RooAdaptiveGaussKronrodIntegrator1D.cxx

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/*****************************************************************************
 * Project: RooFit                                                           *
 * Package: RooFitCore                                                       *
 * @(#)root/roofitcore:$Id$
 * Authors:                                                                  *
 *   WV, Wouter Verkerke, UC Santa Barbara, verkerke@slac.stanford.edu       *
 *   DK, David Kirkby,    UC Irvine,         dkirkby@uci.edu                 *
 *                                                                           *
 * Copyright (c) 2000-2005, Regents of the University of California          *
 *                          and Stanford University. All rights reserved.    *
 *                                                                           *
 * Redistribution and use in source and binary forms,                        *
 * with or without modification, are permitted according to the terms        *
 * listed in LICENSE (http://roofit.sourceforge.net/license.txt)             *
 *****************************************************************************/
 
/**
\file RooAdaptiveGaussKronrodIntegrator1D.cxx
\class RooAdaptiveGaussKronrodIntegrator1D
\ingroup Roofitcore
 
RooAdaptiveGaussKronrodIntegrator1D implements the Gauss-Kronrod integration algorithm.
 
An adaptive Gaussian quadrature method for numerical integration in
which error is estimated based on evaluation at special points
known as the "Kronrod points".  By suitably picking these points,
abscissas from previous iterations can be reused as part of the new
set of points, whereas usual Gaussian quadrature would require
recomputation of all abscissas at each iteration.
 
This class automatically handles (-inf,+inf) integrals by dividing
the integration in three regions (-inf,-1), (-1,1), (1,inf) and
calculating the 1st and 3rd term using a \f$ x \rightarrow 1/x \f$ coordinate
transformation.
 
This class embeds the adaptive Gauss-Kronrod integrator from the
GNU Scientific Library version 1.5 and applies a chosen rule ( 10-,
21-, 31-, 41, 51- or 61-point). The integration domain is
subdivided and recursively integrated until the required precision
is reached.
 
For integrands with integrable singularities the Wynn epsilon rule
can be selected to speed up the convergence of these integrals.
**/
 
#include <assert.h>
#include <stdlib.h>
#include "TClass.h"
#include "Riostream.h"
#include "RooAdaptiveGaussKronrodIntegrator1D.h"
#include "RooArgSet.h"
#include "RooRealVar.h"
#include "RooNumber.h"
#include "RooNumIntFactory.h"
#include "RooIntegratorBinding.h"
#include "TMath.h"
#include "RooMsgService.h"
 
using namespace std ;
 
 
ClassImp(RooAdaptiveGaussKronrodIntegrator1D);
;
 
// --- From GSL_MATH.h -------------------------------------------
struct gsl_function_struct
{
  double (* function) (double x, void * params);
  void * params;
};
typedef struct gsl_function_struct gsl_function ;
#define GSL_FN_EVAL(F,x) (*((F)->function))(x,(F)->params)
 
//----From GSL_INTEGRATION.h ---------------------------------------
typedef struct
  {
    size_t limit;
    size_t size;
    size_t nrmax;
    size_t i;
    size_t maximum_level;
    double *alist;
    double *blist;
    double *rlist;
    double *elist;
    size_t *order;
    size_t *level;
  }
gsl_integration_workspace;
 
gsl_integration_workspace *
  gsl_integration_workspace_alloc (const size_t n);
 
void
  gsl_integration_workspace_free (gsl_integration_workspace * w);
 
int gsl_integration_qag (const gsl_function * f,
                         double a, double b,
                         double epsabs, double epsrel, size_t limit,
                         int key,
                         gsl_integration_workspace * workspace,
                         double *result, double *abserr);
 
int
gsl_integration_qags (const gsl_function *f,
                      double a, double b,
                      double epsabs, double epsrel, size_t limit,
                      gsl_integration_workspace * workspace,
                      double * result, double * abserr) ;
 
int
gsl_integration_qagi (gsl_function * f,
                      double epsabs, double epsrel, size_t limit,
                      gsl_integration_workspace * workspace,
                      double *result, double *abserr) ;
 
int
gsl_integration_qagil (gsl_function * f,
                       double b,
                       double epsabs, double epsrel, size_t limit,
                       gsl_integration_workspace * workspace,
                       double *result, double *abserr) ;
 
int
gsl_integration_qagiu (gsl_function * f,
                       double a,
                       double epsabs, double epsrel, size_t limit,
                       gsl_integration_workspace * workspace,
                       double *result, double *abserr) ;
 
 
//-------------------------------------------------------------------
 
// register integrator class
// create a derived class in order to call the protected method of the
// RoodaptiveGaussKronrodIntegrator1D
namespace RooFit_internal {
struct Roo_internal_AGKInteg1D : public RooAdaptiveGaussKronrodIntegrator1D {
 
   static void registerIntegrator()
   {
      auto &intFactory = RooNumIntFactory::instance();
      RooAdaptiveGaussKronrodIntegrator1D::registerIntegrator(intFactory);
   }
};
// class used to register integrator at loafing time
struct Roo_reg_AGKInteg1D {
   Roo_reg_AGKInteg1D() { Roo_internal_AGKInteg1D::registerIntegrator(); }
};
 
static Roo_reg_AGKInteg1D instance;
} // namespace RooFit_internal
////////////////////////////////////////////////////////////////////////////////
/// Register this class with RooNumIntConfig as a possible choice of numeric
/// integrator for one-dimensional integrals over finite and infinite domains
void RooAdaptiveGaussKronrodIntegrator1D::registerIntegrator(RooNumIntFactory& fact)
{
     RooRealVar maxSeg("maxSeg", "maximum number of segments", 100);
     RooCategory method("method", "Integration method for each segment");
     method.defineType("WynnEpsilon", 0);
     method.defineType("15Points", 1);
     method.defineType("21Points", 2);
     method.defineType("31Points", 3);
     method.defineType("41Points", 4);
     method.defineType("51Points", 5);
     method.defineType("61Points", 6);
     method.setIndex(2);
     fact.storeProtoIntegrator(new RooAdaptiveGaussKronrodIntegrator1D(), RooArgSet(maxSeg, method));
     oocoutI(nullptr,Integration)  << "RooAdaptiveGaussKronrodIntegrator1D has been registered " << std::endl;
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Constructor taking a function binding and a configuration object
 
RooAdaptiveGaussKronrodIntegrator1D::RooAdaptiveGaussKronrodIntegrator1D(const RooAbsFunc& function,
                            const RooNumIntConfig& config) :
  RooAbsIntegrator(function),
  _epsAbs(config.epsRel()),
  _epsRel(config.epsAbs())
{
  // Use this form of the constructor to integrate over the function's default range.
  const RooArgSet& confSet = config.getConfigSection(ClassName()) ;
  _maxSeg = (Int_t) confSet.getRealValue("maxSeg",100) ;
  _methodKey = confSet.getCatIndex("method",2) ;
 
  _useIntegrandLimits= true;
  _valid= initialize();
}
 
 
 
////////////////////////////////////////////////////////////////////////////////
/// Constructor taking a function binding, an integration range and a configuration object
 
RooAdaptiveGaussKronrodIntegrator1D::RooAdaptiveGaussKronrodIntegrator1D(const RooAbsFunc& function,
                            double xmin, double xmax,
                            const RooNumIntConfig& config) :
  RooAbsIntegrator(function),
  _epsAbs(config.epsRel()),
  _epsRel(config.epsAbs()),
  _xmin(xmin),
  _xmax(xmax)
{
  // Use this form of the constructor to integrate over the function's default range.
  const RooArgSet& confSet = config.getConfigSection(ClassName()) ;
  _maxSeg = (Int_t) confSet.getRealValue("maxSeg",100) ;
  _methodKey = confSet.getCatIndex("method",2) ;
 
  _useIntegrandLimits= false;
  _valid= initialize();
}
 
 
 
////////////////////////////////////////////////////////////////////////////////
/// Virtual constructor
 
RooAbsIntegrator* RooAdaptiveGaussKronrodIntegrator1D::clone(const RooAbsFunc& function, const RooNumIntConfig& config) const
{
  return new RooAdaptiveGaussKronrodIntegrator1D(function,config) ;
}
 
 
 
////////////////////////////////////////////////////////////////////////////////
/// Initialize integrator allocate buffers and setup GSL workspace
 
bool RooAdaptiveGaussKronrodIntegrator1D::initialize()
{
  // Allocate coordinate buffer size after number of function dimensions
  _x.resize(_function->getDimension());
  _workspace = gsl_integration_workspace_alloc (_maxSeg)  ;
 
  return checkLimits();
}
 
 
 
////////////////////////////////////////////////////////////////////////////////
/// Destructor.
 
RooAdaptiveGaussKronrodIntegrator1D::~RooAdaptiveGaussKronrodIntegrator1D()
{
  if (_workspace) {
    gsl_integration_workspace_free ((gsl_integration_workspace*) _workspace) ;
  }
}
 
 
 
////////////////////////////////////////////////////////////////////////////////
/// Change our integration limits. Return true if the new limits are
/// ok, or otherwise false. Always returns false and does nothing
/// if this object was constructed to always use our integrand's limits.
 
bool RooAdaptiveGaussKronrodIntegrator1D::setLimits(double* xmin, double* xmax)
{
  if(_useIntegrandLimits) {
    coutE(Integration) << "RooAdaptiveGaussKronrodIntegrator1D::setLimits: cannot override integrand's limits" << endl;
    return false;
  }
 
  _xmin= *xmin;
  _xmax= *xmax;
  return checkLimits();
}
 
 
 
////////////////////////////////////////////////////////////////////////////////
/// Check that our integration range is finite and otherwise return false.
/// Update the limits from the integrand if requested.
 
bool RooAdaptiveGaussKronrodIntegrator1D::checkLimits() const
{
  if(_useIntegrandLimits) {
    assert(0 != integrand() && integrand()->isValid());
    _xmin= integrand()->getMinLimit(0);
    _xmax= integrand()->getMaxLimit(0);
  }
 
  // Determine domain type
  bool infLo= RooNumber::isInfinite(_xmin);
  bool infHi= RooNumber::isInfinite(_xmax);
 
  if (!infLo && !infHi) {
    _domainType = Closed ;
  } else if (infLo && infHi) {
    _domainType = Open ;
  } else if (infLo && !infHi) {
    _domainType = OpenLo ;
  } else {
    _domainType = OpenHi ;
  }
 
 
  return true ;
}
 
 
 
////////////////////////////////////////////////////////////////////////////////
/// Glue function interacing to GSL code
 
double RooAdaptiveGaussKronrodIntegrator1D_GSL_GlueFunction(double x, void *data)
{
  RooAdaptiveGaussKronrodIntegrator1D* instance = (RooAdaptiveGaussKronrodIntegrator1D*) data ;
  return instance->integrand(instance->xvec(x)) ;
}
 
 
 
////////////////////////////////////////////////////////////////////////////////
/// Calculate and return integral at at given parameter values
 
double RooAdaptiveGaussKronrodIntegrator1D::integral(const double *yvec)
{
  assert(isValid());
 
  // Copy yvec to xvec if provided
  if (yvec) {
    UInt_t i ; for (i=0 ; i<_function->getDimension()-1 ; i++) {
      _x[i+1] = yvec[i] ;
    }
  }
 
  // Setup glue function
  gsl_function F;
  F.function = &RooAdaptiveGaussKronrodIntegrator1D_GSL_GlueFunction ;
  F.params = this ;
 
  // Return values
  double result, error;
 
  // Call GSL implementation of integeator
  switch(_domainType) {
  case Closed:
    if (_methodKey==0) {
      gsl_integration_qags (&F, _xmin, _xmax, _epsAbs, _epsRel, _maxSeg, (gsl_integration_workspace*)_workspace,&result, &error);
    } else {
      gsl_integration_qag (&F, _xmin, _xmax, _epsAbs, _epsRel, _maxSeg, _methodKey, (gsl_integration_workspace*)_workspace,&result, &error);
    }
    break ;
  case OpenLo:
    gsl_integration_qagil (&F, _xmax, _epsAbs, _epsRel, _maxSeg, (gsl_integration_workspace*)_workspace,&result, &error);
    break ;
  case OpenHi:
    gsl_integration_qagiu (&F, _xmin, _epsAbs, _epsRel, _maxSeg, (gsl_integration_workspace*)_workspace,&result, &error);
    break ;
  case Open:
    gsl_integration_qagi (&F, _epsAbs, _epsRel, _maxSeg, (gsl_integration_workspace*)_workspace,&result, &error);
    break ;
  }
 
  return result;
}
 
 
// ----------------------------------------------------------------------------
// ---------- Code below imported from GSL version 1.5 ------------------------
// ----------------------------------------------------------------------------
 
/*
 *
 * Copyright (C) 1996, 1997, 1998, 1999, 2000 Brian Gough
 *
 * 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; 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.
 *
 * You should have received a copy of the GNU General Public License
 * along with this program; if not, write to the Free Software
 * Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
 */
 
#define GSL_SUCCESS 0
#define GSL_EDOM     1  /* input domain error, e.g sqrt(-1) */
#define GSL_ENOMEM   8  /* malloc failed */
#define GSL_EBADTOL 13  /* user specified an invalid tolerance */
#define GSL_ETOL    14  /* failed to reach the specified tolerance */
#define GSL_ERROR(a,b) oocoutE(nullptr,Integration) << "RooAdaptiveGaussKronrodIntegrator1D::integral() ERROR: " << a << endl ; return b ;
#define GSL_DBL_MIN        2.2250738585072014e-308
#define GSL_DBL_MAX        1.7976931348623157e+308
#define GSL_DBL_EPSILON    2.2204460492503131e-16
 
#define GSL_EINVAL 2
#define GSL_EMAXITER 3
#define GSL_ESING 4
#define GSL_EFAILED 5
#define GSL_EDIVERGE 6
#define GSL_EROUND 7
 
#define GSL_ERROR_VAL(reason, gsl_errno, value) return value ;
 
#define GSL_MAX(a,b) ((a) > (b) ? (a) : (b))
extern inline double
GSL_MAX_DBL (double a, double b)
{
  return GSL_MAX (a, b);
}
 
double gsl_coerce_double (const double x);
 
double
gsl_coerce_double (const double x)
{
  volatile double y;
  y = x;
  return y;
}
#define GSL_COERCE_DBL(x) (gsl_coerce_double(x))
 
// INCLUDED BELOW #include <gsl/gsl_integration.h>
 
 
/* Workspace for adaptive integrators */
// WVE MOVED TO HEAD OF FILE
 
 
/* Definition of an integration rule */
 
typedef void gsl_integration_rule (const gsl_function * f,
                                   double a, double b,
                                   double *result, double *abserr,
                                   double *defabs, double *resabs);
 
void gsl_integration_qk15 (const gsl_function * f, double a, double b,
                           double *result, double *abserr,
                           double *resabs, double *resasc);
 
void gsl_integration_qk21 (const gsl_function * f, double a, double b,
                           double *result, double *abserr,
                           double *resabs, double *resasc);
 
void gsl_integration_qk31 (const gsl_function * f, double a, double b,
                           double *result, double *abserr,
                           double *resabs, double *resasc);
 
void gsl_integration_qk41 (const gsl_function * f, double a, double b,
                           double *result, double *abserr,
                           double *resabs, double *resasc);
 
void gsl_integration_qk51 (const gsl_function * f, double a, double b,
                           double *result, double *abserr,
                           double *resabs, double *resasc);
 
void gsl_integration_qk61 (const gsl_function * f, double a, double b,
                           double *result, double *abserr,
                           double *resabs, double *resasc);
 
void gsl_integration_qcheb (gsl_function * f, double a, double b,
                            double *cheb12, double *cheb24);
 
/* The low-level integration rules in QUADPACK are identified by small
   integers (1-6). We'll use symbolic constants to refer to them.  */
 
enum
  {
    GSL_INTEG_GAUSS15 = 1,      /* 15 point Gauss-Kronrod rule */
    GSL_INTEG_GAUSS21 = 2,      /* 21 point Gauss-Kronrod rule */
    GSL_INTEG_GAUSS31 = 3,      /* 31 point Gauss-Kronrod rule */
    GSL_INTEG_GAUSS41 = 4,      /* 41 point Gauss-Kronrod rule */
    GSL_INTEG_GAUSS51 = 5,      /* 51 point Gauss-Kronrod rule */
    GSL_INTEG_GAUSS61 = 6       /* 61 point Gauss-Kronrod rule */
  };
 
 
void
gsl_integration_qk (const int n, const double xgk[],
                    const double wg[], const double wgk[],
                    double fv1[], double fv2[],
                    const gsl_function *f, double a, double b,
                    double * result, double * abserr,
                    double * resabs, double * resasc);
 
 
int gsl_integration_qag (const gsl_function * f,
                         double a, double b,
                         double epsabs, double epsrel, size_t limit,
                         int key,
                         gsl_integration_workspace * workspace,
                         double *result, double *abserr);
 
 
// INCLUDED BELOW #include "initialise.c"
static inline
void initialise (gsl_integration_workspace * workspace, double a, double b);
 
static inline
void initialise (gsl_integration_workspace * workspace, double a, double b)
{
  workspace->size = 0;
  workspace->nrmax = 0;
  workspace->i = 0;
  workspace->alist[0] = a;
  workspace->blist[0] = b;
  workspace->rlist[0] = 0.0;
  workspace->elist[0] = 0.0;
  workspace->order[0] = 0;
  workspace->level[0] = 0;
 
  workspace->maximum_level = 0;
}
 
// INCLUDED BELOW #include "set_initial.c"
static inline
void set_initial_result (gsl_integration_workspace * workspace,
                         double result, double error);
 
static inline
void set_initial_result (gsl_integration_workspace * workspace,
                         double result, double error)
{
  workspace->size = 1;
  workspace->rlist[0] = result;
  workspace->elist[0] = error;
}
 
// INCLUDED BELOW #include "qpsrt.c"
static inline void
qpsrt (gsl_integration_workspace * workspace);
 
static inline
void qpsrt (gsl_integration_workspace * workspace)
{
  const size_t last = workspace->size - 1;
  const size_t limit = workspace->limit;
 
  double * elist = workspace->elist;
  size_t * order = workspace->order;
 
  double errmax ;
  double errmin ;
  int i, k, top;
 
  size_t i_nrmax = workspace->nrmax;
  size_t i_maxerr = order[i_nrmax] ;
 
  /* Check whether the list contains more than two error estimates */
 
  if (last < 2)
    {
      order[0] = 0 ;
      order[1] = 1 ;
      workspace->i = i_maxerr ;
      return ;
    }
 
  errmax = elist[i_maxerr] ;
 
  /* This part of the routine is only executed if, due to a difficult
     integrand, subdivision increased the error estimate. In the normal
     case the insert procedure should start after the nrmax-th largest
     error estimate. */
 
  while (i_nrmax > 0 && errmax > elist[order[i_nrmax - 1]])
    {
      order[i_nrmax] = order[i_nrmax - 1] ;
      i_nrmax-- ;
    }
 
  /* Compute the number of elements in the list to be maintained in
     descending order. This number depends on the number of
     subdivisions still allowed. */
 
  if(last < (limit/2 + 2))
    {
      top = last ;
    }
  else
    {
      top = limit - last + 1;
    }
 
  /* Insert errmax by traversing the list top-down, starting
     comparison from the element elist(order(i_nrmax+1)). */
 
  i = i_nrmax + 1 ;
 
  /* The order of the tests in the following line is important to
     prevent a segmentation fault */
 
  while (i < top && errmax < elist[order[i]])
    {
      order[i-1] = order[i] ;
      i++ ;
    }
 
  order[i-1] = i_maxerr ;
 
  /* Insert errmin by traversing the list bottom-up */
 
  errmin = elist[last] ;
 
  k = top - 1 ;
 
  while (k > i - 2 && errmin >= elist[order[k]])
    {
      order[k+1] = order[k] ;
      k-- ;
    }
 
  order[k+1] = last ;
 
  /* Set i_max and e_max */
 
  i_maxerr = order[i_nrmax] ;
 
  workspace->i = i_maxerr ;
  workspace->nrmax = i_nrmax ;
}
 
 
 
// INCLUDED BELOW #include "util.c"
static inline
void update (gsl_integration_workspace * workspace,
                 double a1, double b1, double area1, double error1,
                 double a2, double b2, double area2, double error2);
 
static inline void
retrieve (const gsl_integration_workspace * workspace,
          double * a, double * b, double * r, double * e);
 
 
 
static inline
void update (gsl_integration_workspace * workspace,
             double a1, double b1, double area1, double error1,
             double a2, double b2, double area2, double error2)
{
  double * alist = workspace->alist ;
  double * blist = workspace->blist ;
  double * rlist = workspace->rlist ;
  double * elist = workspace->elist ;
  size_t * level = workspace->level ;
 
  const size_t i_max = workspace->i ;
  const size_t i_new = workspace->size ;
 
  const size_t new_level = workspace->level[i_max] + 1;
 
  /* append the newly-created intervals to the list */
 
  if (error2 > error1)
    {
      alist[i_max] = a2;        /* blist[maxerr] is already == b2 */
      rlist[i_max] = area2;
      elist[i_max] = error2;
      level[i_max] = new_level;
 
      alist[i_new] = a1;
      blist[i_new] = b1;
      rlist[i_new] = area1;
      elist[i_new] = error1;
      level[i_new] = new_level;
    }
  else
    {
      blist[i_max] = b1;        /* alist[maxerr] is already == a1 */
      rlist[i_max] = area1;
      elist[i_max] = error1;
      level[i_max] = new_level;
 
      alist[i_new] = a2;
      blist[i_new] = b2;
      rlist[i_new] = area2;
      elist[i_new] = error2;
      level[i_new] = new_level;
    }
 
  workspace->size++;
 
  if (new_level > workspace->maximum_level)
    {
      workspace->maximum_level = new_level;
    }
 
  qpsrt (workspace) ;
}
 
static inline void
retrieve (const gsl_integration_workspace * workspace,
          double * a, double * b, double * r, double * e)
{
  const size_t i = workspace->i;
  double * alist = workspace->alist;
  double * blist = workspace->blist;
  double * rlist = workspace->rlist;
  double * elist = workspace->elist;
 
  *a = alist[i] ;
  *b = blist[i] ;
  *r = rlist[i] ;
  *e = elist[i] ;
}
 
static inline double
sum_results (const gsl_integration_workspace * workspace);
 
static inline double
sum_results (const gsl_integration_workspace * workspace)
{
  const double * const rlist = workspace->rlist ;
  const size_t n = workspace->size;
 
  size_t k;
  double result_sum = 0;
 
  for (k = 0; k < n; k++)
    {
      result_sum += rlist[k];
    }
 
  return result_sum;
}
 
static inline int
subinterval_too_small (double a1, double a2, double b2);
 
static inline int
subinterval_too_small (double a1, double a2, double b2)
{
  const double e = GSL_DBL_EPSILON;
  const double u = GSL_DBL_MIN;
 
  double tmp = (1 + 100 * e) * (fabs (a2) + 1000 * u);
 
  int status = fabs (a1) <= tmp && fabs (b2) <= tmp;
 
  return status;
}
 
 
static int
qag (const gsl_function *f,
     const double a, const double b,
     const double epsabs, const double epsrel,
     const size_t limit,
     gsl_integration_workspace * workspace,
     double * result, double * abserr,
     gsl_integration_rule * q) ;
 
int
gsl_integration_qag (const gsl_function *f,
                     double a, double b,
                     double epsabs, double epsrel, size_t limit,
                     int key,
                     gsl_integration_workspace * workspace,
                     double * result, double * abserr)
{
  int status ;
  gsl_integration_rule * integration_rule = gsl_integration_qk15 ;
 
  if (key < GSL_INTEG_GAUSS15)
    {
      key = GSL_INTEG_GAUSS15 ;
    }
  else if (key > GSL_INTEG_GAUSS61)
    {
      key = GSL_INTEG_GAUSS61 ;
    }
 
  switch (key)
    {
    case GSL_INTEG_GAUSS15:
      integration_rule = gsl_integration_qk15 ;
      break ;
    case GSL_INTEG_GAUSS21:
      integration_rule = gsl_integration_qk21 ;
      break ;
    case GSL_INTEG_GAUSS31:
      integration_rule = gsl_integration_qk31 ;
      break ;
    case GSL_INTEG_GAUSS41:
      integration_rule = gsl_integration_qk41 ;
      break ;
    case GSL_INTEG_GAUSS51:
      integration_rule = gsl_integration_qk51 ;
      break ;
    case GSL_INTEG_GAUSS61:
      integration_rule = gsl_integration_qk61 ;
      break ;
    }
 
  status = qag (f, a, b, epsabs, epsrel, limit,
                workspace,
                result, abserr,
                integration_rule) ;
 
  return status ;
}
 
static int
qag (const gsl_function * f,
     const double a, const double b,
     const double epsabs, const double epsrel,
     const size_t limit,
     gsl_integration_workspace * workspace,
     double *result, double *abserr,
     gsl_integration_rule * q)
{
  double area, errsum;
  double result0, abserr0, resabs0, resasc0;
  double tolerance;
  size_t iteration = 0;
  int roundoff_type1 = 0, roundoff_type2 = 0, error_type = 0;
 
  double round_off;
 
  /* Initialize results */
 
  initialise (workspace, a, b);
 
  *result = 0;
  *abserr = 0;
 
  if (limit > workspace->limit)
    {
      GSL_ERROR ("iteration limit exceeds available workspace", GSL_EINVAL) ;
    }
 
  if (epsabs <= 0 && (epsrel < 50 * GSL_DBL_EPSILON || epsrel < 0.5e-28))
    {
      GSL_ERROR ("tolerance cannot be acheived with given epsabs and epsrel",
                 GSL_EBADTOL);
    }
 
  /* perform the first integration */
 
  q (f, a, b, &result0, &abserr0, &resabs0, &resasc0);
 
  set_initial_result (workspace, result0, abserr0);
 
  /* Test on accuracy */
 
  tolerance = GSL_MAX_DBL (epsabs, epsrel * fabs (result0));
 
  /* need IEEE rounding here to match original quadpack behavior */
 
  round_off = GSL_COERCE_DBL (50 * GSL_DBL_EPSILON * resabs0);
 
  if (abserr0 <= round_off && abserr0 > tolerance)
    {
      *result = result0;
      *abserr = abserr0;
 
      GSL_ERROR ("cannot reach tolerance because of roundoff error "
                 "on first attempt", GSL_EROUND);
    }
  else if ((abserr0 <= tolerance && abserr0 != resasc0) || abserr0 == 0.0)
    {
      *result = result0;
      *abserr = abserr0;
 
      return GSL_SUCCESS;
    }
  else if (limit == 1)
    {
      *result = result0;
      *abserr = abserr0;
 
      GSL_ERROR ("a maximum of one iteration was insufficient", GSL_EMAXITER);
    }
 
  area = result0;
  errsum = abserr0;
 
  iteration = 1;
 
  do
    {
      double a1, b1, a2, b2;
      double a_i, b_i, r_i, e_i;
      double area1 = 0, area2 = 0, area12 = 0;
      double error1 = 0, error2 = 0, error12 = 0;
      double resasc1, resasc2;
      double resabs1, resabs2;
 
      /* Bisect the subinterval with the largest error estimate */
 
      retrieve (workspace, &a_i, &b_i, &r_i, &e_i);
 
      a1 = a_i;
      b1 = 0.5 * (a_i + b_i);
      a2 = b1;
      b2 = b_i;
 
      q (f, a1, b1, &area1, &error1, &resabs1, &resasc1);
      q (f, a2, b2, &area2, &error2, &resabs2, &resasc2);
 
      area12 = area1 + area2;
      error12 = error1 + error2;
 
      errsum += (error12 - e_i);
      area += area12 - r_i;
 
      if (resasc1 != error1 && resasc2 != error2)
        {
          double delta = r_i - area12;
 
          if (fabs (delta) <= 1.0e-5 * fabs (area12) && error12 >= 0.99 * e_i)
            {
              roundoff_type1++;
            }
          if (iteration >= 10 && error12 > e_i)
            {
              roundoff_type2++;
            }
        }
 
      tolerance = GSL_MAX_DBL (epsabs, epsrel * fabs (area));
 
      if (errsum > tolerance)
        {
          if (roundoff_type1 >= 6 || roundoff_type2 >= 20)
            {
              error_type = 2;   /* round off error */
            }
 
          /* set error flag in the case of bad integrand behaviour at
             a point of the integration range */
 
          if (subinterval_too_small (a1, a2, b2))
            {
              error_type = 3;
            }
        }
 
      update (workspace, a1, b1, area1, error1, a2, b2, area2, error2);
 
      retrieve (workspace, &a_i, &b_i, &r_i, &e_i);
 
      iteration++;
 
    }
  while (iteration < limit && !error_type && errsum > tolerance);
 
  *result = sum_results (workspace);
  *abserr = errsum;
 
  if (errsum <= tolerance)
    {
      return GSL_SUCCESS;
    }
  else if (error_type == 2)
    {
      GSL_ERROR ("roundoff error prevents tolerance from being achieved",
                 GSL_EROUND);
    }
  else if (error_type == 3)
    {
      GSL_ERROR ("bad integrand behavior found in the integration interval",
                 GSL_ESING);
    }
  else if (iteration == limit)
    {
      GSL_ERROR ("maximum number of subdivisions reached", GSL_EMAXITER);
    }
 
  GSL_ERROR ("could not integrate function", GSL_EFAILED);
}
 
 
// INCLUDED BELOW #include "err.c"
static double rescale_error (double err, const double result_abs, const double result_asc) ;
 
static double
rescale_error (double err, const double result_abs, const double result_asc)
{
  err = fabs(err) ;
 
  if (result_asc != 0 && err != 0)
      {
        double scale = TMath::Power((200 * err / result_asc), 1.5) ;
 
        if (scale < 1)
          {
            err = result_asc * scale ;
          }
        else
          {
            err = result_asc ;
          }
      }
  if (result_abs > GSL_DBL_MIN / (50 * GSL_DBL_EPSILON))
    {
      double min_err = 50 * GSL_DBL_EPSILON * result_abs ;
 
      if (min_err > err)
        {
          err = min_err ;
        }
    }
 
  return err ;
}
 
 
void
gsl_integration_qk (const int n,
                    const double xgk[], const double wg[], const double wgk[],
                    double fv1[], double fv2[],
                    const gsl_function * f, double a, double b,
                    double *result, double *abserr,
                    double *resabs, double *resasc)
{
 
  const double center = 0.5 * (a + b);
  const double half_length = 0.5 * (b - a);
  const double abs_half_length = fabs (half_length);
  const double f_center = GSL_FN_EVAL (f, center);
 
  double result_gauss = 0;
  double result_kronrod = f_center * wgk[n - 1];
 
  double result_abs = fabs (result_kronrod);
  double result_asc = 0;
  double mean = 0, err = 0;
 
  int j;
 
  if (n % 2 == 0)
    {
      result_gauss = f_center * wg[n / 2 - 1];
    }
 
  for (j = 0; j < (n - 1) / 2; j++)
    {
      const int jtw = j * 2 + 1;        /* j=1,2,3 jtw=2,4,6 */
      const double abscissa = half_length * xgk[jtw];
      const double fval1 = GSL_FN_EVAL (f, center - abscissa);
      const double fval2 = GSL_FN_EVAL (f, center + abscissa);
      const double fsum = fval1 + fval2;
      fv1[jtw] = fval1;
      fv2[jtw] = fval2;
      result_gauss += wg[j] * fsum;
      result_kronrod += wgk[jtw] * fsum;
      result_abs += wgk[jtw] * (fabs (fval1) + fabs (fval2));
    }
 
  for (j = 0; j < n / 2; j++)
    {
      int jtwm1 = j * 2;
      const double abscissa = half_length * xgk[jtwm1];
      const double fval1 = GSL_FN_EVAL (f, center - abscissa);
      const double fval2 = GSL_FN_EVAL (f, center + abscissa);
      fv1[jtwm1] = fval1;
      fv2[jtwm1] = fval2;
      result_kronrod += wgk[jtwm1] * (fval1 + fval2);
      result_abs += wgk[jtwm1] * (fabs (fval1) + fabs (fval2));
    };
 
  mean = result_kronrod * 0.5;
 
  result_asc = wgk[n - 1] * fabs (f_center - mean);
 
  for (j = 0; j < n - 1; j++)
    {
      result_asc += wgk[j] * (fabs (fv1[j] - mean) + fabs (fv2[j] - mean));
    }
 
  /* scale by the width of the integration region */
 
  err = (result_kronrod - result_gauss) * half_length;
 
  result_kronrod *= half_length;
  result_abs *= abs_half_length;
  result_asc *= abs_half_length;
 
  *result = result_kronrod;
  *resabs = result_abs;
  *resasc = result_asc;
  *abserr = rescale_error (err, result_abs, result_asc);
 
}
 
/* Gauss quadrature weights and kronrod quadrature abscissae and
   weights as evaluated with 80 decimal digit arithmetic by
   L. W. Fullerton, Bell Labs, Nov. 1981. */
 
static const double xgkA[8] =    /* abscissae of the 15-point kronrod rule */
{
  0.991455371120812639206854697526329,
  0.949107912342758524526189684047851,
  0.864864423359769072789712788640926,
  0.741531185599394439863864773280788,
  0.586087235467691130294144838258730,
  0.405845151377397166906606412076961,
  0.207784955007898467600689403773245,
  0.000000000000000000000000000000000
};
 
/* xgk[1], xgk[3], ... abscissae of the 7-point gauss rule.
   xgk[0], xgk[2], ... abscissae to optimally extend the 7-point gauss rule */
 
static const double wgA[4] =     /* weights of the 7-point gauss rule */
{
  0.129484966168869693270611432679082,
  0.279705391489276667901467771423780,
  0.381830050505118944950369775488975,
  0.417959183673469387755102040816327
};
 
static const double wgkA[8] =    /* weights of the 15-point kronrod rule */
{
  0.022935322010529224963732008058970,
  0.063092092629978553290700663189204,
  0.104790010322250183839876322541518,
  0.140653259715525918745189590510238,
  0.169004726639267902826583426598550,
  0.190350578064785409913256402421014,
  0.204432940075298892414161999234649,
  0.209482141084727828012999174891714
};
 
void
gsl_integration_qk15 (const gsl_function * f, double a, double b,
      double *result, double *abserr,
      double *resabs, double *resasc)
{
  double fv1[8], fv2[8];
  // coverity[UNINIT_CTOR]
  gsl_integration_qk (8, xgkA, wgA, wgkA, fv1, fv2, f, a, b, result, abserr, resabs, resasc);
}
 
/* Gauss quadrature weights and kronrod quadrature abscissae and
   weights as evaluated with 80 decimal digit arithmetic by
   L. W. Fullerton, Bell Labs, Nov. 1981. */
 
static const double xgkB[11] =   /* abscissae of the 21-point kronrod rule */
{
  0.995657163025808080735527280689003,
  0.973906528517171720077964012084452,
  0.930157491355708226001207180059508,
  0.865063366688984510732096688423493,
  0.780817726586416897063717578345042,
  0.679409568299024406234327365114874,
  0.562757134668604683339000099272694,
  0.433395394129247190799265943165784,
  0.294392862701460198131126603103866,
  0.148874338981631210884826001129720,
  0.000000000000000000000000000000000
};
 
/* xgk[1], xgk[3], ... abscissae of the 10-point gauss rule.
   xgk[0], xgk[2], ... abscissae to optimally extend the 10-point gauss rule */
 
static const double wgB[5] =     /* weights of the 10-point gauss rule */
{
  0.066671344308688137593568809893332,
  0.149451349150580593145776339657697,
  0.219086362515982043995534934228163,
  0.269266719309996355091226921569469,
  0.295524224714752870173892994651338
};
 
static const double wgkB[11] =   /* weights of the 21-point kronrod rule */
{
  0.011694638867371874278064396062192,
  0.032558162307964727478818972459390,
  0.054755896574351996031381300244580,
  0.075039674810919952767043140916190,
  0.093125454583697605535065465083366,
  0.109387158802297641899210590325805,
  0.123491976262065851077958109831074,
  0.134709217311473325928054001771707,
  0.142775938577060080797094273138717,
  0.147739104901338491374841515972068,
  0.149445554002916905664936468389821
};
 
 
void
gsl_integration_qk21 (const gsl_function * f, double a, double b,
                      double *result, double *abserr,
                      double *resabs, double *resasc)
{
  double fv1[11], fv2[11];
  // coverity[UNINIT_CTOR]
  gsl_integration_qk (11, xgkB, wgB, wgkB, fv1, fv2, f, a, b, result, abserr, resabs, resasc);
}
 
/* Gauss quadrature weights and kronrod quadrature abscissae and
   weights as evaluated with 80 decimal digit arithmetic by
   L. W. Fullerton, Bell Labs, Nov. 1981. */
 
static const double xgkC[16] =   /* abscissae of the 31-point kronrod rule */
{
  0.998002298693397060285172840152271,
  0.987992518020485428489565718586613,
  0.967739075679139134257347978784337,
  0.937273392400705904307758947710209,
  0.897264532344081900882509656454496,
  0.848206583410427216200648320774217,
  0.790418501442465932967649294817947,
  0.724417731360170047416186054613938,
  0.650996741297416970533735895313275,
  0.570972172608538847537226737253911,
  0.485081863640239680693655740232351,
  0.394151347077563369897207370981045,
  0.299180007153168812166780024266389,
  0.201194093997434522300628303394596,
  0.101142066918717499027074231447392,
  0.000000000000000000000000000000000
};
 
/* xgk[1], xgk[3], ... abscissae of the 15-point gauss rule.
   xgk[0], xgk[2], ... abscissae to optimally extend the 15-point gauss rule */
 
static const double wgC[8] =     /* weights of the 15-point gauss rule */
{
  0.030753241996117268354628393577204,
  0.070366047488108124709267416450667,
  0.107159220467171935011869546685869,
  0.139570677926154314447804794511028,
  0.166269205816993933553200860481209,
  0.186161000015562211026800561866423,
  0.198431485327111576456118326443839,
  0.202578241925561272880620199967519
};
 
static const double wgkC[16] =   /* weights of the 31-point kronrod rule */
{
  0.005377479872923348987792051430128,
  0.015007947329316122538374763075807,
  0.025460847326715320186874001019653,
  0.035346360791375846222037948478360,
  0.044589751324764876608227299373280,
  0.053481524690928087265343147239430,
  0.062009567800670640285139230960803,
  0.069854121318728258709520077099147,
  0.076849680757720378894432777482659,
  0.083080502823133021038289247286104,
  0.088564443056211770647275443693774,
  0.093126598170825321225486872747346,
  0.096642726983623678505179907627589,
  0.099173598721791959332393173484603,
  0.100769845523875595044946662617570,
  0.101330007014791549017374792767493
};
 
void
gsl_integration_qk31 (const gsl_function * f, double a, double b,
      double *result, double *abserr,
      double *resabs, double *resasc)
{
  double fv1[16], fv2[16];
  // coverity[UNINIT_CTOR]
  gsl_integration_qk (16, xgkC, wgC, wgkC, fv1, fv2, f, a, b, result, abserr, resabs, resasc);
}
 
/* Gauss quadrature weights and kronrod quadrature abscissae and
   weights as evaluated with 80 decimal digit arithmetic by
   L. W. Fullerton, Bell Labs, Nov. 1981. */
 
static const double xgkD[21] =   /* abscissae of the 41-point kronrod rule */
{
  0.998859031588277663838315576545863,
  0.993128599185094924786122388471320,
  0.981507877450250259193342994720217,
  0.963971927277913791267666131197277,
  0.940822633831754753519982722212443,
  0.912234428251325905867752441203298,
  0.878276811252281976077442995113078,
  0.839116971822218823394529061701521,
  0.795041428837551198350638833272788,
  0.746331906460150792614305070355642,
  0.693237656334751384805490711845932,
  0.636053680726515025452836696226286,
  0.575140446819710315342946036586425,
  0.510867001950827098004364050955251,
  0.443593175238725103199992213492640,
  0.373706088715419560672548177024927,
  0.301627868114913004320555356858592,
  0.227785851141645078080496195368575,
  0.152605465240922675505220241022678,
  0.076526521133497333754640409398838,
  0.000000000000000000000000000000000
};
 
/* xgk[1], xgk[3], ... abscissae of the 20-point gauss rule.
   xgk[0], xgk[2], ... abscissae to optimally extend the 20-point gauss rule */
 
static const double wgD[11] =    /* weights of the 20-point gauss rule */
{
  0.017614007139152118311861962351853,
  0.040601429800386941331039952274932,
  0.062672048334109063569506535187042,
  0.083276741576704748724758143222046,
  0.101930119817240435036750135480350,
  0.118194531961518417312377377711382,
  0.131688638449176626898494499748163,
  0.142096109318382051329298325067165,
  0.149172986472603746787828737001969,
  0.152753387130725850698084331955098
};
 
static const double wgkD[21] =   /* weights of the 41-point kronrod rule */
{
  0.003073583718520531501218293246031,
  0.008600269855642942198661787950102,
  0.014626169256971252983787960308868,
  0.020388373461266523598010231432755,
  0.025882133604951158834505067096153,
  0.031287306777032798958543119323801,
  0.036600169758200798030557240707211,
  0.041668873327973686263788305936895,
  0.046434821867497674720231880926108,
  0.050944573923728691932707670050345,
  0.055195105348285994744832372419777,
  0.059111400880639572374967220648594,
  0.062653237554781168025870122174255,
  0.065834597133618422111563556969398,
  0.068648672928521619345623411885368,
  0.071054423553444068305790361723210,
  0.073030690332786667495189417658913,
  0.074582875400499188986581418362488,
  0.075704497684556674659542775376617,
  0.076377867672080736705502835038061,
  0.076600711917999656445049901530102
};
 
void
gsl_integration_qk41 (const gsl_function * f, double a, double b,
                      double *result, double *abserr,
                      double *resabs, double *resasc)
{
  double fv1[21], fv2[21];
  // coverity[UNINIT]
  gsl_integration_qk (21, xgkD, wgD, wgkD, fv1, fv2, f, a, b, result, abserr, resabs, resasc);
}
 
/* Gauss quadrature weights and kronrod quadrature abscissae and
   weights as evaluated with 80 decimal digit arithmetic by
   L. W. Fullerton, Bell Labs, Nov. 1981. */
 
static const double xgkE[26] =   /* abscissae of the 51-point kronrod rule */
{
  0.999262104992609834193457486540341,
  0.995556969790498097908784946893902,
  0.988035794534077247637331014577406,
  0.976663921459517511498315386479594,
  0.961614986425842512418130033660167,
  0.942974571228974339414011169658471,
  0.920747115281701561746346084546331,
  0.894991997878275368851042006782805,
  0.865847065293275595448996969588340,
  0.833442628760834001421021108693570,
  0.797873797998500059410410904994307,
  0.759259263037357630577282865204361,
  0.717766406813084388186654079773298,
  0.673566368473468364485120633247622,
  0.626810099010317412788122681624518,
  0.577662930241222967723689841612654,
  0.526325284334719182599623778158010,
  0.473002731445714960522182115009192,
  0.417885382193037748851814394594572,
  0.361172305809387837735821730127641,
  0.303089538931107830167478909980339,
  0.243866883720988432045190362797452,
  0.183718939421048892015969888759528,
  0.122864692610710396387359818808037,
  0.061544483005685078886546392366797,
  0.000000000000000000000000000000000
};
 
/* xgk[1], xgk[3], ... abscissae of the 25-point gauss rule.
   xgk[0], xgk[2], ... abscissae to optimally extend the 25-point gauss rule */
 
static const double wgE[13] =    /* weights of the 25-point gauss rule */
{
  0.011393798501026287947902964113235,
  0.026354986615032137261901815295299,
  0.040939156701306312655623487711646,
  0.054904695975835191925936891540473,
  0.068038333812356917207187185656708,
  0.080140700335001018013234959669111,
  0.091028261982963649811497220702892,
  0.100535949067050644202206890392686,
  0.108519624474263653116093957050117,
  0.114858259145711648339325545869556,
  0.119455763535784772228178126512901,
  0.122242442990310041688959518945852,
  0.123176053726715451203902873079050
};
 
static const double wgkE[26] =   /* weights of the 51-point kronrod rule */
{
  0.001987383892330315926507851882843,
  0.005561932135356713758040236901066,
  0.009473973386174151607207710523655,
  0.013236229195571674813656405846976,
  0.016847817709128298231516667536336,
  0.020435371145882835456568292235939,
  0.024009945606953216220092489164881,
  0.027475317587851737802948455517811,
  0.030792300167387488891109020215229,
  0.034002130274329337836748795229551,
  0.037116271483415543560330625367620,
  0.040083825504032382074839284467076,
  0.042872845020170049476895792439495,
  0.045502913049921788909870584752660,
  0.047982537138836713906392255756915,
  0.050277679080715671963325259433440,
  0.052362885806407475864366712137873,
  0.054251129888545490144543370459876,
  0.055950811220412317308240686382747,
  0.057437116361567832853582693939506,
  0.058689680022394207961974175856788,
  0.059720340324174059979099291932562,
  0.060539455376045862945360267517565,
  0.061128509717053048305859030416293,
  0.061471189871425316661544131965264,
  0.061580818067832935078759824240066
};
 
/* wgk[25] was calculated from the values of wgk[0..24] */
 
void
gsl_integration_qk51 (const gsl_function * f, double a, double b,
                      double *result, double *abserr,
                      double *resabs, double *resasc)
{
  double fv1[26], fv2[26];
  //coverity[UNINIT]
  gsl_integration_qk (26, xgkE, wgE, wgkE, fv1, fv2, f, a, b, result, abserr, resabs, resasc);
}
 
/* Gauss quadrature weights and kronrod quadrature abscissae and
   weights as evaluated with 80 decimal digit arithmetic by
   L. W. Fullerton, Bell Labs, Nov. 1981. */
 
static const double xgkF[31] =   /* abscissae of the 61-point kronrod rule */
{
  0.999484410050490637571325895705811,
  0.996893484074649540271630050918695,
  0.991630996870404594858628366109486,
  0.983668123279747209970032581605663,
  0.973116322501126268374693868423707,
  0.960021864968307512216871025581798,
  0.944374444748559979415831324037439,
  0.926200047429274325879324277080474,
  0.905573307699907798546522558925958,
  0.882560535792052681543116462530226,
  0.857205233546061098958658510658944,
  0.829565762382768397442898119732502,
  0.799727835821839083013668942322683,
  0.767777432104826194917977340974503,
  0.733790062453226804726171131369528,
  0.697850494793315796932292388026640,
  0.660061064126626961370053668149271,
  0.620526182989242861140477556431189,
  0.579345235826361691756024932172540,
  0.536624148142019899264169793311073,
  0.492480467861778574993693061207709,
  0.447033769538089176780609900322854,
  0.400401254830394392535476211542661,
  0.352704725530878113471037207089374,
  0.304073202273625077372677107199257,
  0.254636926167889846439805129817805,
  0.204525116682309891438957671002025,
  0.153869913608583546963794672743256,
  0.102806937966737030147096751318001,
  0.051471842555317695833025213166723,
  0.000000000000000000000000000000000
};
 
/* xgk[1], xgk[3], ... abscissae of the 30-point gauss rule.
   xgk[0], xgk[2], ... abscissae to optimally extend the 30-point gauss rule */
 
static const double wgF[15] =    /* weights of the 30-point gauss rule */
{
  0.007968192496166605615465883474674,
  0.018466468311090959142302131912047,
  0.028784707883323369349719179611292,
  0.038799192569627049596801936446348,
  0.048402672830594052902938140422808,
  0.057493156217619066481721689402056,
  0.065974229882180495128128515115962,
  0.073755974737705206268243850022191,
  0.080755895229420215354694938460530,
  0.086899787201082979802387530715126,
  0.092122522237786128717632707087619,
  0.096368737174644259639468626351810,
  0.099593420586795267062780282103569,
  0.101762389748405504596428952168554,
  0.102852652893558840341285636705415
};
 
static const double wgkF[31] =   /* weights of the 61-point kronrod rule */
{
  0.001389013698677007624551591226760,
  0.003890461127099884051267201844516,
  0.006630703915931292173319826369750,
  0.009273279659517763428441146892024,
  0.011823015253496341742232898853251,
  0.014369729507045804812451432443580,
  0.016920889189053272627572289420322,
  0.019414141193942381173408951050128,
  0.021828035821609192297167485738339,
  0.024191162078080601365686370725232,
  0.026509954882333101610601709335075,
  0.028754048765041292843978785354334,
  0.030907257562387762472884252943092,
  0.032981447057483726031814191016854,
  0.034979338028060024137499670731468,
  0.036882364651821229223911065617136,
  0.038678945624727592950348651532281,
  0.040374538951535959111995279752468,
  0.041969810215164246147147541285970,
  0.043452539701356069316831728117073,
  0.044814800133162663192355551616723,
  0.046059238271006988116271735559374,
  0.047185546569299153945261478181099,
  0.048185861757087129140779492298305,
  0.049055434555029778887528165367238,
  0.049795683427074206357811569379942,
  0.050405921402782346840893085653585,
  0.050881795898749606492297473049805,
  0.051221547849258772170656282604944,
  0.051426128537459025933862879215781,
  0.051494729429451567558340433647099
};
 
void
gsl_integration_qk61 (const gsl_function * f, double a, double b,
                      double *result, double *abserr,
                      double *resabs, double *resasc)
{
  double fv1[31], fv2[31];
  //coverity[UNINIT]
  gsl_integration_qk (31, xgkF, wgF, wgkF, fv1, fv2, f, a, b, result, abserr, resabs, resasc);
}
 
gsl_integration_workspace*
gsl_integration_workspace_alloc (const size_t n)
{
  gsl_integration_workspace* w ;
 
  if (n == 0)
    {
      GSL_ERROR_VAL ("workspace length n must be positive integer",
                        GSL_EDOM, 0);
    }
 
  w = (gsl_integration_workspace *)
    malloc (sizeof (gsl_integration_workspace));
 
  if (w == 0)
    {
      GSL_ERROR_VAL ("failed to allocate space for workspace struct",
                        GSL_ENOMEM, 0);
    }
 
  w->alist = (double *) malloc (n * sizeof (double));
 
  if (w->alist == 0)
    {
      free (w);         /* exception in constructor, avoid memory leak */
 
      GSL_ERROR_VAL ("failed to allocate space for alist ranges",
                        GSL_ENOMEM, 0);
    }
 
  w->blist = (double *) malloc (n * sizeof (double));
 
  if (w->blist == 0)
    {
      free (w->alist);
      free (w);         /* exception in constructor, avoid memory leak */
 
      GSL_ERROR_VAL ("failed to allocate space for blist ranges",
                        GSL_ENOMEM, 0);
    }
 
  w->rlist = (double *) malloc (n * sizeof (double));
 
  if (w->rlist == 0)
    {
      free (w->blist);
      free (w->alist);
      free (w);         /* exception in constructor, avoid memory leak */
 
      GSL_ERROR_VAL ("failed to allocate space for rlist ranges",
                        GSL_ENOMEM, 0);
    }
 
 
  w->elist = (double *) malloc (n * sizeof (double));
 
  if (w->elist == 0)
    {
      free (w->rlist);
      free (w->blist);
      free (w->alist);
      free (w);         /* exception in constructor, avoid memory leak */
 
      GSL_ERROR_VAL ("failed to allocate space for elist ranges",
                        GSL_ENOMEM, 0);
    }
 
  w->order = (size_t *) malloc (n * sizeof (size_t));
 
  if (w->order == 0)
    {
      free (w->elist);
      free (w->rlist);
      free (w->blist);
      free (w->alist);
      free (w);         /* exception in constructor, avoid memory leak */
 
      GSL_ERROR_VAL ("failed to allocate space for order ranges",
                        GSL_ENOMEM, 0);
    }
 
  w->level = (size_t *) malloc (n * sizeof (size_t));
 
  if (w->level == 0)
    {
      free (w->order);
      free (w->elist);
      free (w->rlist);
      free (w->blist);
      free (w->alist);
      free (w);         /* exception in constructor, avoid memory leak */
 
      GSL_ERROR_VAL ("failed to allocate space for order ranges",
                        GSL_ENOMEM, 0);
    }
 
  w->size = 0 ;
  w->limit = n ;
  w->maximum_level = 0 ;
 
  return w ;
}
 
void
gsl_integration_workspace_free (gsl_integration_workspace * w)
{
  free (w->level) ;
  free (w->order) ;
  free (w->elist) ;
  free (w->rlist) ;
  free (w->blist) ;
  free (w->alist) ;
  free (w) ;
}
 
 
 
// INCLUDED BELOW #include "reset.c"
static inline void
reset_nrmax (gsl_integration_workspace * workspace);
 
static inline void
reset_nrmax (gsl_integration_workspace * workspace)
{
  workspace->nrmax = 0;
  workspace->i = workspace->order[0] ;
}
 
 
// INCLUDED BELOW #include "qpsrt2.c"
/* The smallest interval has the largest error.  Before bisecting
   decrease the sum of the errors over the larger intervals
   (error_over_large_intervals) and perform extrapolation. */
 
static int
increase_nrmax (gsl_integration_workspace * workspace);
 
static int
increase_nrmax (gsl_integration_workspace * workspace)
{
  int k;
  int id = workspace->nrmax;
  int jupbnd;
 
  const size_t * level = workspace->level;
  const size_t * order = workspace->order;
 
  size_t limit = workspace->limit ;
  size_t last = workspace->size - 1 ;
 
  if (last > (1 + limit / 2))
    {
      jupbnd = limit + 1 - last;
    }
  else
    {
      jupbnd = last;
    }
 
  for (k = id; k <= jupbnd; k++)
    {
      size_t i_max = order[workspace->nrmax];
 
      workspace->i = i_max ;
 
      if (level[i_max] < workspace->maximum_level)
        {
          return 1;
        }
 
      workspace->nrmax++;
 
    }
  return 0;
}
 
static int
large_interval (gsl_integration_workspace * workspace)
{
  size_t i = workspace->i ;
  const size_t * level = workspace->level;
 
  if (level[i] < workspace->maximum_level)
    {
      return 1 ;
    }
  else
    {
      return 0 ;
    }
}
 
 
// INCLUDED BELOW #include "qelg.c"
struct extrapolation_table
  {
    size_t n;
    double rlist2[52];
    size_t nres;
    double res3la[3];
  };
 
static void
  initialise_table (struct extrapolation_table *table);
 
static void
  append_table (struct extrapolation_table *table, double y);
 
static void
initialise_table (struct extrapolation_table *table)
{
  table->n = 0;
  table->nres = 0;
}
#ifdef JUNK
static void
initialise_table (struct extrapolation_table *table, double y)
{
  table->n = 0;
  table->rlist2[0] = y;
  table->nres = 0;
}
#endif
static void
append_table (struct extrapolation_table *table, double y)
{
  size_t n;
  n = table->n;
  table->rlist2[n] = y;
  table->n++;
}
 
/* static inline void
   qelg (size_t * n, double epstab[],
   double * result, double * abserr,
   double res3la[], size_t * nres); */
 
static inline void
  qelg (struct extrapolation_table *table, double *result, double *abserr);
 
static inline void
qelg (struct extrapolation_table *table, double *result, double *abserr)
{
  double *epstab = table->rlist2;
  double *res3la = table->res3la;
  const size_t n = table->n - 1;
 
  const double current = epstab[n];
 
  double absolute = GSL_DBL_MAX;
  double relative = 5 * GSL_DBL_EPSILON * fabs (current);
 
  const size_t newelm = n / 2;
  const size_t n_orig = n;
  size_t n_final = n;
  size_t i;
 
  const size_t nres_orig = table->nres;
 
  *result = current;
  *abserr = GSL_DBL_MAX;
 
  if (n < 2)
    {
      *result = current;
      *abserr = GSL_MAX_DBL (absolute, relative);
      return;
    }
 
  epstab[n + 2] = epstab[n];
  epstab[n] = GSL_DBL_MAX;
 
  for (i = 0; i < newelm; i++)
    {
      double res = epstab[n - 2 * i + 2];
      double e0 = epstab[n - 2 * i - 2];
      double e1 = epstab[n - 2 * i - 1];
      double e2 = res;
 
      double e1abs = fabs (e1);
      double delta2 = e2 - e1;
      double err2 = fabs (delta2);
      double tol2 = GSL_MAX_DBL (fabs (e2), e1abs) * GSL_DBL_EPSILON;
      double delta3 = e1 - e0;
      double err3 = fabs (delta3);
      double tol3 = GSL_MAX_DBL (e1abs, fabs (e0)) * GSL_DBL_EPSILON;
 
      double e3, delta1, err1, tol1, ss;
 
      if (err2 <= tol2 && err3 <= tol3)
        {
          /* If e0, e1 and e2 are equal to within machine accuracy,
             convergence is assumed.  */
 
          *result = res;
          absolute = err2 + err3;
          relative = 5 * GSL_DBL_EPSILON * fabs (res);
          *abserr = GSL_MAX_DBL (absolute, relative);
          return;
        }
 
      e3 = epstab[n - 2 * i];
      epstab[n - 2 * i] = e1;
      delta1 = e1 - e3;
      err1 = fabs (delta1);
      tol1 = GSL_MAX_DBL (e1abs, fabs (e3)) * GSL_DBL_EPSILON;
 
      /* If two elements are very close to each other, omit a part of
         the table by adjusting the value of n */
 
      if (err1 <= tol1 || err2 <= tol2 || err3 <= tol3)
        {
          n_final = 2 * i;
          break;
        }
 
      ss = (1 / delta1 + 1 / delta2) - 1 / delta3;
 
      /* Test to detect irregular behaviour in the table, and
         eventually omit a part of the table by adjusting the value of
         n. */
 
      if (fabs (ss * e1) <= 0.0001)
        {
          n_final = 2 * i;
          break;
        }
 
      /* Compute a new element and eventually adjust the value of
         result. */
 
      res = e1 + 1 / ss;
      epstab[n - 2 * i] = res;
 
      {
        const double error = err2 + fabs (res - e2) + err3;
 
        if (error <= *abserr)
          {
            *abserr = error;
            *result = res;
          }
      }
    }
 
  /* Shift the table */
 
  {
    const size_t limexp = 50 - 1;
 
    if (n_final == limexp)
      {
        n_final = 2 * (limexp / 2);
      }
  }
 
  if (n_orig % 2 == 1)
    {
      for (i = 0; i <= newelm; i++)
        {
          epstab[1 + i * 2] = epstab[i * 2 + 3];
        }
    }
  else
    {
      for (i = 0; i <= newelm; i++)
        {
          epstab[i * 2] = epstab[i * 2 + 2];
        }
    }
 
  if (n_orig != n_final)
    {
      for (i = 0; i <= n_final; i++)
        {
          epstab[i] = epstab[n_orig - n_final + i];
        }
    }
 
  table->n = n_final + 1;
 
  if (nres_orig < 3)
    {
      res3la[nres_orig] = *result;
      *abserr = GSL_DBL_MAX;
    }
  else
    {                           /* Compute error estimate */
      *abserr = (fabs (*result - res3la[2]) + fabs (*result - res3la[1])
                 + fabs (*result - res3la[0]));
 
      res3la[0] = res3la[1];
      res3la[1] = res3la[2];
      res3la[2] = *result;
    }
 
  /* In QUADPACK the variable table->nres is incremented at the top of
     qelg, so it increases on every call. This leads to the array
     res3la being accessed when its elements are still undefined, so I
     have moved the update to this point so that its value more
     useful. */
 
  table->nres = nres_orig + 1;
 
  *abserr = GSL_MAX_DBL (*abserr, 5 * GSL_DBL_EPSILON * fabs (*result));
 
  return;
}
 
 
// INCLUDED BELOW #include "positivity.c"
/* Compare the integral of f(x) with the integral of |f(x)|
   to determine if f(x) covers both positive and negative values */
 
static inline int
test_positivity (double result, double resabs);
 
static inline int
test_positivity (double result, double resabs)
{
  int status = (fabs (result) >= (1 - 50 * GSL_DBL_EPSILON) * resabs);
 
  return status;
}
 
static int qags (const gsl_function * f, const double a, const double
  b, const double epsabs, const double epsrel, const size_t limit,
  gsl_integration_workspace * workspace, double *result, double *abserr,
  gsl_integration_rule * q);
 
int
gsl_integration_qags (const gsl_function *f,
                      double a, double b,
                      double epsabs, double epsrel, size_t limit,
                      gsl_integration_workspace * workspace,
                      double * result, double * abserr)
{
  int status = qags (f, a, b, epsabs, epsrel, limit,
                     workspace,
                     result, abserr,
                     &gsl_integration_qk21) ;
  return status ;
}
 
/* QAGI: evaluate an integral over an infinite range using the
   transformation
 
   integrate(f(x),-Inf,Inf) = integrate((f((1-t)/t) + f(-(1-t)/t))/t^2,0,1)
 
   */
 
static double i_transform (double t, void *params);
 
int
gsl_integration_qagi (gsl_function * f,
                      double epsabs, double epsrel, size_t limit,
                      gsl_integration_workspace * workspace,
                      double *result, double *abserr)
{
  int status;
 
  gsl_function f_transform;
 
  f_transform.function = &i_transform;
  f_transform.params = f;
 
  status = qags (&f_transform, 0.0, 1.0,
                 epsabs, epsrel, limit,
                 workspace,
                 result, abserr,
                 &gsl_integration_qk15);
 
  return status;
}
 
static double
i_transform (double t, void *params)
{
  gsl_function *f = (gsl_function *) params;
  double x = (1 - t) / t;
  double y = GSL_FN_EVAL (f, x) + GSL_FN_EVAL (f, -x);
  return (y / t) / t;
}
 
 
/* QAGIL: Evaluate an integral over an infinite range using the
   transformation,
 
   integrate(f(x),-Inf,b) = integrate(f(b-(1-t)/t)/t^2,0,1)
 
   */
 
struct il_params { double b ; gsl_function * f ; } ;
 
static double il_transform (double t, void *params);
 
int
gsl_integration_qagil (gsl_function * f,
                       double b,
                       double epsabs, double epsrel, size_t limit,
                       gsl_integration_workspace * workspace,
                       double *result, double *abserr)
{
  int status;
 
  gsl_function f_transform;
  struct il_params transform_params  ;
 
  transform_params.b = b ;
  transform_params.f = f ;
 
  f_transform.function = &il_transform;
  f_transform.params = &transform_params;
 
  status = qags (&f_transform, 0.0, 1.0,
                 epsabs, epsrel, limit,
                 workspace,
                 result, abserr,
                 &gsl_integration_qk15);
 
  return status;
}
 
static double
il_transform (double t, void *params)
{
  struct il_params *p = (struct il_params *) params;
  double b = p->b;
  gsl_function * f = p->f;
  double x = b - (1 - t) / t;
  double y = GSL_FN_EVAL (f, x);
  return (y / t) / t;
}
 
/* QAGIU: Evaluate an integral over an infinite range using the
   transformation
 
   integrate(f(x),a,Inf) = integrate(f(a+(1-t)/t)/t^2,0,1)
 
   */
 
struct iu_params { double a ; gsl_function * f ; } ;
 
static double iu_transform (double t, void *params);
 
int
gsl_integration_qagiu (gsl_function * f,
                       double a,
                       double epsabs, double epsrel, size_t limit,
                       gsl_integration_workspace * workspace,
                       double *result, double *abserr)
{
  int status;
 
  gsl_function f_transform;
  struct iu_params transform_params  ;
 
  transform_params.a = a ;
  transform_params.f = f ;
 
  f_transform.function = &iu_transform;
  f_transform.params = &transform_params;
 
  status = qags (&f_transform, 0.0, 1.0,
                 epsabs, epsrel, limit,
                 workspace,
                 result, abserr,
                 &gsl_integration_qk15);
 
  return status;
}
 
static double
iu_transform (double t, void *params)
{
  struct iu_params *p = (struct iu_params *) params;
  double a = p->a;
  gsl_function * f = p->f;
  double x = a + (1 - t) / t;
  double y = GSL_FN_EVAL (f, x);
  return (y / t) / t;
}
 
/* Main integration function */
 
static int
qags (const gsl_function * f,
      const double a, const double b,
      const double epsabs, const double epsrel,
      const size_t limit,
      gsl_integration_workspace * workspace,
      double *result, double *abserr,
      gsl_integration_rule * q)
{
  double area, errsum;
  double res_ext, err_ext;
  double result0, abserr0, resabs0, resasc0;
  double tolerance;
 
  double ertest = 0;
  double error_over_large_intervals = 0;
  double reseps = 0, abseps = 0, correc = 0;
  size_t ktmin = 0;
  int roundoff_type1 = 0, roundoff_type2 = 0, roundoff_type3 = 0;
  int error_type = 0, error_type2 = 0;
 
  size_t iteration = 0;
 
  int positive_integrand = 0;
  int extrapolate = 0;
  int disallow_extrapolation = 0;
 
  struct extrapolation_table table;
 
  /* Initialize results */
 
  initialise (workspace, a, b);
 
  *result = 0;
  *abserr = 0;
 
  if (limit > workspace->limit)
    {
      GSL_ERROR ("iteration limit exceeds available workspace", GSL_EINVAL) ;
    }
 
  /* Test on accuracy */
 
  if (epsabs <= 0 && (epsrel < 50 * GSL_DBL_EPSILON || epsrel < 0.5e-28))
    {
      GSL_ERROR ("tolerance cannot be acheived with given epsabs and epsrel",
                 GSL_EBADTOL);
    }
 
  /* Perform the first integration */
 
  q (f, a, b, &result0, &abserr0, &resabs0, &resasc0);
 
  set_initial_result (workspace, result0, abserr0);
 
  tolerance = GSL_MAX_DBL (epsabs, epsrel * fabs (result0));
 
  if (abserr0 <= 100 * GSL_DBL_EPSILON * resabs0 && abserr0 > tolerance)
    {
      *result = result0;
      *abserr = abserr0;
 
      GSL_ERROR ("cannot reach tolerance because of roundoff error"
                 "on first attempt", GSL_EROUND);
    }
  else if ((abserr0 <= tolerance && abserr0 != resasc0) || abserr0 == 0.0)
    {
      *result = result0;
      *abserr = abserr0;
 
      return GSL_SUCCESS;
    }
  else if (limit == 1)
    {
      *result = result0;
      *abserr = abserr0;
 
      GSL_ERROR ("a maximum of one iteration was insufficient", GSL_EMAXITER);
    }
 
  /* Initialization */
 
  initialise_table (&table);
  append_table (&table, result0);
 
  area = result0;
  errsum = abserr0;
 
  res_ext = result0;
  err_ext = GSL_DBL_MAX;
 
  positive_integrand = test_positivity (result0, resabs0);
 
  iteration = 1;
 
  do
    {
      size_t current_level;
      double a1, b1, a2, b2;
      double a_i, b_i, r_i, e_i;
      double area1 = 0, area2 = 0, area12 = 0;
      double error1 = 0, error2 = 0, error12 = 0;
      double resasc1, resasc2;
      double resabs1, resabs2;
      double last_e_i;
 
      /* Bisect the subinterval with the largest error estimate */
 
      retrieve (workspace, &a_i, &b_i, &r_i, &e_i);
 
      current_level = workspace->level[workspace->i] + 1;
 
      a1 = a_i;
      b1 = 0.5 * (a_i + b_i);
      a2 = b1;
      b2 = b_i;
 
      iteration++;
 
      q (f, a1, b1, &area1, &error1, &resabs1, &resasc1);
      q (f, a2, b2, &area2, &error2, &resabs2, &resasc2);
 
      area12 = area1 + area2;
      error12 = error1 + error2;
      last_e_i = e_i;
 
      /* Improve previous approximations to the integral and test for
         accuracy.
 
         We write these expressions in the same way as the original
         QUADPACK code so that the rounding errors are the same, which
         makes testing easier. */
 
      errsum = errsum + error12 - e_i;
      area = area + area12 - r_i;
 
      tolerance = GSL_MAX_DBL (epsabs, epsrel * fabs (area));
 
      if (resasc1 != error1 && resasc2 != error2)
        {
          double delta = r_i - area12;
 
          if (fabs (delta) <= 1.0e-5 * fabs (area12) && error12 >= 0.99 * e_i)
            {
              if (!extrapolate)
                {
                  roundoff_type1++;
                }
              else
                {
                  roundoff_type2++;
                }
            }
          if (iteration > 10 && error12 > e_i)
            {
              roundoff_type3++;
            }
        }
 
      /* Test for roundoff and eventually set error flag */
 
      if (roundoff_type1 + roundoff_type2 >= 10 || roundoff_type3 >= 20)
        {
          error_type = 2;       /* round off error */
        }
 
      if (roundoff_type2 >= 5)
        {
          error_type2 = 1;
        }
 
      /* set error flag in the case of bad integrand behaviour at
         a point of the integration range */
 
      if (subinterval_too_small (a1, a2, b2))
        {
          error_type = 4;
        }
 
      /* append the newly-created intervals to the list */
 
      update (workspace, a1, b1, area1, error1, a2, b2, area2, error2);
 
      if (errsum <= tolerance)
        {
          goto compute_result;
        }
 
      if (error_type)
        {
          break;
        }
 
      if (iteration >= limit - 1)
        {
          error_type = 1;
          break;
        }
 
      if (iteration == 2)       /* set up variables on first iteration */
        {
          error_over_large_intervals = errsum;
          ertest = tolerance;
          append_table (&table, area);
          continue;
        }
 
      if (disallow_extrapolation)
        {
          continue;
        }
 
      error_over_large_intervals += -last_e_i;
 
      if (current_level < workspace->maximum_level)
        {
          error_over_large_intervals += error12;
        }
 
      if (!extrapolate)
        {
          /* test whether the interval to be bisected next is the
             smallest interval. */
 
          if (large_interval (workspace))
            continue;
 
          extrapolate = 1;
          workspace->nrmax = 1;
        }
 
      if (!error_type2 && error_over_large_intervals > ertest)
        {
          if (increase_nrmax (workspace))
            continue;
        }
 
      /* Perform extrapolation */
 
      append_table (&table, area);
 
      qelg (&table, &reseps, &abseps);
 
      ktmin++;
 
      if (ktmin > 5 && err_ext < 0.001 * errsum)
        {
          error_type = 5;
        }
 
      if (abseps < err_ext)
        {
          ktmin = 0;
          err_ext = abseps;
          res_ext = reseps;
          correc = error_over_large_intervals;
          ertest = GSL_MAX_DBL (epsabs, epsrel * fabs (reseps));
          if (err_ext <= ertest)
            break;
        }
 
      /* Prepare bisection of the smallest interval. */
 
      if (table.n == 1)
        {
          disallow_extrapolation = 1;
        }
 
      if (error_type == 5)
        {
          break;
        }
 
      /* work on interval with largest error */
 
      reset_nrmax (workspace);
      extrapolate = 0;
      error_over_large_intervals = errsum;
 
    }
  while (iteration < limit);
 
  *result = res_ext;
  *abserr = err_ext;
 
  if (err_ext == GSL_DBL_MAX)
    goto compute_result;
 
  if (error_type || error_type2)
    {
      if (error_type2)
        {
          err_ext += correc;
        }
 
//       if (error_type == 0)
//         error_type = 3;
 
      if (res_ext != 0.0 && area != 0.0)
        {
          if (err_ext / fabs (res_ext) > errsum / fabs (area))
            goto compute_result;
        }
      else if (err_ext > errsum)
        {
          goto compute_result;
        }
      else if (area == 0.0)
        {
          goto return_error;
        }
    }
 
  /*  Test on divergence. */
 
  {
    double max_area = GSL_MAX_DBL (fabs (res_ext), fabs (area));
 
    if (!positive_integrand && max_area < 0.01 * resabs0)
      goto return_error;
  }
 
  {
    double ratio = res_ext / area;
 
    if (ratio < 0.01 || ratio > 100.0 || errsum > fabs (area))
      error_type = 6;
  }
 
  goto return_error;
 
compute_result:
 
  *result = sum_results (workspace);
  *abserr = errsum;
 
return_error:
 
  if (error_type > 2)
    error_type--;
 
 
 
  if (error_type == 0)
    {
      return GSL_SUCCESS;
    }
  else if (error_type == 1)
    {
      GSL_ERROR ("number of iterations was insufficient", GSL_EMAXITER);
    }
  else if (error_type == 2)
    {
      GSL_ERROR ("cannot reach tolerance because of roundoff error",
                 GSL_EROUND);
    }
  else if (error_type == 3)
    {
      GSL_ERROR ("bad integrand behavior found in the integration interval",
                 GSL_ESING);
    }
  else if (error_type == 4)
    {
      GSL_ERROR ("roundoff error detected in the extrapolation table",
                 GSL_EROUND);
    }
  else if (error_type == 5)
    {
      GSL_ERROR ("integral is divergent, or slowly convergent",
                 GSL_EDIVERGE);
    }
 
  GSL_ERROR ("could not integrate function", GSL_EFAILED);
}