source: trunk/source/global/HEPNumerics/include/G4Integrator.icc@ 1107

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//
// $Id: G4Integrator.icc,v 1.13 2006/06/29 18:59:47 gunter Exp $
// GEANT4 tag $Name: geant4-09-02-ref-02 $
//
// Implementation of G4Integrator methods. 
//
// 

/////////////////////////////////////////////////////////////////////
//
// Sympson integration method
//
/////////////////////////////////////////////////////////////////////
//
// Integration of class member functions T::f by Simpson method. 

template <class T, class F> 
G4double G4Integrator<T,F>::Simpson( T&       typeT, 
                                     F        f,
                                     G4double xInitial,
                                     G4double xFinal,
                                     G4int    iterationNumber ) 
{
   G4int    i ;
   G4double step = (xFinal - xInitial)/iterationNumber ;
   G4double x = xInitial ;
   G4double xPlus = xInitial + 0.5*step ;
   G4double mean = ( (typeT.*f)(xInitial) + (typeT.*f)(xFinal) )*0.5 ;
   G4double sum = (typeT.*f)(xPlus) ;

   for(i=1;i<iterationNumber;i++)
   {
      x     += step ;
      xPlus += step ;
      mean  += (typeT.*f)(x) ;
      sum   += (typeT.*f)(xPlus) ;
   }
   mean += 2.0*sum ;

   return mean*step/3.0 ;   
}

/////////////////////////////////////////////////////////////////////
//
// Integration of class member functions T::f by Simpson method.
// Convenient to use with 'this' pointer

template <class T, class F> 
G4double G4Integrator<T,F>::Simpson( T*       ptrT, 
                                F        f,
                                G4double xInitial,
                                G4double xFinal,
                                G4int    iterationNumber ) 
{
   G4int    i ;
   G4double step = (xFinal - xInitial)/iterationNumber ;
   G4double x = xInitial ;
   G4double xPlus = xInitial + 0.5*step ;
   G4double mean = ( (ptrT->*f)(xInitial) + (ptrT->*f)(xFinal) )*0.5 ;
   G4double sum = (ptrT->*f)(xPlus) ;

   for(i=1;i<iterationNumber;i++)
   {
      x     += step ;
      xPlus += step ;
      mean  += (ptrT->*f)(x) ;
      sum   += (ptrT->*f)(xPlus) ;
   }
   mean += 2.0*sum ;

   return mean*step/3.0 ;   
}

/////////////////////////////////////////////////////////////////////
//
// Integration of class member functions T::f by Simpson method.
// Convenient to use, when function f is defined in global scope, i.e. in main()
// program

template <class T, class F> 
G4double G4Integrator<T,F>::Simpson( G4double (*f)(G4double),
                                G4double xInitial,
                                G4double xFinal,
                                G4int    iterationNumber ) 
{
   G4int    i ;
   G4double step = (xFinal - xInitial)/iterationNumber ;
   G4double x = xInitial ;
   G4double xPlus = xInitial + 0.5*step ;
   G4double mean = ( (*f)(xInitial) + (*f)(xFinal) )*0.5 ;
   G4double sum = (*f)(xPlus) ;

   for(i=1;i<iterationNumber;i++)
   {
      x     += step ;
      xPlus += step ;
      mean  += (*f)(x) ;
      sum   += (*f)(xPlus) ;
   }
   mean += 2.0*sum ;

   return mean*step/3.0 ;   
}

//////////////////////////////////////////////////////////////////////////
//
// Adaptive Gauss method
//
//////////////////////////////////////////////////////////////////////////
//
//

template <class T, class F> 
G4double G4Integrator<T,F>::Gauss( T& typeT, F f,
                                   G4double xInitial, G4double xFinal   ) 
{
   static G4double root = 1.0/std::sqrt(3.0) ;
   
   G4double xMean = (xInitial + xFinal)/2.0 ;
   G4double Step = (xFinal - xInitial)/2.0 ;
   G4double delta = Step*root ;
   G4double sum = ((typeT.*f)(xMean + delta) + 
                   (typeT.*f)(xMean - delta)) ;
   
   return sum*Step ;   
}

//////////////////////////////////////////////////////////////////////
//
//

template <class T, class F> G4double 
G4Integrator<T,F>::Gauss( T* ptrT, F f, G4double a, G4double b )
{
  return Gauss(*ptrT,f,a,b) ;
}

///////////////////////////////////////////////////////////////////////
//
//

template <class T, class F>
G4double G4Integrator<T,F>::Gauss( G4double (*f)(G4double), 
                              G4double xInitial, G4double xFinal) 
{
   static G4double root = 1.0/std::sqrt(3.0) ;
   
   G4double xMean = (xInitial + xFinal)/2.0 ;
   G4double Step  = (xFinal - xInitial)/2.0 ;
   G4double delta = Step*root ;
   G4double sum   = ( (*f)(xMean + delta) + (*f)(xMean - delta) ) ;
   
   return sum*Step ;   
}

///////////////////////////////////////////////////////////////////////////
//
//

template <class T, class F>  
void G4Integrator<T,F>::AdaptGauss( T& typeT, F f, G4double  xInitial,
                               G4double  xFinal, G4double fTolerance,
                               G4double& sum,
                               G4int&    depth      ) 
{
   if(depth > 100)
   {
     G4cout<<"G4Integrator<T,F>::AdaptGauss: WARNING !!!"<<G4endl  ;
     G4cout<<"Function varies too rapidly to get stated accuracy in 100 steps "
           <<G4endl ;

     return ;
   }
   G4double xMean = (xInitial + xFinal)/2.0 ;
   G4double leftHalf  = Gauss(typeT,f,xInitial,xMean) ;
   G4double rightHalf = Gauss(typeT,f,xMean,xFinal) ;
   G4double full = Gauss(typeT,f,xInitial,xFinal) ;
   if(std::fabs(leftHalf+rightHalf-full) < fTolerance)
   {
      sum += full ;
   }
   else
   {
      depth++ ;
      AdaptGauss(typeT,f,xInitial,xMean,fTolerance,sum,depth) ;
      AdaptGauss(typeT,f,xMean,xFinal,fTolerance,sum,depth) ;
   }
}

template <class T, class F>  
void G4Integrator<T,F>::AdaptGauss( T* ptrT, F f, G4double  xInitial,
                               G4double  xFinal, G4double fTolerance,
                               G4double& sum,
                               G4int&    depth      ) 
{
  AdaptGauss(*ptrT,f,xInitial,xFinal,fTolerance,sum,depth) ;
}

/////////////////////////////////////////////////////////////////////////
//
//
template <class T, class F>
void G4Integrator<T,F>::AdaptGauss( G4double (*f)(G4double), 
                               G4double xInitial, G4double xFinal, 
                               G4double fTolerance, G4double& sum, 
                               G4int& depth ) 
{
   if(depth > 100)
   {
     G4cout<<"G4SimpleIntegration::AdaptGauss: WARNING !!!"<<G4endl  ;
     G4cout<<"Function varies too rapidly to get stated accuracy in 100 steps "
           <<G4endl ;

     return ;
   }
   G4double xMean = (xInitial + xFinal)/2.0 ;
   G4double leftHalf  = Gauss(f,xInitial,xMean) ;
   G4double rightHalf = Gauss(f,xMean,xFinal) ;
   G4double full = Gauss(f,xInitial,xFinal) ;
   if(std::fabs(leftHalf+rightHalf-full) < fTolerance)
   {
      sum += full ;
   }
   else
   {
      depth++ ;
      AdaptGauss(f,xInitial,xMean,fTolerance,sum,depth) ;
      AdaptGauss(f,xMean,xFinal,fTolerance,sum,depth) ;
   }
}

////////////////////////////////////////////////////////////////////////
//
// Adaptive Gauss integration with accuracy 'e'
// Convenient for using with class object typeT
       
template<class T, class F>
G4double G4Integrator<T,F>::AdaptiveGauss(  T& typeT, F f, G4double xInitial,
                                            G4double xFinal, G4double e   ) 
{
   G4int depth = 0 ;
   G4double sum = 0.0 ;
   AdaptGauss(typeT,f,xInitial,xFinal,e,sum,depth) ;
   return sum ;
}

////////////////////////////////////////////////////////////////////////
//
// Adaptive Gauss integration with accuracy 'e'
// Convenient for using with 'this' pointer
       
template<class T, class F>
G4double G4Integrator<T,F>::AdaptiveGauss(  T* ptrT, F f, G4double xInitial,
                                            G4double xFinal, G4double e   ) 
{
  return AdaptiveGauss(*ptrT,f,xInitial,xFinal,e) ;
}

////////////////////////////////////////////////////////////////////////
//
// Adaptive Gauss integration with accuracy 'e'
// Convenient for using with global scope function f
       
template <class T, class F>
G4double G4Integrator<T,F>::AdaptiveGauss( G4double (*f)(G4double), 
                            G4double xInitial, G4double xFinal, G4double e ) 
{
   G4int depth = 0 ;
   G4double sum = 0.0 ;
   AdaptGauss(f,xInitial,xFinal,e,sum,depth) ;
   return sum ;
}

////////////////////////////////////////////////////////////////////////////
// Gauss integration methods involving ortogonal polynomials
////////////////////////////////////////////////////////////////////////////
//
// Methods involving Legendre polynomials  
//
/////////////////////////////////////////////////////////////////////////
//
// The value nLegendre set the accuracy required, i.e the number of points
// where the function pFunction will be evaluated during integration.
// The function creates the arrays for abscissas and weights that used 
// in Gauss-Legendre quadrature method. 
// The values a and b are the limits of integration of the function  f .
// nLegendre MUST BE EVEN !!!
// Returns the integral of the function f between a and b, by 2*fNumber point 
// Gauss-Legendre integration: the function is evaluated exactly
// 2*fNumber times at interior points in the range of integration. 
// Since the weights and abscissas are, in this case, symmetric around 
// the midpoint of the range of integration, there are actually only 
// fNumber distinct values of each.
// Convenient for using with some class object dataT

template <class T, class F>
G4double G4Integrator<T,F>::Legendre( T& typeT, F f, G4double a, G4double b,
                                      G4int nLegendre )
{
   G4double newton, newton1, temp1, temp2, temp3, temp ;
   G4double xDiff, xMean, dx, integral ;

   const G4double tolerance = 1.6e-10 ;
   G4int i, j,   k = nLegendre ;
   G4int fNumber = (nLegendre + 1)/2 ;

   if(2*fNumber != k)
   {
      G4Exception("G4Integrator<T,F>::Legendre(T&,F, ...)", "InvalidCall",
                  FatalException, "Invalid (odd) nLegendre in constructor.");
   }

   G4double* fAbscissa = new G4double[fNumber] ;
   G4double* fWeight   = new G4double[fNumber] ;
      
   for(i=1;i<=fNumber;i++)      // Loop over the desired roots
   {
      newton = std::cos(pi*(i - 0.25)/(k + 0.5)) ;  // Initial root approximation

      do     // loop of Newton's method  
      {                           
         temp1 = 1.0 ;
         temp2 = 0.0 ;
         for(j=1;j<=k;j++)
         {
            temp3 = temp2 ;
            temp2 = temp1 ;
            temp1 = ((2.0*j - 1.0)*newton*temp2 - (j - 1.0)*temp3)/j ;
         }
         temp = k*(newton*temp1 - temp2)/(newton*newton - 1.0) ;
         newton1 = newton ;
         newton  = newton1 - temp1/temp ;       // Newton's method
      }
      while(std::fabs(newton - newton1) > tolerance) ;
         
      fAbscissa[fNumber-i] =  newton ;
      fWeight[fNumber-i] = 2.0/((1.0 - newton*newton)*temp*temp) ;
   }

   //
   // Now we ready to get integral 
   //
   
   xMean = 0.5*(a + b) ;
   xDiff = 0.5*(b - a) ;
   integral = 0.0 ;
   for(i=0;i<fNumber;i++)
   {
      dx = xDiff*fAbscissa[i] ;
      integral += fWeight[i]*( (typeT.*f)(xMean + dx) + 
                               (typeT.*f)(xMean - dx)    ) ;
   }
   delete[] fAbscissa;
   delete[] fWeight;
   return integral *= xDiff ;
} 

///////////////////////////////////////////////////////////////////////
//
// Convenient for using with the pointer 'this'

template <class T, class F>
G4double G4Integrator<T,F>::Legendre( T* ptrT, F f, G4double a,
                                      G4double b, G4int nLegendre ) 
{
  return Legendre(*ptrT,f,a,b,nLegendre) ;
}

///////////////////////////////////////////////////////////////////////
//
// Convenient for using with global scope function f

template <class T, class F>
G4double G4Integrator<T,F>::Legendre( G4double (*f)(G4double),
                          G4double a, G4double b, G4int nLegendre) 
{
   G4double newton, newton1, temp1, temp2, temp3, temp ;
   G4double xDiff, xMean, dx, integral ;

   const G4double tolerance = 1.6e-10 ;
   G4int i, j,   k = nLegendre ;
   G4int fNumber = (nLegendre + 1)/2 ;

   if(2*fNumber != k)
   {
      G4Exception("G4Integrator<T,F>::Legendre(...)", "InvalidCall",
                  FatalException, "Invalid (odd) nLegendre in constructor.");
   }

   G4double* fAbscissa = new G4double[fNumber] ;
   G4double* fWeight   = new G4double[fNumber] ;
      
   for(i=1;i<=fNumber;i++)      // Loop over the desired roots
   {
      newton = std::cos(pi*(i - 0.25)/(k + 0.5)) ;  // Initial root approximation

      do     // loop of Newton's method  
      {                           
         temp1 = 1.0 ;
         temp2 = 0.0 ;
         for(j=1;j<=k;j++)
         {
            temp3 = temp2 ;
            temp2 = temp1 ;
            temp1 = ((2.0*j - 1.0)*newton*temp2 - (j - 1.0)*temp3)/j ;
         }
         temp = k*(newton*temp1 - temp2)/(newton*newton - 1.0) ;
         newton1 = newton ;
         newton  = newton1 - temp1/temp ;       // Newton's method
      }
      while(std::fabs(newton - newton1) > tolerance) ;
         
      fAbscissa[fNumber-i] =  newton ;
      fWeight[fNumber-i] = 2.0/((1.0 - newton*newton)*temp*temp) ;
   }

   //
   // Now we ready to get integral 
   //
   
   xMean = 0.5*(a + b) ;
   xDiff = 0.5*(b - a) ;
   integral = 0.0 ;
   for(i=0;i<fNumber;i++)
   {
      dx = xDiff*fAbscissa[i] ;
      integral += fWeight[i]*( (*f)(xMean + dx) + (*f)(xMean - dx)    ) ;
   }
   delete[] fAbscissa;
   delete[] fWeight;

   return integral *= xDiff ;
} 

////////////////////////////////////////////////////////////////////////////
//
// Returns the integral of the function to be pointed by T::f between a and b,
// by ten point Gauss-Legendre integration: the function is evaluated exactly
// ten times at interior points in the range of integration. Since the weights
// and abscissas are, in this case, symmetric around the midpoint of the 
// range of integration, there are actually only five distinct values of each
// Convenient for using with class object typeT

template <class T, class F>  
G4double G4Integrator<T,F>::Legendre10( T& typeT, F f,G4double a, G4double b) 
{
   G4int i ;
   G4double xDiff, xMean, dx, integral ;
   
   // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 916
   
   static G4double abscissa[] = { 0.148874338981631, 0.433395394129247,
                                  0.679409568299024, 0.865063366688985,
                                  0.973906528517172                      } ;
   
   static G4double weight[] =   { 0.295524224714753, 0.269266719309996, 
                                  0.219086362515982, 0.149451349150581,
                                  0.066671344308688                      } ;
   xMean = 0.5*(a + b) ;
   xDiff = 0.5*(b - a) ;
   integral = 0.0 ;
   for(i=0;i<5;i++)
   {
     dx = xDiff*abscissa[i] ;
     integral += weight[i]*( (typeT.*f)(xMean + dx) + (typeT.*f)(xMean - dx)) ;
   }
   return integral *= xDiff ;
}

///////////////////////////////////////////////////////////////////////////
//
// Convenient for using with the pointer 'this'

template <class T, class F>  
G4double G4Integrator<T,F>::Legendre10( T* ptrT, F f,G4double a, G4double b)
{
  return Legendre10(*ptrT,f,a,b) ;
} 

//////////////////////////////////////////////////////////////////////////
//
// Convenient for using with global scope functions

template <class T, class F>
G4double G4Integrator<T,F>::Legendre10( G4double (*f)(G4double),
                                        G4double a, G4double b ) 
{
   G4int i ;
   G4double xDiff, xMean, dx, integral ;
   
   // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 916
   
   static G4double abscissa[] = { 0.148874338981631, 0.433395394129247,
                                  0.679409568299024, 0.865063366688985,
                                  0.973906528517172                      } ;
   
   static G4double weight[] =   { 0.295524224714753, 0.269266719309996, 
                                  0.219086362515982, 0.149451349150581,
                                  0.066671344308688                      } ;
   xMean = 0.5*(a + b) ;
   xDiff = 0.5*(b - a) ;
   integral = 0.0 ;
   for(i=0;i<5;i++)
   {
     dx = xDiff*abscissa[i] ;
     integral += weight[i]*( (*f)(xMean + dx) + (*f)(xMean - dx)) ;
   }
   return integral *= xDiff ;
}

///////////////////////////////////////////////////////////////////////
//
// Returns the integral of the function to be pointed by T::f between a and b,
// by 96 point Gauss-Legendre integration: the function is evaluated exactly
// ten Times at interior points in the range of integration. Since the weights
// and abscissas are, in this case, symmetric around the midpoint of the 
// range of integration, there are actually only five distinct values of each
// Convenient for using with some class object typeT

template <class T, class F>  
G4double G4Integrator<T,F>::Legendre96( T& typeT, F f,G4double a, G4double b) 
{
   G4int i ;
   G4double xDiff, xMean, dx, integral ;
   
   // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 919
   
   static G4double 
   abscissa[] = { 
                  0.016276744849602969579, 0.048812985136049731112,
                  0.081297495464425558994, 0.113695850110665920911,
                  0.145973714654896941989, 0.178096882367618602759,  // 6
                           
                  0.210031310460567203603, 0.241743156163840012328,
                  0.273198812591049141487, 0.304364944354496353024,
                  0.335208522892625422616, 0.365696861472313635031,  // 12
                           
                  0.395797649828908603285, 0.425478988407300545365,
                  0.454709422167743008636, 0.483457973920596359768,
                  0.511694177154667673586, 0.539388108324357436227,  // 18
                           
                  0.566510418561397168404, 0.593032364777572080684,
                  0.618925840125468570386, 0.644163403784967106798,
                  0.668718310043916153953, 0.692564536642171561344,  // 24
                           
                  0.715676812348967626225, 0.738030643744400132851,
                  0.759602341176647498703, 0.780369043867433217604,
                  0.800308744139140817229, 0.819400310737931675539,  // 30
                           
                  0.837623511228187121494, 0.854959033434601455463,
                  0.871388505909296502874, 0.886894517402420416057,
                  0.901460635315852341319, 0.915071423120898074206,  // 36
                           
                  0.927712456722308690965, 0.939370339752755216932,
                  0.950032717784437635756, 0.959688291448742539300,
                  0.968326828463264212174, 0.975939174585136466453,  // 42
                           
                  0.982517263563014677447, 0.988054126329623799481,
                  0.992543900323762624572, 0.995981842987209290650,
                  0.998364375863181677724, 0.999689503883230766828   // 48
                                                                            } ;
   
   static G4double 
   weight[] = {  
                  0.032550614492363166242, 0.032516118713868835987,
                  0.032447163714064269364, 0.032343822568575928429,
                  0.032206204794030250669, 0.032034456231992663218,  // 6
                           
                  0.031828758894411006535, 0.031589330770727168558,
                  0.031316425596862355813, 0.031010332586313837423,
                  0.030671376123669149014, 0.030299915420827593794,  // 12
                           
                  0.029896344136328385984, 0.029461089958167905970,
                  0.028994614150555236543, 0.028497411065085385646,
                  0.027970007616848334440, 0.027412962726029242823,  // 18
                           
                  0.026826866725591762198, 0.026212340735672413913,
                  0.025570036005349361499, 0.024900633222483610288,
                  0.024204841792364691282, 0.023483399085926219842,  // 24
                           
                  0.022737069658329374001, 0.021966644438744349195,
                  0.021172939892191298988, 0.020356797154333324595,
                  0.019519081140145022410, 0.018660679627411467385,  // 30
                           
                  0.017782502316045260838, 0.016885479864245172450,
                  0.015970562902562291381, 0.015038721026994938006,
                  0.014090941772314860916, 0.013128229566961572637,  // 36
                           
                  0.012151604671088319635, 0.011162102099838498591,
                  0.010160770535008415758, 0.009148671230783386633,
                  0.008126876925698759217, 0.007096470791153865269,  // 42
                           
                  0.006058545504235961683, 0.005014202742927517693,
                  0.003964554338444686674, 0.002910731817934946408,
                  0.001853960788946921732, 0.000796792065552012429   // 48
                                                                            } ;
   xMean = 0.5*(a + b) ;
   xDiff = 0.5*(b - a) ;
   integral = 0.0 ;
   for(i=0;i<48;i++)
   {
      dx = xDiff*abscissa[i] ;
      integral += weight[i]*((typeT.*f)(xMean + dx) + (typeT.*f)(xMean - dx)) ;
   }
   return integral *= xDiff ;
}

///////////////////////////////////////////////////////////////////////
//
// Convenient for using with the pointer 'this'

template <class T, class F>  
G4double G4Integrator<T,F>::Legendre96( T* ptrT, F f,G4double a, G4double b)
{
  return Legendre96(*ptrT,f,a,b) ;
} 

///////////////////////////////////////////////////////////////////////
//
// Convenient for using with global scope function f 

template <class T, class F>
G4double G4Integrator<T,F>::Legendre96( G4double (*f)(G4double),
                                        G4double a, G4double b ) 
{
   G4int i ;
   G4double xDiff, xMean, dx, integral ;
   
   // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 919
   
   static G4double 
   abscissa[] = { 
                  0.016276744849602969579, 0.048812985136049731112,
                  0.081297495464425558994, 0.113695850110665920911,
                  0.145973714654896941989, 0.178096882367618602759,  // 6
                           
                  0.210031310460567203603, 0.241743156163840012328,
                  0.273198812591049141487, 0.304364944354496353024,
                  0.335208522892625422616, 0.365696861472313635031,  // 12
                           
                  0.395797649828908603285, 0.425478988407300545365,
                  0.454709422167743008636, 0.483457973920596359768,
                  0.511694177154667673586, 0.539388108324357436227,  // 18
                           
                  0.566510418561397168404, 0.593032364777572080684,
                  0.618925840125468570386, 0.644163403784967106798,
                  0.668718310043916153953, 0.692564536642171561344,  // 24
                           
                  0.715676812348967626225, 0.738030643744400132851,
                  0.759602341176647498703, 0.780369043867433217604,
                  0.800308744139140817229, 0.819400310737931675539,  // 30
                           
                  0.837623511228187121494, 0.854959033434601455463,
                  0.871388505909296502874, 0.886894517402420416057,
                  0.901460635315852341319, 0.915071423120898074206,  // 36
                           
                  0.927712456722308690965, 0.939370339752755216932,
                  0.950032717784437635756, 0.959688291448742539300,
                  0.968326828463264212174, 0.975939174585136466453,  // 42
                           
                  0.982517263563014677447, 0.988054126329623799481,
                  0.992543900323762624572, 0.995981842987209290650,
                  0.998364375863181677724, 0.999689503883230766828   // 48
                                                                            } ;
   
   static G4double 
   weight[] = {  
                  0.032550614492363166242, 0.032516118713868835987,
                  0.032447163714064269364, 0.032343822568575928429,
                  0.032206204794030250669, 0.032034456231992663218,  // 6
                           
                  0.031828758894411006535, 0.031589330770727168558,
                  0.031316425596862355813, 0.031010332586313837423,
                  0.030671376123669149014, 0.030299915420827593794,  // 12
                           
                  0.029896344136328385984, 0.029461089958167905970,
                  0.028994614150555236543, 0.028497411065085385646,
                  0.027970007616848334440, 0.027412962726029242823,  // 18
                           
                  0.026826866725591762198, 0.026212340735672413913,
                  0.025570036005349361499, 0.024900633222483610288,
                  0.024204841792364691282, 0.023483399085926219842,  // 24
                           
                  0.022737069658329374001, 0.021966644438744349195,
                  0.021172939892191298988, 0.020356797154333324595,
                  0.019519081140145022410, 0.018660679627411467385,  // 30
                           
                  0.017782502316045260838, 0.016885479864245172450,
                  0.015970562902562291381, 0.015038721026994938006,
                  0.014090941772314860916, 0.013128229566961572637,  // 36
                           
                  0.012151604671088319635, 0.011162102099838498591,
                  0.010160770535008415758, 0.009148671230783386633,
                  0.008126876925698759217, 0.007096470791153865269,  // 42
                           
                  0.006058545504235961683, 0.005014202742927517693,
                  0.003964554338444686674, 0.002910731817934946408,
                  0.001853960788946921732, 0.000796792065552012429   // 48
                                                                            } ;
   xMean = 0.5*(a + b) ;
   xDiff = 0.5*(b - a) ;
   integral = 0.0 ;
   for(i=0;i<48;i++)
   {
      dx = xDiff*abscissa[i] ;
      integral += weight[i]*((*f)(xMean + dx) + (*f)(xMean - dx)) ;
   }
   return integral *= xDiff ;
}

//////////////////////////////////////////////////////////////////////////////
//
// Methods involving Chebyshev polynomials 
//
///////////////////////////////////////////////////////////////////////////
//
// Integrates function pointed by T::f from a to b by Gauss-Chebyshev 
// quadrature method.
// Convenient for using with class object typeT

template <class T, class F>
G4double G4Integrator<T,F>::Chebyshev( T& typeT, F f, G4double a, 
                                       G4double b, G4int nChebyshev ) 
{
   G4int i ;
   G4double xDiff, xMean, dx, integral = 0.0 ;
   
   G4int fNumber = nChebyshev  ;   // Try to reduce fNumber twice ??
   G4double cof = pi/fNumber ;
   G4double* fAbscissa = new G4double[fNumber] ;
   G4double* fWeight   = new G4double[fNumber] ;
   for(i=0;i<fNumber;i++)
   {
      fAbscissa[i] = std::cos(cof*(i + 0.5)) ;
      fWeight[i] = cof*std::sqrt(1 - fAbscissa[i]*fAbscissa[i]) ;
   }

   //
   // Now we ready to estimate the integral
   //

   xMean = 0.5*(a + b) ;
   xDiff = 0.5*(b - a) ;
   for(i=0;i<fNumber;i++)
   {
      dx = xDiff*fAbscissa[i] ;
      integral += fWeight[i]*(typeT.*f)(xMean + dx)  ;
   }
   delete[] fAbscissa;
   delete[] fWeight;
   return integral *= xDiff ;
}

///////////////////////////////////////////////////////////////////////
//
// Convenient for using with 'this' pointer

template <class T, class F>
G4double G4Integrator<T,F>::Chebyshev( T* ptrT, F f, G4double a,
                                       G4double b, G4int n )
{
  return Chebyshev(*ptrT,f,a,b,n) ;
} 

////////////////////////////////////////////////////////////////////////
//
// For use with global scope functions f 

template <class T, class F>
G4double G4Integrator<T,F>::Chebyshev( G4double (*f)(G4double), 
                           G4double a, G4double b, G4int nChebyshev ) 
{
   G4int i ;
   G4double xDiff, xMean, dx, integral = 0.0 ;
   
   G4int fNumber = nChebyshev  ;   // Try to reduce fNumber twice ??
   G4double cof = pi/fNumber ;
   G4double* fAbscissa = new G4double[fNumber] ;
   G4double* fWeight   = new G4double[fNumber] ;
   for(i=0;i<fNumber;i++)
   {
      fAbscissa[i] = std::cos(cof*(i + 0.5)) ;
      fWeight[i] = cof*std::sqrt(1 - fAbscissa[i]*fAbscissa[i]) ;
   }

   //
   // Now we ready to estimate the integral
   //

   xMean = 0.5*(a + b) ;
   xDiff = 0.5*(b - a) ;
   for(i=0;i<fNumber;i++)
   {
      dx = xDiff*fAbscissa[i] ;
      integral += fWeight[i]*(*f)(xMean + dx)  ;
   }
   delete[] fAbscissa;
   delete[] fWeight;
   return integral *= xDiff ;
}

//////////////////////////////////////////////////////////////////////
//
// Method involving Laguerre polynomials
//
//////////////////////////////////////////////////////////////////////
//
// Integral from zero to infinity of std::pow(x,alpha)*std::exp(-x)*f(x). 
// The value of nLaguerre sets the accuracy.
// The function creates arrays fAbscissa[0,..,nLaguerre-1] and 
// fWeight[0,..,nLaguerre-1] . 
// Convenient for using with class object 'typeT' and (typeT.*f) function
// (T::f)

template <class T, class F>
G4double G4Integrator<T,F>::Laguerre( T& typeT, F f, G4double alpha,
                                      G4int nLaguerre ) 
{
   const G4double tolerance = 1.0e-10 ;
   const G4int maxNumber = 12 ;
   G4int i, j, k ;
   G4double newton=0., newton1, temp1, temp2, temp3, temp, cofi ;
   G4double integral = 0.0 ;

   G4int fNumber = nLaguerre ;
   G4double* fAbscissa = new G4double[fNumber] ;
   G4double* fWeight   = new G4double[fNumber] ;
      
   for(i=1;i<=fNumber;i++)      // Loop over the desired roots
   {
      if(i == 1)
      {
         newton = (1.0 + alpha)*(3.0 + 0.92*alpha)
                / (1.0 + 2.4*fNumber + 1.8*alpha) ;
      }
      else if(i == 2)
      {
         newton += (15.0 + 6.25*alpha)/(1.0 + 0.9*alpha + 2.5*fNumber) ;
      }
      else
      {
         cofi = i - 2 ;
         newton += ((1.0+2.55*cofi)/(1.9*cofi)
                 + 1.26*cofi*alpha/(1.0+3.5*cofi))
                 * (newton - fAbscissa[i-3])/(1.0 + 0.3*alpha) ;
      }
      for(k=1;k<=maxNumber;k++)
      {
         temp1 = 1.0 ;
         temp2 = 0.0 ;

         for(j=1;j<=fNumber;j++)
         {
            temp3 = temp2 ;
            temp2 = temp1 ;
         temp1 = ((2*j - 1 + alpha - newton)*temp2 - (j - 1 + alpha)*temp3)/j ;
         }
         temp = (fNumber*temp1 - (fNumber +alpha)*temp2)/newton ;
         newton1 = newton ;
         newton  = newton1 - temp1/temp ;

         if(std::fabs(newton - newton1) <= tolerance) 
         {
            break ;
         }
      }
      if(k > maxNumber)
      {
         G4Exception("G4Integrator<T,F>::Laguerre(T,F, ...)", "Error",
                     FatalException, "Too many (>12) iterations.");
      }
         
      fAbscissa[i-1] =  newton ;
      fWeight[i-1] = -std::exp(GammaLogarithm(alpha + fNumber) - 
                GammaLogarithm((G4double)fNumber))/(temp*fNumber*temp2) ;
   }

   //
   // Integral evaluation
   //

   for(i=0;i<fNumber;i++)
   {
      integral += fWeight[i]*(typeT.*f)(fAbscissa[i]) ;
   }
   delete[] fAbscissa;
   delete[] fWeight;
   return integral ;
}



//////////////////////////////////////////////////////////////////////
//
//

template <class T, class F> G4double 
G4Integrator<T,F>::Laguerre( T* ptrT, F f, G4double alpha, G4int nLaguerre ) 
{
  return Laguerre(*ptrT,f,alpha,nLaguerre) ;
}

////////////////////////////////////////////////////////////////////////
//
// For use with global scope functions f 

template <class T, class F> G4double 
G4Integrator<T,F>::Laguerre( G4double (*f)(G4double), 
                             G4double alpha, G4int nLaguerre ) 
{
   const G4double tolerance = 1.0e-10 ;
   const G4int maxNumber = 12 ;
   G4int i, j, k ;
   G4double newton=0., newton1, temp1, temp2, temp3, temp, cofi ;
   G4double integral = 0.0 ;

   G4int fNumber = nLaguerre ;
   G4double* fAbscissa = new G4double[fNumber] ;
   G4double* fWeight   = new G4double[fNumber] ;
      
   for(i=1;i<=fNumber;i++)      // Loop over the desired roots
   {
      if(i == 1)
      {
         newton = (1.0 + alpha)*(3.0 + 0.92*alpha)
                / (1.0 + 2.4*fNumber + 1.8*alpha) ;
      }
      else if(i == 2)
      {
         newton += (15.0 + 6.25*alpha)/(1.0 + 0.9*alpha + 2.5*fNumber) ;
      }
      else
      {
         cofi = i - 2 ;
         newton += ((1.0+2.55*cofi)/(1.9*cofi)
                    + 1.26*cofi*alpha/(1.0+3.5*cofi))
                 * (newton - fAbscissa[i-3])/(1.0 + 0.3*alpha) ;
      }
      for(k=1;k<=maxNumber;k++)
      {
         temp1 = 1.0 ;
         temp2 = 0.0 ;

         for(j=1;j<=fNumber;j++)
         {
            temp3 = temp2 ;
            temp2 = temp1 ;
         temp1 = ((2*j - 1 + alpha - newton)*temp2 - (j - 1 + alpha)*temp3)/j ;
         }
         temp = (fNumber*temp1 - (fNumber +alpha)*temp2)/newton ;
         newton1 = newton ;
         newton  = newton1 - temp1/temp ;

         if(std::fabs(newton - newton1) <= tolerance) 
         {
            break ;
         }
      }
      if(k > maxNumber)
      {
         G4Exception("G4Integrator<T,F>::Laguerre( ...)", "Error",
                     FatalException, "Too many (>12) iterations.");
      }
         
      fAbscissa[i-1] =  newton ;
      fWeight[i-1] = -std::exp(GammaLogarithm(alpha + fNumber) - 
                GammaLogarithm((G4double)fNumber))/(temp*fNumber*temp2) ;
   }

   //
   // Integral evaluation
   //

   for(i=0;i<fNumber;i++)
   {
      integral += fWeight[i]*(*f)(fAbscissa[i]) ;
   }
   delete[] fAbscissa;
   delete[] fWeight;
   return integral ;
}

///////////////////////////////////////////////////////////////////////
//
// Auxiliary function which returns the value of std::log(gamma-function(x))
// Returns the value ln(Gamma(xx) for xx > 0.  Full accuracy is obtained for 
// xx > 1. For 0 < xx < 1. the reflection formula (6.1.4) can be used first.
// (Adapted from Numerical Recipes in C)
//

template <class T, class F>
G4double G4Integrator<T,F>::GammaLogarithm(G4double xx)
{
  static G4double cof[6] = { 76.18009172947146,     -86.50532032941677,
                             24.01409824083091,      -1.231739572450155,
                              0.1208650973866179e-2, -0.5395239384953e-5  } ;
  register G4int j;
  G4double x = xx - 1.0 ;
  G4double tmp = x + 5.5 ;
  tmp -= (x + 0.5) * std::log(tmp) ;
  G4double ser = 1.000000000190015 ;

  for ( j = 0; j <= 5; j++ )
  {
    x += 1.0 ;
    ser += cof[j]/x ;
  }
  return -tmp + std::log(2.5066282746310005*ser) ;
}

///////////////////////////////////////////////////////////////////////
//
// Method involving Hermite polynomials
//
///////////////////////////////////////////////////////////////////////
//
//
// Gauss-Hermite method for integration of std::exp(-x*x)*f(x) 
// from minus infinity to plus infinity . 
//

template <class T, class F>    
G4double G4Integrator<T,F>::Hermite( T& typeT, F f, G4int nHermite ) 
{
   const G4double tolerance = 1.0e-12 ;
   const G4int maxNumber = 12 ;
   
   G4int i, j, k ;
   G4double integral = 0.0 ;
   G4double newton=0., newton1, temp1, temp2, temp3, temp ;

   G4double piInMinusQ = std::pow(pi,-0.25) ;    // 1.0/std::sqrt(std::sqrt(pi)) ??

   G4int fNumber = (nHermite +1)/2 ;
   G4double* fAbscissa = new G4double[fNumber] ;
   G4double* fWeight   = new G4double[fNumber] ;

   for(i=1;i<=fNumber;i++)
   {
      if(i == 1)
      {
         newton = std::sqrt((G4double)(2*nHermite + 1)) - 
                  1.85575001*std::pow((G4double)(2*nHermite + 1),-0.16666999) ;
      }
      else if(i == 2)
      {
         newton -= 1.14001*std::pow((G4double)nHermite,0.425999)/newton ;
      }
      else if(i == 3)
      {
         newton = 1.86002*newton - 0.86002*fAbscissa[0] ;
      }
      else if(i == 4)
      {
         newton = 1.91001*newton - 0.91001*fAbscissa[1] ;
      }
      else 
      {
         newton = 2.0*newton - fAbscissa[i - 3] ;
      }
      for(k=1;k<=maxNumber;k++)
      {
         temp1 = piInMinusQ ;
         temp2 = 0.0 ;

         for(j=1;j<=nHermite;j++)
         {
            temp3 = temp2 ;
            temp2 = temp1 ;
            temp1 = newton*std::sqrt(2.0/j)*temp2 - 
                    std::sqrt(((G4double)(j - 1))/j)*temp3 ;
         }
         temp = std::sqrt((G4double)2*nHermite)*temp2 ;
         newton1 = newton ;
         newton = newton1 - temp1/temp ;

         if(std::fabs(newton - newton1) <= tolerance) 
         {
            break ;
         }
      }
      if(k > maxNumber)
      {
         G4Exception("G4Integrator<T,F>::Hermite(T,F, ...)", "Error",
                     FatalException, "Too many (>12) iterations.");
      }
      fAbscissa[i-1] =  newton ;
      fWeight[i-1] = 2.0/(temp*temp) ;
   }

   //
   // Integral calculation
   //

   for(i=0;i<fNumber;i++)
   {
     integral += fWeight[i]*( (typeT.*f)(fAbscissa[i]) + 
                              (typeT.*f)(-fAbscissa[i])   ) ;
   }
   delete[] fAbscissa;
   delete[] fWeight;
   return integral ;
}


////////////////////////////////////////////////////////////////////////
//
// For use with 'this' pointer

template <class T, class F>    
G4double G4Integrator<T,F>::Hermite( T* ptrT, F f, G4int n )
{
  return Hermite(*ptrT,f,n) ;
} 

////////////////////////////////////////////////////////////////////////
//
// For use with global scope f

template <class T, class F>
G4double G4Integrator<T,F>::Hermite( G4double (*f)(G4double), G4int nHermite) 
{
   const G4double tolerance = 1.0e-12 ;
   const G4int maxNumber = 12 ;
   
   G4int i, j, k ;
   G4double integral = 0.0 ;
   G4double newton=0., newton1, temp1, temp2, temp3, temp ;

   G4double piInMinusQ = std::pow(pi,-0.25) ;    // 1.0/std::sqrt(std::sqrt(pi)) ??

   G4int fNumber = (nHermite +1)/2 ;
   G4double* fAbscissa = new G4double[fNumber] ;
   G4double* fWeight   = new G4double[fNumber] ;

   for(i=1;i<=fNumber;i++)
   {
      if(i == 1)
      {
         newton = std::sqrt((G4double)(2*nHermite + 1)) - 
                  1.85575001*std::pow((G4double)(2*nHermite + 1),-0.16666999) ;
      }
      else if(i == 2)
      {
         newton -= 1.14001*std::pow((G4double)nHermite,0.425999)/newton ;
      }
      else if(i == 3)
      {
         newton = 1.86002*newton - 0.86002*fAbscissa[0] ;
      }
      else if(i == 4)
      {
         newton = 1.91001*newton - 0.91001*fAbscissa[1] ;
      }
      else 
      {
         newton = 2.0*newton - fAbscissa[i - 3] ;
      }
      for(k=1;k<=maxNumber;k++)
      {
         temp1 = piInMinusQ ;
         temp2 = 0.0 ;

         for(j=1;j<=nHermite;j++)
         {
            temp3 = temp2 ;
            temp2 = temp1 ;
            temp1 = newton*std::sqrt(2.0/j)*temp2 - 
                    std::sqrt(((G4double)(j - 1))/j)*temp3 ;
         }
         temp = std::sqrt((G4double)2*nHermite)*temp2 ;
         newton1 = newton ;
         newton = newton1 - temp1/temp ;

         if(std::fabs(newton - newton1) <= tolerance) 
         {
            break ;
         }
      }
      if(k > maxNumber)
      {
         G4Exception("G4Integrator<T,F>::Hermite(...)", "Error",
                     FatalException, "Too many (>12) iterations.");
      }
      fAbscissa[i-1] =  newton ;
      fWeight[i-1] = 2.0/(temp*temp) ;
   }

   //
   // Integral calculation
   //

   for(i=0;i<fNumber;i++)
   {
     integral += fWeight[i]*( (*f)(fAbscissa[i]) + (*f)(-fAbscissa[i])   ) ;
   }
   delete[] fAbscissa;
   delete[] fWeight;
   return integral ;
}

////////////////////////////////////////////////////////////////////////////
//
// Method involving Jacobi polynomials
//
////////////////////////////////////////////////////////////////////////////
//
// Gauss-Jacobi method for integration of ((1-x)^alpha)*((1+x)^beta)*f(x)
// from minus unit to plus unit .
//

template <class T, class F> 
G4double G4Integrator<T,F>::Jacobi( T& typeT, F f, G4double alpha, 
                                    G4double beta, G4int nJacobi) 
{
  const G4double tolerance = 1.0e-12 ;
  const G4double maxNumber = 12 ;
  G4int i, k, j ;
  G4double alphaBeta, alphaReduced, betaReduced, root1=0., root2=0., root3=0. ;
  G4double a, b, c, newton1, newton2, newton3, newton, temp, root=0., rootTemp ;

  G4int     fNumber   = nJacobi ;
  G4double* fAbscissa = new G4double[fNumber] ;
  G4double* fWeight   = new G4double[fNumber] ;

  for (i=1;i<=nJacobi;i++)
  {
     if (i == 1)
     {
        alphaReduced = alpha/nJacobi ;
        betaReduced = beta/nJacobi ;
        root1 = (1.0+alpha)*(2.78002/(4.0+nJacobi*nJacobi)+
              0.767999*alphaReduced/nJacobi) ;
        root2 = 1.0+1.48*alphaReduced+0.96002*betaReduced +
                0.451998*alphaReduced*alphaReduced +
                0.83001*alphaReduced*betaReduced      ;
        root  = 1.0-root1/root2 ;
     } 
     else if (i == 2)
     {
        root1=(4.1002+alpha)/((1.0+alpha)*(1.0+0.155998*alpha)) ;
        root2=1.0+0.06*(nJacobi-8.0)*(1.0+0.12*alpha)/nJacobi ;
        root3=1.0+0.012002*beta*(1.0+0.24997*std::fabs(alpha))/nJacobi ;
        root -= (1.0-root)*root1*root2*root3 ;
     } 
     else if (i == 3) 
     {
        root1=(1.67001+0.27998*alpha)/(1.0+0.37002*alpha) ;
        root2=1.0+0.22*(nJacobi-8.0)/nJacobi ;
        root3=1.0+8.0*beta/((6.28001+beta)*nJacobi*nJacobi) ;
        root -= (fAbscissa[0]-root)*root1*root2*root3 ;
     }
     else if (i == nJacobi-1)
     {
        root1=(1.0+0.235002*beta)/(0.766001+0.118998*beta) ;
        root2=1.0/(1.0+0.639002*(nJacobi-4.0)/(1.0+0.71001*(nJacobi-4.0))) ;
        root3=1.0/(1.0+20.0*alpha/((7.5+alpha)*nJacobi*nJacobi)) ;
        root += (root-fAbscissa[nJacobi-4])*root1*root2*root3 ;
     } 
     else if (i == nJacobi) 
     {
        root1 = (1.0+0.37002*beta)/(1.67001+0.27998*beta) ;
        root2 = 1.0/(1.0+0.22*(nJacobi-8.0)/nJacobi) ;
        root3 = 1.0/(1.0+8.0*alpha/((6.28002+alpha)*nJacobi*nJacobi)) ;
        root += (root-fAbscissa[nJacobi-3])*root1*root2*root3 ;
     } 
     else
     {
        root = 3.0*fAbscissa[i-2]-3.0*fAbscissa[i-3]+fAbscissa[i-4] ;
     }
     alphaBeta = alpha + beta ;
     for (k=1;k<=maxNumber;k++)
     {
        temp = 2.0 + alphaBeta ;
        newton1 = (alpha-beta+temp*root)/2.0 ;
        newton2 = 1.0 ;
        for (j=2;j<=nJacobi;j++)
        {
           newton3 = newton2 ;
           newton2 = newton1 ;
           temp = 2*j+alphaBeta ;
           a = 2*j*(j+alphaBeta)*(temp-2.0) ;
            b = (temp-1.0)*(alpha*alpha-beta*beta+temp*(temp-2.0)*root) ;
           c = 2.0*(j-1+alpha)*(j-1+beta)*temp ;
           newton1 = (b*newton2-c*newton3)/a ;
        }
        newton = (nJacobi*(alpha - beta - temp*root)*newton1 +
              2.0*(nJacobi + alpha)*(nJacobi + beta)*newton2)/
             (temp*(1.0 - root*root)) ;
        rootTemp = root ;
        root = rootTemp - newton1/newton ;
        if (std::fabs(root-rootTemp) <= tolerance)
        {
           break ;
        }
     }
     if (k > maxNumber) 
     {
        G4Exception("G4Integrator<T,F>::Jacobi(T,F, ...)", "Error",
                    FatalException, "Too many (>12) iterations.");
     }
     fAbscissa[i-1] = root ;
     fWeight[i-1] = std::exp(GammaLogarithm((G4double)(alpha+nJacobi)) + 
                        GammaLogarithm((G4double)(beta+nJacobi)) - 
                        GammaLogarithm((G4double)(nJacobi+1.0)) -
                        GammaLogarithm((G4double)(nJacobi + alphaBeta + 1.0)))
                        *temp*std::pow(2.0,alphaBeta)/(newton*newton2)             ;
   }

   //
   // Calculation of the integral
   //

   G4double integral = 0.0 ;
   for(i=0;i<fNumber;i++)
   {
      integral += fWeight[i]*(typeT.*f)(fAbscissa[i]) ;
   }
   delete[] fAbscissa;
   delete[] fWeight;
   return integral ;
}


/////////////////////////////////////////////////////////////////////////
//
// For use with 'this' pointer

template <class T, class F>    
G4double G4Integrator<T,F>::Jacobi( T* ptrT, F f, G4double alpha, 
                                             G4double beta, G4int n)
{
  return Jacobi(*ptrT,f,alpha,beta,n) ;
} 

/////////////////////////////////////////////////////////////////////////
//
// For use with global scope f 

template <class T, class F>
G4double G4Integrator<T,F>::Jacobi( G4double (*f)(G4double), G4double alpha, 
                                    G4double beta, G4int nJacobi) 
{
  const G4double tolerance = 1.0e-12 ;
  const G4double maxNumber = 12 ;
  G4int i, k, j ;
  G4double alphaBeta, alphaReduced, betaReduced, root1=0., root2=0., root3=0. ;
  G4double a, b, c, newton1, newton2, newton3, newton, temp, root=0., rootTemp ;

  G4int     fNumber   = nJacobi ;
  G4double* fAbscissa = new G4double[fNumber] ;
  G4double* fWeight   = new G4double[fNumber] ;

  for (i=1;i<=nJacobi;i++)
  {
     if (i == 1)
     {
        alphaReduced = alpha/nJacobi ;
        betaReduced = beta/nJacobi ;
        root1 = (1.0+alpha)*(2.78002/(4.0+nJacobi*nJacobi)+
              0.767999*alphaReduced/nJacobi) ;
        root2 = 1.0+1.48*alphaReduced+0.96002*betaReduced +
                0.451998*alphaReduced*alphaReduced +
                0.83001*alphaReduced*betaReduced      ;
        root  = 1.0-root1/root2 ;
     } 
     else if (i == 2)
     {
        root1=(4.1002+alpha)/((1.0+alpha)*(1.0+0.155998*alpha)) ;
        root2=1.0+0.06*(nJacobi-8.0)*(1.0+0.12*alpha)/nJacobi ;
        root3=1.0+0.012002*beta*(1.0+0.24997*std::fabs(alpha))/nJacobi ;
        root -= (1.0-root)*root1*root2*root3 ;
     } 
     else if (i == 3) 
     {
        root1=(1.67001+0.27998*alpha)/(1.0+0.37002*alpha) ;
        root2=1.0+0.22*(nJacobi-8.0)/nJacobi ;
        root3=1.0+8.0*beta/((6.28001+beta)*nJacobi*nJacobi) ;
        root -= (fAbscissa[0]-root)*root1*root2*root3 ;
     }
     else if (i == nJacobi-1)
     {
        root1=(1.0+0.235002*beta)/(0.766001+0.118998*beta) ;
        root2=1.0/(1.0+0.639002*(nJacobi-4.0)/(1.0+0.71001*(nJacobi-4.0))) ;
        root3=1.0/(1.0+20.0*alpha/((7.5+alpha)*nJacobi*nJacobi)) ;
        root += (root-fAbscissa[nJacobi-4])*root1*root2*root3 ;
     } 
     else if (i == nJacobi) 
     {
        root1 = (1.0+0.37002*beta)/(1.67001+0.27998*beta) ;
        root2 = 1.0/(1.0+0.22*(nJacobi-8.0)/nJacobi) ;
        root3 = 1.0/(1.0+8.0*alpha/((6.28002+alpha)*nJacobi*nJacobi)) ;
        root += (root-fAbscissa[nJacobi-3])*root1*root2*root3 ;
     } 
     else
     {
        root = 3.0*fAbscissa[i-2]-3.0*fAbscissa[i-3]+fAbscissa[i-4] ;
     }
     alphaBeta = alpha + beta ;
     for (k=1;k<=maxNumber;k++)
     {
        temp = 2.0 + alphaBeta ;
        newton1 = (alpha-beta+temp*root)/2.0 ;
        newton2 = 1.0 ;
        for (j=2;j<=nJacobi;j++)
        {
           newton3 = newton2 ;
           newton2 = newton1 ;
           temp = 2*j+alphaBeta ;
           a = 2*j*(j+alphaBeta)*(temp-2.0) ;
           b = (temp-1.0)*(alpha*alpha-beta*beta+temp*(temp-2.0)*root) ;
           c = 2.0*(j-1+alpha)*(j-1+beta)*temp ;
           newton1 = (b*newton2-c*newton3)/a ;
        }
        newton = (nJacobi*(alpha - beta - temp*root)*newton1 +
             2.0*(nJacobi + alpha)*(nJacobi + beta)*newton2) /
             (temp*(1.0 - root*root)) ;
        rootTemp = root ;
        root = rootTemp - newton1/newton ;
        if (std::fabs(root-rootTemp) <= tolerance)
        {
           break ;
        }
     }
     if (k > maxNumber) 
     {
        G4Exception("G4Integrator<T,F>::Jacobi(...)", "Error",
                    FatalException, "Too many (>12) iterations.");
     }
     fAbscissa[i-1] = root ;
     fWeight[i-1] =
        std::exp(GammaLogarithm((G4double)(alpha+nJacobi)) + 
                 GammaLogarithm((G4double)(beta+nJacobi)) - 
                 GammaLogarithm((G4double)(nJacobi+1.0)) -
                 GammaLogarithm((G4double)(nJacobi + alphaBeta + 1.0)))
        *temp*std::pow(2.0,alphaBeta)/(newton*newton2);
   }

   //
   // Calculation of the integral
   //

   G4double integral = 0.0 ;
   for(i=0;i<fNumber;i++)
   {
      integral += fWeight[i]*(*f)(fAbscissa[i]) ;
   }
   delete[] fAbscissa;
   delete[] fWeight;
   return integral ;
}

//
//
///////////////////////////////////////////////////////////////////