G4DataInterpolation.cc

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//
//
// $Id: G4DataInterpolation.cc 67970 2013-03-13 10:10:06Z gcosmo $
//
#include "G4DataInterpolation.hh"

//////////////////////////////////////////////////////////////////////////////
//
// Constructor for initializing of fArgument, fFunction and fNumber
// data members

G4DataInterpolation::G4DataInterpolation( G4double pX[], 
                                          G4double pY[], 
                                          G4int number    )
   : fArgument(new G4double[number]),
     fFunction(new G4double[number]),
     fSecondDerivative(0),
     fNumber(number)
{
   for(G4int i=0;i<fNumber;i++)
   {
      fArgument[i] = pX[i] ;
      fFunction[i] = pY[i] ;
   }
} 

////////////////////////////////////////////////////////////////////////////
//
// Constructor for cubic spline interpolation. It creates the array 
// fSecondDerivative[0,...fNumber-1] which is used in this interpolation by
// the function 


G4DataInterpolation::G4DataInterpolation( G4double pX[], 
                                          G4double pY[], 
                                          G4int number,
                                          G4double pFirstDerStart,
                                          G4double pFirstDerFinish  ) 
   : fArgument(new G4double[number]),
     fFunction(new G4double[number]),
     fSecondDerivative(new G4double[number]),
     fNumber(number)
{
   G4int i=0 ;
   G4double p=0.0, qn=0.0, sig=0.0, un=0.0 ;
   const G4double maxDerivative = 0.99e30 ;
   G4double* u = new G4double[fNumber - 1] ;

   for(i=0;i<fNumber;i++)
   {
      fArgument[i] = pX[i] ;
      fFunction[i] = pY[i] ;
   }
   if(pFirstDerStart > maxDerivative)
   {
      fSecondDerivative[0] = 0.0 ;
      u[0] = 0.0 ;
   }
   else
   {
      fSecondDerivative[0] = -0.5 ;
      u[0] = (3.0/(fArgument[1]-fArgument[0]))
           * ((fFunction[1]-fFunction[0])/(fArgument[1]-fArgument[0])
           - pFirstDerStart) ;
   }
   
   // Decomposition loop for tridiagonal algorithm. fSecondDerivative[i]
   // and u[i] are used for temporary storage of the decomposed factors.
   
   for(i=1;i<fNumber-1;i++)
   {
      sig = (fArgument[i]-fArgument[i-1])/(fArgument[i+1]-fArgument[i-1]) ;
      p = sig*fSecondDerivative[i-1] + 2.0 ;
      fSecondDerivative[i] = (sig - 1.0)/p ;
      u[i] = (fFunction[i+1]-fFunction[i])/(fArgument[i+1]-fArgument[i]) -
             (fFunction[i]-fFunction[i-1])/(fArgument[i]-fArgument[i-1]) ;
      u[i] =(6.0*u[i]/(fArgument[i+1]-fArgument[i-1]) - sig*u[i-1])/p ;
   }
   if(pFirstDerFinish > maxDerivative)
   {
      qn = 0.0 ;
      un = 0.0 ;
   }
   else
   {
      qn = 0.5 ;
      un = (3.0/(fArgument[fNumber-1]-fArgument[fNumber-2]))
         * (pFirstDerFinish - (fFunction[fNumber-1]-fFunction[fNumber-2])
         / (fArgument[fNumber-1]-fArgument[fNumber-2])) ;
   }
   fSecondDerivative[fNumber-1] = (un - qn*u[fNumber-2])/
                                  (qn*fSecondDerivative[fNumber-2] + 1.0) ;
   
   // The backsubstitution loop for the triagonal algorithm of solving
   // a linear system of equations.
   
   for(G4int k=fNumber-2;k>=0;k--)
   {
      fSecondDerivative[k] = fSecondDerivative[k]*fSecondDerivative[k+1] + u[k];
   }
   delete[] u ;
} 

/////////////////////////////////////////////////////////////////////////////
//
// Destructor deletes dynamically created arrays for data members: fArgument,
// fFunction and fSecondDerivative, all have dimension of fNumber
      
G4DataInterpolation::~G4DataInterpolation()
{
   delete [] fArgument ;
   delete [] fFunction ;
   if(fSecondDerivative)  { delete [] fSecondDerivative; }
}

/////////////////////////////////////////////////////////////////////////////
//
// This function returns the value P(pX), where P(x) is polynom of fNumber-1
// degree such that P(fArgument[i]) = fFunction[i], for i = 0, ..., fNumber-1.
// This is Lagrange's form of interpolation and it is based on Neville's
// algorithm

G4double 
G4DataInterpolation::PolynomInterpolation(G4double pX,
                                          G4double& deltaY ) const
{
   G4int i=0, j=1, k=0 ;
   G4double mult=0.0, difi=0.0, deltaLow=0.0, deltaUp=0.0, cd=0.0, y=0.0 ;
   G4double* c = new G4double[fNumber] ;
   G4double* d = new G4double[fNumber] ;
   G4double diff = std::fabs(pX-fArgument[0]) ;
   for(i=0;i<fNumber;i++)
   {
      difi = std::fabs(pX-fArgument[i]) ;
      if(difi <diff)
      {
         k = i ;
         diff = difi ;
      }
      c[i] = fFunction[i] ;
      d[i] = fFunction[i] ;   
   }
   y = fFunction[k--] ;     
   for(j=1;j<fNumber;j++)
   {
      for(i=0;i<fNumber-j;i++)
      {
         deltaLow = fArgument[i] - pX ;
         deltaUp = fArgument[i+j] - pX ;
         cd = c[i+1] - d[i] ;
         mult = deltaLow - deltaUp ;
         if (!(mult != 0.0))
         {
            G4Exception("G4DataInterpolation::PolynomInterpolation()",
                        "Error", FatalException, "Coincident nodes !") ;
         }
         mult  = cd/mult ;
         d[i] = deltaUp*mult ;
         c[i] = deltaLow*mult ;
      }
      y += (deltaY = (2*k < (fNumber - j -1) ? c[k+1] : d[k--] )) ;
   }
   delete[] c ;
   delete[] d ;
   
   return y ;
}

////////////////////////////////////////////////////////////////////////////
//
// Given arrays fArgument[0,..,fNumber-1] and fFunction[0,..,fNumber-1], this
// function calculates an array of coefficients. The coefficients don't provide
// usually (fNumber>10) better accuracy for polynom interpolation, as compared
// with PolynomInterpolation function. They could be used instead for derivate 
// calculations and some other applications.

void 
G4DataInterpolation::PolIntCoefficient( G4double cof[]) const 
{
   G4int i=0, j=0 ;
   G4double factor;
   G4double reducedY=0.0, mult=1.0 ;
   G4double* tempArgument = new G4double[fNumber] ;
   
   for(i=0;i<fNumber;i++)
   {
      tempArgument[i] = cof[i] = 0.0 ;
   }
   tempArgument[fNumber-1] = -fArgument[0] ;
   
   for(i=1;i<fNumber;i++)
   {
      for(j=fNumber-1-i;j<fNumber-1;j++)
      {
         tempArgument[j] -= fArgument[i]*tempArgument[j+1] ;
      }
      tempArgument[fNumber-1] -= fArgument[i] ;
   }
   for(i=0;i<fNumber;i++)
   {
      factor = fNumber ;
      for(j=fNumber-1;j>=1;j--)
      {
         factor = j*tempArgument[j] + factor*fArgument[i] ;
      }
      reducedY = fFunction[i]/factor ;
      mult = 1.0 ;
      for(j=fNumber-1;j>=0;j--)
      {
         cof[j] += mult*reducedY ;
         mult = tempArgument[j] + mult*fArgument[i] ;
      }
   }
   delete[] tempArgument ;
}

/////////////////////////////////////////////////////////////////////////////
//
// The function returns diagonal rational function (Bulirsch and Stoer
// algorithm of Neville type) Pn(x)/Qm(x) where P and Q are polynoms.
// Tests showed the method is not stable and hasn't advantage if compared
// with polynomial interpolation ?!

G4double
G4DataInterpolation::RationalPolInterpolation(G4double pX,
                                              G4double& deltaY ) const 
{
   G4int i=0, j=1, k=0 ;
   const G4double tolerance = 1.6e-24 ;
   G4double mult=0.0, difi=0.0, cd=0.0, y=0.0, cof=0.0 ;
   G4double* c = new G4double[fNumber] ;
   G4double* d = new G4double[fNumber] ;
   G4double diff = std::fabs(pX-fArgument[0]) ;
   for(i=0;i<fNumber;i++)
   {
      difi = std::fabs(pX-fArgument[i]) ;
      if (!(difi != 0.0))
      {
         y = fFunction[i] ;
         deltaY = 0.0 ;
         delete[] c ;
         delete[] d ;
         return y ;
      }
      else if(difi < diff)
      {
         k = i ;
         diff = difi ;
      }
      c[i] = fFunction[i] ;
      d[i] = fFunction[i] + tolerance ;   // to prevent rare zero/zero cases
   }
   y = fFunction[k--] ;    
   for(j=1;j<fNumber;j++)
   {
      for(i=0;i<fNumber-j;i++)
      {
         cd  = c[i+1] - d[i] ;
         difi  = fArgument[i+j] - pX ;
         cof  = (fArgument[i] - pX)*d[i]/difi ;
         mult = cof - c[i+1] ;
         if (!(mult != 0.0))    // function to be interpolated has pole at pX
         {
            G4Exception("G4DataInterpolation::RationalPolInterpolation()",
                        "Error", FatalException, "Coincident nodes !") ;
         }
         mult  = cd/mult ;
         d[i] = c[i+1]*mult ;
         c[i] = cof*mult ;
      }
      y += (deltaY = (2*k < (fNumber - j - 1) ? c[k+1] : d[k--] )) ;
   }
   delete[] c ;
   delete[] d ;
   
   return y ;
}

/////////////////////////////////////////////////////////////////////////////
//
// Cubic spline interpolation in point pX for function given by the table:
// fArgument, fFunction. The constructor, which creates fSecondDerivative,
// must be called before. The function works optimal, if sequential calls
// are in random values of pX.

G4double 
G4DataInterpolation::CubicSplineInterpolation(G4double pX) const 
{
   G4int kLow=0, kHigh=fNumber-1, k=0 ;
   
   // Searching in the table by means of bisection method. 
   // fArgument must be monotonic, either increasing or decreasing
   
   while((kHigh - kLow) > 1)
   {
      k = (kHigh + kLow) >> 1 ; // compute midpoint 'bisection'
      if(fArgument[k] > pX)
      {
         kHigh = k ;
      }
      else
      {
         kLow = k ;
      }
   }        // kLow and kHigh now bracket the input value of pX
   G4double deltaHL = fArgument[kHigh] - fArgument[kLow] ;
   if (!(deltaHL != 0.0))
   {
      G4Exception("G4DataInterpolation::CubicSplineInterpolation()",
                  "Error", FatalException, "Bad fArgument input !") ;
   }
   G4double a = (fArgument[kHigh] - pX)/deltaHL ;
   G4double b = (pX - fArgument[kLow])/deltaHL ;
   
   // Final evaluation of cubic spline polynomial for return
   
   return a*fFunction[kLow] + b*fFunction[kHigh] + 
          ((a*a*a - a)*fSecondDerivative[kLow] +
           (b*b*b - b)*fSecondDerivative[kHigh])*deltaHL*deltaHL/6.0  ;
}

///////////////////////////////////////////////////////////////////////////
//
// Return cubic spline interpolation in the point pX which is located between
// fArgument[index] and fArgument[index+1]. It is usually called in sequence
// of known from external analysis values of index.

G4double 
G4DataInterpolation::FastCubicSpline(G4double pX, 
                                     G4int index) const 
{
   G4double delta = fArgument[index+1] - fArgument[index] ;
   if (!(delta != 0.0))
   {
      G4Exception("G4DataInterpolation::FastCubicSpline()",
                  "Error", FatalException, "Bad fArgument input !") ;
   }
   G4double a = (fArgument[index+1] - pX)/delta ;
   G4double b = (pX - fArgument[index])/delta ;
   
   // Final evaluation of cubic spline polynomial for return
   
   return a*fFunction[index] + b*fFunction[index+1] + 
          ((a*a*a - a)*fSecondDerivative[index] +
           (b*b*b - b)*fSecondDerivative[index+1])*delta*delta/6.0  ;
}

////////////////////////////////////////////////////////////////////////////
//
// Given argument pX, returns index k, so that pX bracketed by fArgument[k]
// and fArgument[k+1]

G4int 
G4DataInterpolation::LocateArgument(G4double pX) const 
{
   G4int kLow=-1, kHigh=fNumber, k=0 ; 
   G4bool ascend=(fArgument[fNumber-1] >= fArgument[0]) ;
   while((kHigh - kLow) > 1)
   {
      k = (kHigh + kLow) >> 1 ; // compute midpoint 'bisection'
      if( (pX >= fArgument[k]) == ascend)
      {
         kLow = k ;
      }
      else
      {
         kHigh = k ;
      }
   }
   if (!(pX != fArgument[0]))
   {
      return 1 ;
   }
   else if (!(pX != fArgument[fNumber-1]))
   {
      return fNumber - 2 ;
   }
   else  return kLow ;
}

/////////////////////////////////////////////////////////////////////////////
//
// Given a value pX, returns a value 'index' such that pX is between
// fArgument[index] and fArgument[index+1]. fArgument MUST BE MONOTONIC,
// either increasing or decreasing. If index = -1 or fNumber, this indicates
// that pX is out of range. The value index on input is taken as the initial
// approximation for index on output.

void 
G4DataInterpolation::CorrelatedSearch( G4double pX,
                                       G4int& index ) const 
{
   G4int kHigh=0, k=0, Increment=0 ;
   // ascend = true for ascending order of table, false otherwise
   G4bool ascend = (fArgument[fNumber-1] >= fArgument[0]) ;
   if(index < 0 || index > fNumber-1)
   {
      index = -1 ;
      kHigh = fNumber ;
   }
   else
   {
      Increment = 1 ;   //   What value would be the best ?
      if((pX >= fArgument[index]) == ascend)
      {
         if(index == fNumber -1)
         {
            index = fNumber ;
            return ;
         }
         kHigh = index + 1 ;
         while((pX >= fArgument[kHigh]) == ascend)
         {
            index = kHigh ;
            Increment += Increment ;      // double the Increment
            kHigh = index + Increment ;
            if(kHigh > (fNumber - 1))
            {
               kHigh = fNumber ;
               break ;
            }
         }
      }
      else
      {
         if(index == 0)
         {
            index = -1 ;
            return ;
         }
         kHigh = index-- ;
         while((pX < fArgument[index]) == ascend)
         {
            kHigh = index ;
            Increment <<= 1 ;      // double the Increment
            if(Increment >= kHigh)
            {
               index = -1 ;
               break ;
            }
            else
            {
               index = kHigh - Increment ;
            }
         }
      }        // Value bracketed
   }
   // final bisection searching

   while((kHigh - index) != 1)
   {
      k = (kHigh + index) >> 1 ;
      if((pX >= fArgument[k]) == ascend)
      {
         index = k ;
      }
      else
      {
         kHigh = k ;
      }
   }
   if (!(pX != fArgument[fNumber-1]))
   {
      index = fNumber - 2 ;
   }
   if (!(pX != fArgument[0]))
   {
      index = 0 ;
   }
   return ;
}

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