File: integration.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 _integration_pkg_h
#define _integration_pkg_h
#include "ap.h"
#include "alglibinternal.h"
#include "linalg.h"
#include "specialfunctions.h"

/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS COMPUTATIONAL CORE DECLARATIONS (DATATYPES)
//
/////////////////////////////////////////////////////////////////////////
namespace alglib_impl
{
typedef struct
{
    ae_int_t terminationtype;
    ae_int_t nfev;
    ae_int_t nintervals;
} autogkreport;
typedef struct
{
    double a;
    double b;
    double eps;
    double xwidth;
    double x;
    double f;
    ae_int_t info;
    double r;
    ae_matrix heap;
    ae_int_t heapsize;
    ae_int_t heapwidth;
    ae_int_t heapused;
    double sumerr;
    double sumabs;
    ae_vector qn;
    ae_vector wg;
    ae_vector wk;
    ae_vector wr;
    ae_int_t n;
    rcommstate rstate;
} autogkinternalstate;
typedef struct
{
    double a;
    double b;
    double alpha;
    double beta;
    double xwidth;
    double x;
    double xminusa;
    double bminusx;
    ae_bool needf;
    double f;
    ae_int_t wrappermode;
    autogkinternalstate internalstate;
    rcommstate rstate;
    double v;
    ae_int_t terminationtype;
    ae_int_t nfev;
    ae_int_t nintervals;
} autogkstate;

}

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





/*************************************************************************
Integration report:
* TerminationType = completetion code:
    * -5    non-convergence of Gauss-Kronrod nodes
            calculation subroutine.
    * -1    incorrect parameters were specified
    *  1    OK
* Rep.NFEV countains number of function calculations
* Rep.NIntervals contains number of intervals [a,b]
  was partitioned into.
*************************************************************************/
class _autogkreport_owner
{
public:
    _autogkreport_owner();
    _autogkreport_owner(const _autogkreport_owner &rhs);
    _autogkreport_owner& operator=(const _autogkreport_owner &rhs);
    virtual ~_autogkreport_owner();
    alglib_impl::autogkreport* c_ptr();
    alglib_impl::autogkreport* c_ptr() const;
protected:
    alglib_impl::autogkreport *p_struct;
};
class autogkreport : public _autogkreport_owner
{
public:
    autogkreport();
    autogkreport(const autogkreport &rhs);
    autogkreport& operator=(const autogkreport &rhs);
    virtual ~autogkreport();
    ae_int_t &terminationtype;
    ae_int_t &nfev;
    ae_int_t &nintervals;

};


/*************************************************************************
This structure stores state of the integration algorithm.

Although this class has public fields,  they are not intended for external
use. You should use ALGLIB functions to work with this class:
* autogksmooth()/AutoGKSmoothW()/... to create objects
* autogkintegrate() to begin integration
* autogkresults() to get results
*************************************************************************/
class _autogkstate_owner
{
public:
    _autogkstate_owner();
    _autogkstate_owner(const _autogkstate_owner &rhs);
    _autogkstate_owner& operator=(const _autogkstate_owner &rhs);
    virtual ~_autogkstate_owner();
    alglib_impl::autogkstate* c_ptr();
    alglib_impl::autogkstate* c_ptr() const;
protected:
    alglib_impl::autogkstate *p_struct;
};
class autogkstate : public _autogkstate_owner
{
public:
    autogkstate();
    autogkstate(const autogkstate &rhs);
    autogkstate& operator=(const autogkstate &rhs);
    virtual ~autogkstate();
    ae_bool &needf;
    double &x;
    double &xminusa;
    double &bminusx;
    double &f;

};

/*************************************************************************
Computation of nodes and weights for a Gauss quadrature formula

The algorithm generates the N-point Gauss quadrature formula  with  weight
function given by coefficients alpha and beta  of  a  recurrence  relation
which generates a system of orthogonal polynomials:

P-1(x)   =  0
P0(x)    =  1
Pn+1(x)  =  (x-alpha(n))*Pn(x)  -  beta(n)*Pn-1(x)

and zeroth moment Mu0

Mu0 = integral(W(x)dx,a,b)

INPUT PARAMETERS:
    Alpha      array[0..N-1], alpha coefficients
    Beta       array[0..N-1], beta coefficients
                Zero-indexed element is not used and may be arbitrary.
                Beta[I]>0.
    Mu0        zeroth moment of the weight function.
    N          number of nodes of the quadrature formula, N>=1

OUTPUT PARAMETERS:
    Info    -   error code:
                * -3    internal eigenproblem solver hasn't converged
                * -2    Beta[i]<=0
                * -1    incorrect N was passed
                *  1    OK
    X       -   array[0..N-1] - array of quadrature nodes,
                in ascending order.
    W       -   array[0..N-1] - array of quadrature weights.

  -- ALGLIB --
     Copyright 2005-2009 by Bochkanov Sergey
*************************************************************************/
void gqgeneraterec(const real_1d_array &alpha, const real_1d_array &beta, const double mu0, const ae_int_t n, ae_int_t &info, real_1d_array &x, real_1d_array &w);


/*************************************************************************
Computation of nodes and weights for a Gauss-Lobatto quadrature formula

The algorithm generates the N-point Gauss-Lobatto quadrature formula  with
weight function given by coefficients alpha and beta of a recurrence which
generates a system of orthogonal polynomials.

P-1(x)   =  0
P0(x)    =  1
Pn+1(x)  =  (x-alpha(n))*Pn(x)  -  beta(n)*Pn-1(x)

and zeroth moment Mu0

Mu0 = integral(W(x)dx,a,b)

INPUT PARAMETERS:
    Alpha      array[0..N-2], alpha coefficients
    Beta       array[0..N-2], beta coefficients.
                Zero-indexed element is not used, may be arbitrary.
                Beta[I]>0
    Mu0        zeroth moment of the weighting function.
    A          left boundary of the integration interval.
    B          right boundary of the integration interval.
    N          number of nodes of the quadrature formula, N>=3
                (including the left and right boundary nodes).

OUTPUT PARAMETERS:
    Info    -   error code:
                * -3    internal eigenproblem solver hasn't converged
                * -2    Beta[i]<=0
                * -1    incorrect N was passed
                *  1    OK
    X       -   array[0..N-1] - array of quadrature nodes,
                in ascending order.
    W       -   array[0..N-1] - array of quadrature weights.

  -- ALGLIB --
     Copyright 2005-2009 by Bochkanov Sergey
*************************************************************************/
void gqgenerategausslobattorec(const real_1d_array &alpha, const real_1d_array &beta, const double mu0, const double a, const double b, const ae_int_t n, ae_int_t &info, real_1d_array &x, real_1d_array &w);


/*************************************************************************
Computation of nodes and weights for a Gauss-Radau quadrature formula

The algorithm generates the N-point Gauss-Radau  quadrature  formula  with
weight function given by the coefficients alpha and  beta  of a recurrence
which generates a system of orthogonal polynomials.

P-1(x)   =  0
P0(x)    =  1
Pn+1(x)  =  (x-alpha(n))*Pn(x)  -  beta(n)*Pn-1(x)

and zeroth moment Mu0

Mu0 = integral(W(x)dx,a,b)

INPUT PARAMETERS:
    Alpha      array[0..N-2], alpha coefficients.
    Beta       array[0..N-1], beta coefficients
                Zero-indexed element is not used.
                Beta[I]>0
    Mu0        zeroth moment of the weighting function.
    A          left boundary of the integration interval.
    N          number of nodes of the quadrature formula, N>=2
                (including the left boundary node).

OUTPUT PARAMETERS:
    Info    -   error code:
                * -3    internal eigenproblem solver hasn't converged
                * -2    Beta[i]<=0
                * -1    incorrect N was passed
                *  1    OK
    X       -   array[0..N-1] - array of quadrature nodes,
                in ascending order.
    W       -   array[0..N-1] - array of quadrature weights.


  -- ALGLIB --
     Copyright 2005-2009 by Bochkanov Sergey
*************************************************************************/
void gqgenerategaussradaurec(const real_1d_array &alpha, const real_1d_array &beta, const double mu0, const double a, const ae_int_t n, ae_int_t &info, real_1d_array &x, real_1d_array &w);


/*************************************************************************
Returns nodes/weights for Gauss-Legendre quadrature on [-1,1] with N
nodes.

INPUT PARAMETERS:
    N           -   number of nodes, >=1

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -4    an  error   was   detected   when  calculating
                            weights/nodes.  N  is  too  large   to  obtain
                            weights/nodes  with  high   enough   accuracy.
                            Try  to   use   multiple   precision  version.
                    * -3    internal eigenproblem solver hasn't  converged
                    * -1    incorrect N was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes,
                    in ascending order.
    W           -   array[0..N-1] - array of quadrature weights.


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void gqgenerategausslegendre(const ae_int_t n, ae_int_t &info, real_1d_array &x, real_1d_array &w);


/*************************************************************************
Returns  nodes/weights  for  Gauss-Jacobi quadrature on [-1,1] with weight
function W(x)=Power(1-x,Alpha)*Power(1+x,Beta).

INPUT PARAMETERS:
    N           -   number of nodes, >=1
    Alpha       -   power-law coefficient, Alpha>-1
    Beta        -   power-law coefficient, Beta>-1

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -4    an  error  was   detected   when   calculating
                            weights/nodes. Alpha or  Beta  are  too  close
                            to -1 to obtain weights/nodes with high enough
                            accuracy, or, may be, N is too large.  Try  to
                            use multiple precision version.
                    * -3    internal eigenproblem solver hasn't converged
                    * -1    incorrect N/Alpha/Beta was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes,
                    in ascending order.
    W           -   array[0..N-1] - array of quadrature weights.


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void gqgenerategaussjacobi(const ae_int_t n, const double alpha, const double beta, ae_int_t &info, real_1d_array &x, real_1d_array &w);


/*************************************************************************
Returns  nodes/weights  for  Gauss-Laguerre  quadrature  on  [0,+inf) with
weight function W(x)=Power(x,Alpha)*Exp(-x)

INPUT PARAMETERS:
    N           -   number of nodes, >=1
    Alpha       -   power-law coefficient, Alpha>-1

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -4    an  error  was   detected   when   calculating
                            weights/nodes. Alpha is too  close  to  -1  to
                            obtain weights/nodes with high enough accuracy
                            or, may  be,  N  is  too  large.  Try  to  use
                            multiple precision version.
                    * -3    internal eigenproblem solver hasn't converged
                    * -1    incorrect N/Alpha was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes,
                    in ascending order.
    W           -   array[0..N-1] - array of quadrature weights.


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void gqgenerategausslaguerre(const ae_int_t n, const double alpha, ae_int_t &info, real_1d_array &x, real_1d_array &w);


/*************************************************************************
Returns  nodes/weights  for  Gauss-Hermite  quadrature on (-inf,+inf) with
weight function W(x)=Exp(-x*x)

INPUT PARAMETERS:
    N           -   number of nodes, >=1

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -4    an  error  was   detected   when   calculating
                            weights/nodes.  May be, N is too large. Try to
                            use multiple precision version.
                    * -3    internal eigenproblem solver hasn't converged
                    * -1    incorrect N/Alpha was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes,
                    in ascending order.
    W           -   array[0..N-1] - array of quadrature weights.


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void gqgenerategausshermite(const ae_int_t n, ae_int_t &info, real_1d_array &x, real_1d_array &w);

/*************************************************************************
Computation of nodes and weights of a Gauss-Kronrod quadrature formula

The algorithm generates the N-point Gauss-Kronrod quadrature formula  with
weight  function  given  by  coefficients  alpha  and beta of a recurrence
relation which generates a system of orthogonal polynomials:

    P-1(x)   =  0
    P0(x)    =  1
    Pn+1(x)  =  (x-alpha(n))*Pn(x)  -  beta(n)*Pn-1(x)

and zero moment Mu0

    Mu0 = integral(W(x)dx,a,b)


INPUT PARAMETERS:
    Alpha          alpha coefficients, array[0..floor(3*K/2)].
    Beta           beta coefficients,  array[0..ceil(3*K/2)].
                    Beta[0] is not used and may be arbitrary.
                    Beta[I]>0.
    Mu0            zeroth moment of the weight function.
    N              number of nodes of the Gauss-Kronrod quadrature formula,
                    N >= 3,
                    N =  2*K+1.

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -5    no real and positive Gauss-Kronrod formula can
                            be created for such a weight function  with  a
                            given number of nodes.
                    * -4    N is too large, task may be ill  conditioned -
                            x[i]=x[i+1] found.
                    * -3    internal eigenproblem solver hasn't converged
                    * -2    Beta[i]<=0
                    * -1    incorrect N was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes,
                    in ascending order.
    WKronrod    -   array[0..N-1] - Kronrod weights
    WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                    corresponding to extended Kronrod nodes).

  -- ALGLIB --
     Copyright 08.05.2009 by Bochkanov Sergey
*************************************************************************/
void gkqgeneraterec(const real_1d_array &alpha, const real_1d_array &beta, const double mu0, const ae_int_t n, ae_int_t &info, real_1d_array &x, real_1d_array &wkronrod, real_1d_array &wgauss);


/*************************************************************************
Returns   Gauss   and   Gauss-Kronrod   nodes/weights  for  Gauss-Legendre
quadrature with N points.

GKQLegendreCalc (calculation) or  GKQLegendreTbl  (precomputed  table)  is
used depending on machine precision and number of nodes.

INPUT PARAMETERS:
    N           -   number of Kronrod nodes, must be odd number, >=3.

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -4    an  error   was   detected   when  calculating
                            weights/nodes.  N  is  too  large   to  obtain
                            weights/nodes  with  high   enough   accuracy.
                            Try  to   use   multiple   precision  version.
                    * -3    internal eigenproblem solver hasn't converged
                    * -1    incorrect N was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes, ordered in
                    ascending order.
    WKronrod    -   array[0..N-1] - Kronrod weights
    WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                    corresponding to extended Kronrod nodes).


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void gkqgenerategausslegendre(const ae_int_t n, ae_int_t &info, real_1d_array &x, real_1d_array &wkronrod, real_1d_array &wgauss);


/*************************************************************************
Returns   Gauss   and   Gauss-Kronrod   nodes/weights   for   Gauss-Jacobi
quadrature on [-1,1] with weight function

    W(x)=Power(1-x,Alpha)*Power(1+x,Beta).

INPUT PARAMETERS:
    N           -   number of Kronrod nodes, must be odd number, >=3.
    Alpha       -   power-law coefficient, Alpha>-1
    Beta        -   power-law coefficient, Beta>-1

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -5    no real and positive Gauss-Kronrod formula can
                            be created for such a weight function  with  a
                            given number of nodes.
                    * -4    an  error  was   detected   when   calculating
                            weights/nodes. Alpha or  Beta  are  too  close
                            to -1 to obtain weights/nodes with high enough
                            accuracy, or, may be, N is too large.  Try  to
                            use multiple precision version.
                    * -3    internal eigenproblem solver hasn't converged
                    * -1    incorrect N was passed
                    * +1    OK
                    * +2    OK, but quadrature rule have exterior  nodes,
                            x[0]<-1 or x[n-1]>+1
    X           -   array[0..N-1] - array of quadrature nodes, ordered in
                    ascending order.
    WKronrod    -   array[0..N-1] - Kronrod weights
    WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                    corresponding to extended Kronrod nodes).


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void gkqgenerategaussjacobi(const ae_int_t n, const double alpha, const double beta, ae_int_t &info, real_1d_array &x, real_1d_array &wkronrod, real_1d_array &wgauss);


/*************************************************************************
Returns Gauss and Gauss-Kronrod nodes for quadrature with N points.

Reduction to tridiagonal eigenproblem is used.

INPUT PARAMETERS:
    N           -   number of Kronrod nodes, must be odd number, >=3.

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -4    an  error   was   detected   when  calculating
                            weights/nodes.  N  is  too  large   to  obtain
                            weights/nodes  with  high   enough   accuracy.
                            Try  to   use   multiple   precision  version.
                    * -3    internal eigenproblem solver hasn't converged
                    * -1    incorrect N was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes, ordered in
                    ascending order.
    WKronrod    -   array[0..N-1] - Kronrod weights
    WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                    corresponding to extended Kronrod nodes).

  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void gkqlegendrecalc(const ae_int_t n, ae_int_t &info, real_1d_array &x, real_1d_array &wkronrod, real_1d_array &wgauss);


/*************************************************************************
Returns Gauss and Gauss-Kronrod nodes for quadrature with N  points  using
pre-calculated table. Nodes/weights were  computed  with  accuracy  up  to
1.0E-32 (if MPFR version of ALGLIB is used). In standard double  precision
accuracy reduces to something about 2.0E-16 (depending  on your compiler's
handling of long floating point constants).

INPUT PARAMETERS:
    N           -   number of Kronrod nodes.
                    N can be 15, 21, 31, 41, 51, 61.

OUTPUT PARAMETERS:
    X           -   array[0..N-1] - array of quadrature nodes, ordered in
                    ascending order.
    WKronrod    -   array[0..N-1] - Kronrod weights
    WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                    corresponding to extended Kronrod nodes).


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void gkqlegendretbl(const ae_int_t n, real_1d_array &x, real_1d_array &wkronrod, real_1d_array &wgauss, double &eps);

/*************************************************************************
Integration of a smooth function F(x) on a finite interval [a,b].

Fast-convergent algorithm based on a Gauss-Kronrod formula is used. Result
is calculated with accuracy close to the machine precision.

Algorithm works well only with smooth integrands.  It  may  be  used  with
continuous non-smooth integrands, but with  less  performance.

It should never be used with integrands which have integrable singularities
at lower or upper limits - algorithm may crash. Use AutoGKSingular in such
cases.

INPUT PARAMETERS:
    A, B    -   interval boundaries (A<B, A=B or A>B)

OUTPUT PARAMETERS
    State   -   structure which stores algorithm state

SEE ALSO
    AutoGKSmoothW, AutoGKSingular, AutoGKResults.


  -- ALGLIB --
     Copyright 06.05.2009 by Bochkanov Sergey
*************************************************************************/
void autogksmooth(const double a, const double b, autogkstate &state);


/*************************************************************************
Integration of a smooth function F(x) on a finite interval [a,b].

This subroutine is same as AutoGKSmooth(), but it guarantees that interval
[a,b] is partitioned into subintervals which have width at most XWidth.

Subroutine  can  be  used  when  integrating nearly-constant function with
narrow "bumps" (about XWidth wide). If "bumps" are too narrow, AutoGKSmooth
subroutine can overlook them.

INPUT PARAMETERS:
    A, B    -   interval boundaries (A<B, A=B or A>B)

OUTPUT PARAMETERS
    State   -   structure which stores algorithm state

SEE ALSO
    AutoGKSmooth, AutoGKSingular, AutoGKResults.


  -- ALGLIB --
     Copyright 06.05.2009 by Bochkanov Sergey
*************************************************************************/
void autogksmoothw(const double a, const double b, const double xwidth, autogkstate &state);


/*************************************************************************
Integration on a finite interval [A,B].
Integrand have integrable singularities at A/B.

F(X) must diverge as "(x-A)^alpha" at A, as "(B-x)^beta" at B,  with known
alpha/beta (alpha>-1, beta>-1).  If alpha/beta  are  not known,  estimates
from below can be used (but these estimates should be greater than -1 too).

One  of  alpha/beta variables (or even both alpha/beta) may be equal to 0,
which means than function F(x) is non-singular at A/B. Anyway (singular at
bounds or not), function F(x) is supposed to be continuous on (A,B).

Fast-convergent algorithm based on a Gauss-Kronrod formula is used. Result
is calculated with accuracy close to the machine precision.

INPUT PARAMETERS:
    A, B    -   interval boundaries (A<B, A=B or A>B)
    Alpha   -   power-law coefficient of the F(x) at A,
                Alpha>-1
    Beta    -   power-law coefficient of the F(x) at B,
                Beta>-1

OUTPUT PARAMETERS
    State   -   structure which stores algorithm state

SEE ALSO
    AutoGKSmooth, AutoGKSmoothW, AutoGKResults.


  -- ALGLIB --
     Copyright 06.05.2009 by Bochkanov Sergey
*************************************************************************/
void autogksingular(const double a, const double b, const double alpha, const double beta, autogkstate &state);


/*************************************************************************
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 autogkiteration(const autogkstate &state);


/*************************************************************************
This function is used to launcn iterations of the 1-dimensional integrator

It accepts following parameters:
    func    -   callback which calculates f(x) for given x
    ptr     -   optional pointer which is passed to func; can be NULL


  -- ALGLIB --
     Copyright 07.05.2009 by Bochkanov Sergey

*************************************************************************/
void autogkintegrate(autogkstate &state,
    void (*func)(double x, double xminusa, double bminusx, double &y, void *ptr),
    void *ptr = NULL);


/*************************************************************************
Adaptive integration results

Called after AutoGKIteration returned False.

Input parameters:
    State   -   algorithm state (used by AutoGKIteration).

Output parameters:
    V       -   integral(f(x)dx,a,b)
    Rep     -   optimization report (see AutoGKReport description)

  -- ALGLIB --
     Copyright 14.11.2007 by Bochkanov Sergey
*************************************************************************/
void autogkresults(const autogkstate &state, double &v, autogkreport &rep);
}

/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS COMPUTATIONAL CORE DECLARATIONS (FUNCTIONS)
//
/////////////////////////////////////////////////////////////////////////
namespace alglib_impl
{
void gqgeneraterec(/* Real    */ ae_vector* alpha,
     /* Real    */ ae_vector* beta,
     double mu0,
     ae_int_t n,
     ae_int_t* info,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* w,
     ae_state *_state);
void gqgenerategausslobattorec(/* Real    */ ae_vector* alpha,
     /* Real    */ ae_vector* beta,
     double mu0,
     double a,
     double b,
     ae_int_t n,
     ae_int_t* info,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* w,
     ae_state *_state);
void gqgenerategaussradaurec(/* Real    */ ae_vector* alpha,
     /* Real    */ ae_vector* beta,
     double mu0,
     double a,
     ae_int_t n,
     ae_int_t* info,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* w,
     ae_state *_state);
void gqgenerategausslegendre(ae_int_t n,
     ae_int_t* info,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* w,
     ae_state *_state);
void gqgenerategaussjacobi(ae_int_t n,
     double alpha,
     double beta,
     ae_int_t* info,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* w,
     ae_state *_state);
void gqgenerategausslaguerre(ae_int_t n,
     double alpha,
     ae_int_t* info,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* w,
     ae_state *_state);
void gqgenerategausshermite(ae_int_t n,
     ae_int_t* info,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* w,
     ae_state *_state);
void gkqgeneraterec(/* Real    */ ae_vector* alpha,
     /* Real    */ ae_vector* beta,
     double mu0,
     ae_int_t n,
     ae_int_t* info,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* wkronrod,
     /* Real    */ ae_vector* wgauss,
     ae_state *_state);
void gkqgenerategausslegendre(ae_int_t n,
     ae_int_t* info,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* wkronrod,
     /* Real    */ ae_vector* wgauss,
     ae_state *_state);
void gkqgenerategaussjacobi(ae_int_t n,
     double alpha,
     double beta,
     ae_int_t* info,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* wkronrod,
     /* Real    */ ae_vector* wgauss,
     ae_state *_state);
void gkqlegendrecalc(ae_int_t n,
     ae_int_t* info,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* wkronrod,
     /* Real    */ ae_vector* wgauss,
     ae_state *_state);
void gkqlegendretbl(ae_int_t n,
     /* Real    */ ae_vector* x,
     /* Real    */ ae_vector* wkronrod,
     /* Real    */ ae_vector* wgauss,
     double* eps,
     ae_state *_state);
void autogksmooth(double a,
     double b,
     autogkstate* state,
     ae_state *_state);
void autogksmoothw(double a,
     double b,
     double xwidth,
     autogkstate* state,
     ae_state *_state);
void autogksingular(double a,
     double b,
     double alpha,
     double beta,
     autogkstate* state,
     ae_state *_state);
ae_bool autogkiteration(autogkstate* state, ae_state *_state);
void autogkresults(autogkstate* state,
     double* v,
     autogkreport* rep,
     ae_state *_state);
ae_bool _autogkreport_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _autogkreport_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _autogkreport_clear(void* _p);
void _autogkreport_destroy(void* _p);
ae_bool _autogkinternalstate_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _autogkinternalstate_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _autogkinternalstate_clear(void* _p);
void _autogkinternalstate_destroy(void* _p);
ae_bool _autogkstate_init(void* _p, ae_state *_state, ae_bool make_automatic);
ae_bool _autogkstate_init_copy(void* _dst, void* _src, ae_state *_state, ae_bool make_automatic);
void _autogkstate_clear(void* _p);
void _autogkstate_destroy(void* _p);

}
#endif