterpolant_owner &rhs);
    _spline1dinterpolant_owner& operator=(const _spline1dinterpolant_owner &rhs);
    virtual ~_spline1dinterpolant_owner();
    alglib_impl::spline1dinterpolant* c_ptr();
    alglib_impl::spline1dinterpolant* c_ptr() const;
protected:
    alglib_impl::spline1dinterpolant *p_struct;
};
class spline1dinterpolant : public _spline1dinterpolant_owner
{
public:
    spline1dinterpolant();
    spline1dinterpolant(const spline1dinterpolant &rhs);
    spline1dinterpolant& operator=(const spline1dinterpolant &rhs);
    virtual ~spline1dinterpolant();

};

/*************************************************************************
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
*************************************************************************/
class _polynomialfitreport_owner
{
public:
    _polynomialfitreport_owner();
    _polynomialfitreport_owner(const _polynomialfitreport_owner &rhs);
    _polynomialfitreport_owner& operator=(const _polynomialfitreport_owner &rhs);
    virtual ~_polynomialfitreport_owner();
    alglib_impl::polynomialfitreport* c_ptr();
    alglib_impl::polynomialfitreport* c_ptr() const;
protected:
    alglib_impl::polynomialfitreport *p_struct;
};
class polynomialfitreport : public _polynomialfitreport_owner
{
public:
    polynomialfitreport();
    polynomialfitreport(const polynomialfitreport &rhs);
    polynomialfitreport& operator=(const polynomialfitreport &rhs);
    virtual ~polynomialfitreport();
    double &taskrcond;
    double &rmserror;
    double &avgerror;
    double &avgrelerror;
    double &maxerror;

};


/*************************************************************************
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
*************************************************************************/
class _barycentricfitreport_owner
{
public:
    _barycentricfitreport_owner();
    _barycentricfitreport_owner(const _barycentricfitreport_owner &rhs);
    _barycentricfitreport_owner& operator=(const _barycentricfitreport_owner &rhs);
    virtual ~_barycentricfitreport_owner();
    alglib_impl::barycentricfitreport* c_ptr();
    alglib_impl::barycentricfitreport* c_ptr() const;
protected:
    alglib_impl::barycentricfitreport *p_struct;
};
class barycentricfitreport : public _barycentricfitreport_owner
{
public:
    barycentricfitreport();
    barycentricfitreport(const barycentricfitreport &rhs);
    barycentricfitreport& operator=(const barycentricfitreport &rhs);
    virtual ~barycentricfitreport();
    double &taskrcond;
    ae_int_t &dbest;
    double &rmserror;
    double &avgerror;
    double &avgrelerror;
    double &maxerror;

};


/*************************************************************************
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
*************************************************************************/
class _spline1dfitreport_owner
{
public:
    _spline1dfitreport_owner();
    _spline1dfitreport_owner(const _spline1dfitreport_owner &rhs);
    _spline1dfitreport_owner& operator=(const _spline1dfitreport_owner &rhs);
    virtual ~_spline1dfitreport_owner();
    alglib_impl::spline1dfitreport* c_ptr();
    alglib_impl::spline1dfitreport* c_ptr() const;
protected:
    alglib_impl::spline1dfitreport *p_struct;
};
class spline1dfitreport : public _spline1dfitreport_owner
{
public:
    spline1dfitreport();
    spline1dfitreport(const spline1dfitreport &rhs);
    spline1dfitreport& operator=(const spline1dfitreport &rhs);
    virtual ~spline1dfitreport();
    double &taskrcond;
    double &rmserror;
    double &avgerror;
    double &avgrelerror;
    double &maxerror;

};


/*************************************************************************
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.
*************************************************************************/
class _lsfitreport_owner
{
public:
    _lsfitreport_owner();
    _lsfitreport_owner(const _lsfitreport_owner &rhs);
    _lsfitreport_owner& operator=(const _lsfitreport_owner &rhs);
    virtual ~_lsfitreport_owner();
    alglib_impl::lsfitreport* c_ptr();
    alglib_impl::lsfitreport* c_ptr() const;
protected:
    alglib_impl::lsfitreport *p_struct;
};
class lsfitreport : public _lsfitreport_owner
{
public:
    lsfitreport();
    lsfitreport(const lsfitreport &rhs);
    lsfitreport& operator=(const lsfitreport &rhs);
    virtual ~lsfitreport();
    double &taskrcond;
    ae_int_t &iterationscount;
    ae_int_t &varidx;
    double &rmserror;
    double &avgerror;
    double &avgrelerror;
    double &maxerror;
    double &wrmserror;
    real_2d_array covpar;
    real_1d_array errpar;
    real_1d_array errcurve;
    real_1d_array noise;
    double &r2;

};


/*************************************************************************
Nonlinear fitter.

You should use ALGLIB functions to work with fitter.
Never try to access its fields directly!
*************************************************************************/
class _lsfitstate_owner
{
public:
    _lsfitstate_owner();
    _lsfitstate_owner(const _lsfitstate_owner &rhs);
    _lsfitstate_owner& operator=(const _lsfitstate_owner &rhs);
    virtual ~_lsfitstate_owner();
    alglib_impl::lsfitstate* c_ptr();
    alglib_impl::lsfitstate* c_ptr() const;
protected:
    alglib_impl::lsfitstate *p_struct;
};
class lsfitstate : public _lsfitstate_owner
{
public:
    lsfitstate();
    lsfitstate(const lsfitstate &rhs);
    lsfitstate& operator=(const lsfitstate &rhs);
    virtual ~lsfitstate();
    ae_bool &needf;
    ae_bool &needfg;
    ae_bool &needfgh;
    ae_bool &xupdated;
    real_1d_array c;
    double &f;
    real_1d_array g;
    real_2d_array h;
    real_1d_array x;

};

/*************************************************************************
Parametric spline inteprolant: 2-dimensional curve.

You should not try to access its members directly - use PSpline2XXXXXXXX()
functions instead.
*************************************************************************/
class _pspline2interpolant_owner
{
public:
    _pspline2interpolant_owner();
    _pspline2interpolant_owner(const _pspline2interpolant_owner &rhs);
    _pspline2interpolant_owner& operator=(const _pspline2interpolant_owner &rhs);
    virtual ~_pspline2interpolant_owner();
    alglib_impl::pspline2interpolant* c_ptr();
    alglib_impl::pspline2interpolant* c_ptr() const;
protected:
    alglib_impl::pspline2interpolant *p_struct;
};
class pspline2interpolant : public _pspline2interpolant_owner
{
public:
    pspline2interpolant();
    pspline2interpolant(const pspline2interpolant &rhs);
    pspline2interpolant& operator=(const pspline2interpolant &rhs);
    virtual ~pspline2interpolant();

};


/*************************************************************************
Parametric spline inteprolant: 3-dimensional curve.

You should not try to access its members directly - use PSpline3XXXXXXXX()
functions instead.
*************************************************************************/
class _pspline3interpolant_owner
{
public:
    _pspline3interpolant_owner();
    _pspline3interpolant_owner(const _pspline3interpolant_owner &rhs);
    _pspline3interpolant_owner& operator=(const _pspline3interpolant_owner &rhs);
    virtual ~_pspline3interpolant_owner();
    alglib_impl::pspline3interpolant* c_ptr();
    alglib_impl::pspline3interpolant* c_ptr() const;
protected:
    alglib_impl::pspline3interpolant *p_struct;
};
class pspline3interpolant : public _pspline3interpolant_owner
{
public:
    pspline3interpolant();
    pspline3interpolant(const pspline3interpolant &rhs);
    pspline3interpolant& operator=(const pspline3interpolant &rhs);
    virtual ~pspline3interpolant();

};

/*************************************************************************
RBF model.

Never try to directly work with fields of this object - always use  ALGLIB
functions to use this object.
*************************************************************************/
class _rbfmodel_owner
{
public:
    _rbfmodel_owner();
    _rbfmodel_owner(const _rbfmodel_owner &rhs);
    _rbfmodel_owner& operator=(const _rbfmodel_owner &rhs);
    virtual ~_rbfmodel_owner();
    alglib_impl::rbfmodel* c_ptr();
    alglib_impl::rbfmodel* c_ptr() const;
protected:
    alglib_impl::rbfmodel *p_struct;
};
class rbfmodel : public _rbfmodel_owner
{
public:
    rbfmodel();
    rbfmodel(const rbfmodel &rhs);
    rbfmodel& operator=(const rbfmodel &rhs);
    virtual ~rbfmodel();

};


/*************************************************************************
RBF solution report:
* TerminationType   -   termination type, positive values - success,
                        non-positive - failure.
*************************************************************************/
class _rbfreport_owner
{
public:
    _rbfreport_owner();
    _rbfreport_owner(const _rbfreport_owner &rhs);
    _rbfreport_owner& operator=(const _rbfreport_owner &rhs);
    virtual ~_rbfreport_owner();
    alglib_impl::rbfreport* c_ptr();
    alglib_impl::rbfreport* c_ptr() const;
protected:
    alglib_impl::rbfreport *p_struct;
};
class rbfreport : public _rbfreport_owner
{
public:
    rbfreport();
    rbfreport(const rbfreport &rhs);
    rbfreport& operator=(const rbfreport &rhs);
    virtual ~rbfreport();
    ae_int_t &arows;
    ae_int_t &acols;
    ae_int_t &annz;
    ae_int_t &iterationscount;
    ae_int_t &nmv;
    ae_int_t &terminationtype;

};

/*************************************************************************
2-dimensional spline inteprolant
*************************************************************************/
class _spline2dinterpolant_owner
{
public:
    _spline2dinterpolant_owner();
    _spline2dinterpolant_owner(const _spline2dinterpolant_owner &rhs);
    _spline2dinterpolant_owner& operator=(const _spline2dinterpolant_owner &rhs);
    virtual ~_spline2dinterpolant_owner();
    alglib_impl::spline2dinterpolant* c_ptr();
    alglib_impl::spline2dinterpolant* c_ptr() const;
protected:
    alglib_impl::spline2dinterpolant *p_struct;
};
class spline2dinterpolant : public _spline2dinterpolant_owner
{
public:
    spline2dinterpolant();
    spline2dinterpolant(const spline2dinterpolant &rhs);
    spline2dinterpolant& operator=(const spline2dinterpolant &rhs);
    virtual ~spline2dinterpolant();

};

/*************************************************************************
3-dimensional spline inteprolant
*************************************************************************/
class _spline3dinterpolant_owner
{
public:
    _spline3dinterpolant_owner();
    _spline3dinterpolant_owner(const _spline3dinterpolant_owner &rhs);
    _spline3dinterpolant_owner& operator=(const _spline3dinterpolant_owner &rhs);
    virtual ~_spline3dinterpolant_owner();
    alglib_impl::spline3dinterpolant* c_ptr();
    alglib_impl::spline3dinterpolant* c_ptr() const;
protected:
    alglib_impl::spline3dinterpolant *p_struct;
};
class spline3dinterpolant : public _spline3dinterpolant_owner
{
public:
    spline3dinterpolant();
    spline3dinterpolant(const spline3dinterpolant &rhs);
    spline3dinterpolant& operator=(const spline3dinterpolant &rhs);
    virtual ~spline3dinterpolant();

};

/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);

/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);

/*************************************************************************
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);


/*************************************************************************
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);
void polynomialcheb2bar(const real_1d_array &t, const double a, const double b, barycentricinterpolant &p);


/*************************************************************************
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);
void polynomialbar2pow(const barycentricinterpolant &p, real_1d_array &a);


/*************************************************************************
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);
void polynomialpow2bar(const real_1d_array &a, barycentricinterpolant &p);


/*************************************************************************
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);
void polynomialbuild(const real_1d_array &x, const real_1d_array &y, barycentricinterpolant &p);


/*************************************************************************
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);
void polynomialbuildeqdist(const double a, const double b, const real_1d_array &y, barycentricinterpolant &p);


/*************************************************************************
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);
void polynomialbuildcheb1(const double a, const double b, const real_1d_array &y, barycentricinterpolant &p);


/*************************************************************************
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);
void polynomialbuildcheb2(const double a, const double b, const real_1d_array &y, barycentricinterpolant &p);


/*************************************************************************
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);
double polynomialcalceqdist(const double a, const double b, const real_1d_array &f, const double t);


/*************************************************************************
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);
double polynomialcalccheb1(const double a, const double b, const real_1d_array &f, const double t);


/*************************************************************************
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);
double polynomialcalccheb2(const double a, const double b, const real_1d_array &f, const double t);

/*************************************************************************
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);
void spline1dbuildlinear(const real_1d_array &x, const real_1d_array &y, spline1dinterpolant &c);


/*************************************************************************
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);
void spline1dbuildcubic(const real_1d_array &x, const real_1d_array &y, spline1dinterpolant &c);


/*************************************************************************
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);
void spline1dgriddiffcubic(const real_1d_array &x, const real_1d_array &y, real_1d_array &d);


/*************************************************************************
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);
void spline1dgriddiff2cubic(const real_1d_array &x, const real_1d_array &y, real_1d_array &d1, real_1d_array &d2);


/*************************************************************************
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);
void spline1dconvcubic(const real_1d_array &x, const real_1d_array &y, const real_1d_array &x2, real_1d_array &y2);


/*************************************************************************
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);
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);


/*************************************************************************
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);
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);


/*************************************************************************
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);
void spline1dbuildcatmullrom(const real_1d_array &x, const real_1d_array &y, spline1dinterpolant &c);


/*************************************************************************
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);
void spline1dbuildhermite(const real_1d_array &x, const real_1d_array &y, const real_1d_array &d, spline1dinterpolant &c);


/*************************************************************************
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);
void spline1dbuildakima(const real_1d_array &x, const real_1d_array &y, spline1dinterpolant &c);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);
void spline1dbuildmonotone(const real_1d_array &x, const real_1d_array &y, spline1dinterpolant &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(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);
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);


/*************************************************************************
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);
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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].
    N   -   number of points (optional):
            * N>0
            * if given, only first N elements of X/Y are processed
            * if not given, automatically determined from X/Y 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 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);
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);


/*************************************************************************
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);
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);


/*************************************************************************
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);
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);


/*************************************************************************
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);
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);


/*************************************************************************
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);
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);


/*************************************************************************
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);
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);


/*************************************************************************
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);
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);


/*************************************************************************
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);
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);


/*************************************************************************
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);
void lsfitlinear(const real_1d_array &y, const real_2d_array &fmatrix, ae_int_t &info, real_1d_array &c, lsfitreport &rep);


/*************************************************************************
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);
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);


/*************************************************************************
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);
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);


/*************************************************************************
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);
void lsfitcreatef(const real_2d_array &x, const real_1d_array &y, const real_1d_array &c, const double diffstep, lsfitstate &state);


/*************************************************************************
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);
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);


/*************************************************************************
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);
void lsfitcreatefg(const real_2d_array &x, const real_1d_array &y, const real_1d_array &c, const bool cheapfg, lsfitstate &state);


/*************************************************************************
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);
void lsfitcreatewfgh(const real_2d_array &x, const real_1d_array &y, const real_1d_array &w, const real_1d_array &c, lsfitstate &state);


/*************************************************************************
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);
void lsfitcreatefgh(const real_2d_array &x, const real_1d_array &y, const real_1d_array &c, lsfitstate &state);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
This family of functions is used to launcn iterations of nonlinear fitter

These functions accept following parameters:
    state   -   algorithm state
    func    -   callback which calculates function (or merit function)
                value func at given point x
    grad    -   callback which calculates function (or merit function)
                value func and gradient grad at given point x
    hess    -   callback which calculates function (or merit function)
                value func, gradient grad and Hessian hess at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL

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

*************************************************************************/
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) = NULL,
    void *ptr = NULL);
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) = NULL,
    void *ptr = NULL);
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) = NULL,
    void *ptr = NULL);


/*************************************************************************
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);


/*************************************************************************
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);

/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);

/*************************************************************************
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);


/*************************************************************************
This function unserializes data structure from string.
*************************************************************************/
void rbfunserialize(std::string &s_in, rbfmodel &obj);


/*************************************************************************
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);


/*************************************************************************
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);
void rbfsetpoints(const rbfmodel &s, const real_2d_array &xy);


/*************************************************************************
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);
void rbfsetalgoqnn(const rbfmodel &s);


/*************************************************************************
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);
void rbfsetalgomultilayer(const rbfmodel &s, const double rbase, const ae_int_t nlayers);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);

/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);

/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);


/*************************************************************************
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);
}

/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS COMPUTATIONAL CORE DECLARATIONS (FUNCTIONS)
//
/////////////////////////////////////////////////////////////////////////
namespace alglib_impl
{
double idwcalc(idwinterpolant* z,
     /* Real    */ ae_vector* x,
     ae_state *_state);
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);
void idwbuildmodifiedshepardr(/* Real    */ ae_matrix* xy,
     ae_int_t n,
     ae_int_t nx,
     double r,
     idwinterpolant* z,
     ae_state *_state);
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_bool _idwinterpolant_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _idwinterpolant_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _idwinterpolant_clear(void* _p);
void _idwinterpolant_destroy(void* _p);
double barycentriccalc(barycentricinterpolant* b,
     double t,
     ae_state *_state);
void barycentricdiff1(barycentricinterpolant* b,
     double t,
     double* f,
     double* df,
     ae_state *_state);
void barycentricdiff2(barycentricinterpolant* b,
     double t,
     double* f,
     double* df,
     double* d2f,
     ae_state *_state);
void barycentriclintransx(barycentricinterpolant* b,
     double ca,
     double cb,
     ae_state *_state);
void barycentriclintransy(barycentricinterpolant* b,
     double ca,
     double cb,
     ae_state *_state);
void barycentricunpack(barycentricinterpolant* b,
     ae_int_t* n,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* y,
     /* Real    */ ae_vector* w,
     ae_state *_state);
void barycentricbuildxyw(/* Real    */ ae_vector* x,
     /* Real    */ ae_vector* y,
     /* Real    */ ae_vector* w,
     ae_int_t n,
     barycentricinterpolant* b,
     ae_state *_state);
void barycentricbuildfloaterhormann(/* Real    */ ae_vector* x,
     /* Real    */ ae_vector* y,
     ae_int_t n,
     ae_int_t d,
     barycentricinterpolant* b,
     ae_state *_state);
void barycentriccopy(barycentricinterpolant* b,
     barycentricinterpolant* b2,
     ae_state *_state);
ae_bool _barycentricinterpolant_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _barycentricinterpolant_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _barycentricinterpolant_clear(void* _p);
void _barycentricinterpolant_destroy(void* _p);
void polynomialbar2cheb(barycentricinterpolant* p,
     double a,
     double b,
     /* Real    */ ae_vector* t,
     ae_state *_state);
void polynomialcheb2bar(/* Real    */ ae_vector* t,
     ae_int_t n,
     double a,
     double b,
     barycentricinterpolant* p,
     ae_state *_state);
void polynomialbar2pow(barycentricinterpolant* p,
     double c,
     double s,
     /* Real    */ ae_vector* a,
     ae_state *_state);
void polynomialpow2bar(/* Real    */ ae_vector* a,
     ae_int_t n,
     double c,
     double s,
     barycentricinterpolant* p,
     ae_state *_state);
void polynomialbuild(/* Real    */ ae_vector* x,
     /* Real    */ ae_vector* y,
     ae_int_t n,
     barycentricinterpolant* p,
     ae_state *_state);
void polynomialbuildeqdist(double a,
     double b,
     /* Real    */ ae_vector* y,
     ae_int_t n,
     barycentricinterpolant* p,
     ae_state *_state);
void polynomialbuildcheb1(double a,
     double b,
     /* Real    */ ae_vector* y,
     ae_int_t n,
     barycentricinterpolant* p,
     ae_state *_state);
void polynomialbuildcheb2(double a,
     double b,
     /* Real    */ ae_vector* y,
     ae_int_t n,
     barycentricinterpolant* p,
     ae_state *_state);
double polynomialcalceqdist(double a,
     double b,
     /* Real    */ ae_vector* f,
     ae_int_t n,
     double t,
     ae_state *_state);
double polynomialcalccheb1(double a,
     double b,
     /* Real    */ ae_vector* f,
     ae_int_t n,
     double t,
     ae_state *_state);
double polynomialcalccheb2(double a,
     double b,
     /* Real    */ ae_vector* f,
     ae_int_t n,
     double t,
     ae_state *_state);
void spline1dbuildlinear(/* Real    */ ae_vector* x,
     /* Real    */ ae_vector* y,
     ae_int_t n,
     spline1dinterpolant* c,
     ae_state *_state);
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);
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);
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);
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);
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);
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);
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);
void spline1dbuildhermite(/* Real    */ ae_vector* x,
     /* Real    */ ae_vector* y,
     /* Real    */ ae_vector* d,
     ae_int_t n,
     spline1dinterpolant* c,
     ae_state *_state);
void spline1dbuildakima(/* Real    */ ae_vector* x,
     /* Real    */ ae_vector* y,
     ae_int_t n,
     spline1dinterpolant* c,
     ae_state *_state);
double spline1dcalc(spline1dinterpolant* c, double x, ae_state *_state);
void spline1ddiff(spline1dinterpolant* c,
     double x,
     double* s,
     double* ds,
     double* d2s,
     ae_state *_state);
void spline1dcopy(spline1dinterpolant* c,
     spline1dinterpolant* cc,
     ae_state *_state);
void spline1dunpack(spline1dinterpolant* c,
     ae_int_t* n,
     /* Real    */ ae_matrix* tbl,
     ae_state *_state);
void spline1dlintransx(spline1dinterpolant* c,
     double a,
     double b,
     ae_state *_state);
void spline1dlintransy(spline1dinterpolant* c,
     double a,
     double b,
     ae_state *_state);
double spline1dintegrate(spline1dinterpolant* c,
     double x,
     ae_state *_state);
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);
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);
void heapsortdpoints(/* Real    */ ae_vector* x,
     /* Real    */ ae_vector* y,
     /* Real    */ ae_vector* d,
     ae_int_t n,
     ae_state *_state);
void solvepolinom2(double p0,
     double m0,
     double p1,
     double m1,
     double* x0,
     double* x1,
     ae_int_t* nr,
     ae_state *_state);
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 bisectmethod(double pa,
     double ma,
     double pb,
     double mb,
     double a,
     double b,
     double* x,
     ae_state *_state);
void spline1dbuildmonotone(/* Real    */ ae_vector* x,
     /* Real    */ ae_vector* y,
     ae_int_t n,
     spline1dinterpolant* c,
     ae_state *_state);
ae_bool _spline1dinterpolant_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _spline1dinterpolant_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _spline1dinterpolant_clear(void* _p);
void _spline1dinterpolant_destroy(void* _p);
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);
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);
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);
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);
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);
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);
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);
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);
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);
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);
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);
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);
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);
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);
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);
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);
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);
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);
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);
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);
void lsfitsetcond(lsfitstate* state,
     double epsf,
     double epsx,
     ae_int_t maxits,
     ae_state *_state);
void lsfitsetstpmax(lsfitstate* state, double stpmax, ae_state *_state);
void lsfitsetxrep(lsfitstate* state, ae_bool needxrep, ae_state *_state);
void lsfitsetscale(lsfitstate* state,
     /* Real    */ ae_vector* s,
     ae_state *_state);
void lsfitsetbc(lsfitstate* state,
     /* Real    */ ae_vector* bndl,
     /* Real    */ ae_vector* bndu,
     ae_state *_state);
ae_bool lsfititeration(lsfitstate* state, ae_state *_state);
void lsfitresults(lsfitstate* state,
     ae_int_t* info,
     /* Real    */ ae_vector* c,
     lsfitreport* rep,
     ae_state *_state);
void lsfitsetgradientcheck(lsfitstate* state,
     double teststep,
     ae_state *_state);
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);
ae_bool _polynomialfitreport_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _polynomialfitreport_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _polynomialfitreport_clear(void* _p);
void _polynomialfitreport_destroy(void* _p);
ae_bool _barycentricfitreport_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _barycentricfitreport_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _barycentricfitreport_clear(void* _p);
void _barycentricfitreport_destroy(void* _p);
ae_bool _spline1dfitreport_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _spline1dfitreport_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _spline1dfitreport_clear(void* _p);
void _spline1dfitreport_destroy(void* _p);
ae_bool _lsfitreport_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _lsfitreport_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _lsfitreport_clear(void* _p);
void _lsfitreport_destroy(void* _p);
ae_bool _lsfitstate_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _lsfitstate_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _lsfitstate_clear(void* _p);
void _lsfitstate_destroy(void* _p);
void pspline2build(/* Real    */ ae_matrix* xy,
     ae_int_t n,
     ae_int_t st,
     ae_int_t pt,
     pspline2interpolant* p,
     ae_state *_state);
void pspline3build(/* Real    */ ae_matrix* xy,
     ae_int_t n,
     ae_int_t st,
     ae_int_t pt,
     pspline3interpolant* p,
     ae_state *_state);
void pspline2buildperiodic(/* Real    */ ae_matrix* xy,
     ae_int_t n,
     ae_int_t st,
     ae_int_t pt,
     pspline2interpolant* p,
     ae_state *_state);
void pspline3buildperiodic(/* Real    */ ae_matrix* xy,
     ae_int_t n,
     ae_int_t st,
     ae_int_t pt,
     pspline3interpolant* p,
     ae_state *_state);
void pspline2parametervalues(pspline2interpolant* p,
     ae_int_t* n,
     /* Real    */ ae_vector* t,
     ae_state *_state);
void pspline3parametervalues(pspline3interpolant* p,
     ae_int_t* n,
     /* Real    */ ae_vector* t,
     ae_state *_state);
void pspline2calc(pspline2interpolant* p,
     double t,
     double* x,
     double* y,
     ae_state *_state);
void pspline3calc(pspline3interpolant* p,
     double t,
     double* x,
     double* y,
     double* z,
     ae_state *_state);
void pspline2tangent(pspline2interpolant* p,
     double t,
     double* x,
     double* y,
     ae_state *_state);
void pspline3tangent(pspline3interpolant* p,
     double t,
     double* x,
     double* y,
     double* z,
     ae_state *_state);
void pspline2diff(pspline2interpolant* p,
     double t,
     double* x,
     double* dx,
     double* y,
     double* dy,
     ae_state *_state);
void pspline3diff(pspline3interpolant* p,
     double t,
     double* x,
     double* dx,
     double* y,
     double* dy,
     double* z,
     double* dz,
     ae_state *_state);
void pspline2diff2(pspline2interpolant* p,
     double t,
     double* x,
     double* dx,
     double* d2x,
     double* y,
     double* dy,
     double* d2y,
     ae_state *_state);
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);
double pspline2arclength(pspline2interpolant* p,
     double a,
     double b,
     ae_state *_state);
double pspline3arclength(pspline3interpolant* p,
     double a,
     double b,
     ae_state *_state);
ae_bool _pspline2interpolant_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _pspline2interpolant_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _pspline2interpolant_clear(void* _p);
void _pspline2interpolant_destroy(void* _p);
ae_bool _pspline3interpolant_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _pspline3interpolant_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _pspline3interpolant_clear(void* _p);
void _pspline3interpolant_destroy(void* _p);
void rbfcreate(ae_int_t nx, ae_int_t ny, rbfmodel* s, ae_state *_state);
void rbfsetpoints(rbfmodel* s,
     /* Real    */ ae_matrix* xy,
     ae_int_t n,
     ae_state *_state);
void rbfsetalgoqnn(rbfmodel* s, double q, double z, ae_state *_state);
void rbfsetalgomultilayer(rbfmodel* s,
     double rbase,
     ae_int_t nlayers,
     double lambdav,
     ae_state *_state);
void rbfsetlinterm(rbfmodel* s, ae_state *_state);
void rbfsetconstterm(rbfmodel* s, ae_state *_state);
void rbfsetzeroterm(rbfmodel* s, ae_state *_state);
void rbfsetcond(rbfmodel* s,
     double epsort,
     double epserr,
     ae_int_t maxits,
     ae_state *_state);
void rbfbuildmodel(rbfmodel* s, rbfreport* rep, ae_state *_state);
double rbfcalc2(rbfmodel* s, double x0, double x1, ae_state *_state);
double rbfcalc3(rbfmodel* s,
     double x0,
     double x1,
     double x2,
     ae_state *_state);
void rbfcalc(rbfmodel* s,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* y,
     ae_state *_state);
void rbfcalcbuf(rbfmodel* s,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* y,
     ae_state *_state);
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);
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);
void rbfalloc(ae_serializer* s, rbfmodel* model, ae_state *_state);
void rbfserialize(ae_serializer* s, rbfmodel* model, ae_state *_state);
void rbfunserialize(ae_serializer* s, rbfmodel* model, ae_state *_state);
ae_bool _rbfmodel_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _rbfmodel_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _rbfmodel_clear(void* _p);
void _rbfmodel_destroy(void* _p);
ae_bool _rbfreport_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _rbfreport_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _rbfreport_clear(void* _p);
void _rbfreport_destroy(void* _p);
double spline2dcalc(spline2dinterpolant* c,
     double x,
     double y,
     ae_state *_state);
void spline2ddiff(spline2dinterpolant* c,
     double x,
     double y,
     double* f,
     double* fx,
     double* fy,
     double* fxy,
     ae_state *_state);
void spline2dlintransxy(spline2dinterpolant* c,
     double ax,
     double bx,
     double ay,
     double by,
     ae_state *_state);
void spline2dlintransf(spline2dinterpolant* c,
     double a,
     double b,
     ae_state *_state);
void spline2dcopy(spline2dinterpolant* c,
     spline2dinterpolant* cc,
     ae_state *_state);
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);
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);
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);
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);
void spline2dcalcvbuf(spline2dinterpolant* c,
     double x,
     double y,
     /* Real    */ ae_vector* f,
     ae_state *_state);
void spline2dcalcv(spline2dinterpolant* c,
     double x,
     double y,
     /* Real    */ ae_vector* f,
     ae_state *_state);
void spline2dunpackv(spline2dinterpolant* c,
     ae_int_t* m,
     ae_int_t* n,
     ae_int_t* d,
     /* Real    */ ae_matrix* tbl,
     ae_state *_state);
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);
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);
void spline2dunpack(spline2dinterpolant* c,
     ae_int_t* m,
     ae_int_t* n,
     /* Real    */ ae_matrix* tbl,
     ae_state *_state);
ae_bool _spline2dinterpolant_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _spline2dinterpolant_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _spline2dinterpolant_clear(void* _p);
void _spline2dinterpolant_destroy(void* _p);
double spline3dcalc(spline3dinterpolant* c,
     double x,
     double y,
     double z,
     ae_state *_state);
void spline3dlintransxyz(spline3dinterpolant* c,
     double ax,
     double bx,
     double ay,
     double by,
     double az,
     double bz,
     ae_state *_state);
void spline3dlintransf(spline3dinterpolant* c,
     double a,
     double b,
     ae_state *_state);
void spline3dcopy(spline3dinterpolant* c,
     spline3dinterpolant* cc,
     ae_state *_state);
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);
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);
void spline3dcalcvbuf(spline3dinterpolant* c,
     double x,
     double y,
     double z,
     /* Real    */ ae_vector* f,
     ae_state *_state);
void spline3dcalcv(spline3dinterpolant* c,
     double x,
     double y,
     double z,
     /* Real    */ ae_vector* f,
     ae_state *_state);
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_bool _spline3dinterpolant_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _spline3dinterpolant_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _spline3dinterpolant_clear(void* _p);
void _spline3dinterpolant_destroy(void* _p);

}
#endif