G4DataInterpolation.cc

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//
// G4DataInterpolation class implementation
//
// Author: V.Grichine, 03.04.1997
// --------------------------------------------------------------------
 
#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])
  , 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;
  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;
}
 
//
//