G4PolynomialPDF.cc

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//
// -------------------------------------------------------------------
//      GEANT4 Class file
//
//
//      File name:     G4PolynomialPDF
//
//      Author:        Jason Detwiler ([email protected])
// 
//      Creation date: Aug 2012
// -------------------------------------------------------------------
 
#include "G4PolynomialPDF.hh"
#include "Randomize.hh"
 
using namespace std;
 
G4PolynomialPDF::G4PolynomialPDF(size_t n, const G4double* coeffs, 
                 G4double x1, G4double x2) :
  fX1(x1), fX2(x2), fChanged(true), fTolerance(1.e-8), fVerbose(0)
{
  if(coeffs != nullptr) SetCoefficients(n, coeffs);
  else if(n > 0) SetNCoefficients(n);
}
 
G4PolynomialPDF::~G4PolynomialPDF() 
{}
 
void G4PolynomialPDF::SetCoefficient(size_t i, G4double value, bool doSimplify)
{
  while(i >= fCoefficients.size()) fCoefficients.push_back(0);  
  /* Loop checking, 30-Oct-2015, G.Folger */
  fCoefficients[i] = value;
  fChanged = true;
  if(doSimplify) Simplify();
}
 
void G4PolynomialPDF::SetCoefficients(size_t nCoeffs, 
                      const G4double* coefficients)
{
  SetNCoefficients(nCoeffs);
  for(size_t i=0; i<GetNCoefficients(); ++i) {
    SetCoefficient(i, coefficients[i], false);
  }
  fChanged = true;
  Simplify();
}
 
void G4PolynomialPDF::Simplify()
{
  while(fCoefficients.size() && fCoefficients[fCoefficients.size()-1] == 0) {
    if(fVerbose > 0) {
      G4cout << "G4PolynomialPDF::Simplify() WARNING: had to pop coefficient "
             << fCoefficients.size()-1 << G4endl;
    }
    fCoefficients.pop_back();
    fChanged = true;
  }
}
 
void G4PolynomialPDF::SetDomain(G4double x1, G4double x2) 
{ 
  if(x2 <= x1) {
    if(fVerbose > 0) {
      G4cout << "G4PolynomialPDF::SetDomain() WARNING: Invalid domain! "
         << "(x1 = " << x1 << ", x2 = " << x2 << ")." << G4endl;
    }
    return;
  }
  fX1 = x1; 
  fX2 = x2; 
  fChanged = true; 
}
 
void G4PolynomialPDF::Normalize()
{
  /// Normalize PDF to 1 over domain fX1 to fX2.
  /// Double-check that the highest-order coefficient is non-zero.
  while(fCoefficients.size()) {  /* Loop checking, 30-Oct-2015, G.Folger */
    if(fCoefficients[fCoefficients.size()-1] == 0.0) fCoefficients.pop_back();
    else break;
  }
 
  G4double x1N = fX1, x2N = fX2;
  G4double sum = 0;
  for(size_t i=0; i<GetNCoefficients(); ++i) {
    sum += GetCoefficient(i)*(x2N - x1N)/G4double(i+1);
    x1N*=fX1;
    x2N*=fX2;
  }
  if(sum <= 0) {
    if(fVerbose > 0) {
      G4cout << "G4PolynomialPDF::Normalize() WARNING: PDF has non-positive area: " 
         << sum << G4endl;
      Dump();
    }
    return;
  }
 
  for(size_t i=0; i<GetNCoefficients(); ++i) {
    SetCoefficient(i, GetCoefficient(i)/sum, false);
  }
  Simplify();
}
 
G4double G4PolynomialPDF::Evaluate(G4double x, G4int ddxPower)
{
  /// Evaluate f(x)
  /// ddxPower = -1: f = CDF
  /// ddxPower = 0: f = PDF
  /// ddxPower = 1: f = (d/dx) PDF
  /// ddxPower = 2: f = (d2/dx2) PDF
  if(ddxPower < -1 || ddxPower > 2) {
    if(fVerbose > 0) {
      G4cout << "G4PolynomialPDF::GetX() WARNING: ddxPower " << ddxPower 
         << " not implemented" << G4endl;
    }
    return 0.0;
  }
 
  double f = 0.; // return value
  double xN = 1.; // x to the power N
  double x1N = 1.; // endpoint x1 to the power N; only used by CDF
  for(size_t i=0; i<=GetNCoefficients(); ++i) {
    if(ddxPower == -1) { // CDF
      if(i>0) f += GetCoefficient(i-1)*(xN - x1N)/i;
      x1N *= fX1;
    }
    else if(ddxPower == 0 && i<GetNCoefficients()) f += GetCoefficient(i)*xN; // PDF
    else if(ddxPower == 1) { // (d/dx) PDF
      if(i<GetNCoefficients()-1) f += GetCoefficient(i+1)*xN*(i+1);
    }
    else if(ddxPower == 2) { // (d2/dx2) PDF
      if(i<GetNCoefficients()-2) f += GetCoefficient(i+2)*xN*((i+2)*(i+1));
    }
    xN *= x;
  }
  return f;
}
 
G4bool G4PolynomialPDF::HasNegativeMinimum(G4double x1, G4double x2)
{
  // ax2 + bx + c = 0
  // p': 2ax + b = 0 -> = 0 at min: x_extreme = -b/2a
 
  if(x1 < fX1 || x2 > fX2 || x2 < x1) {
    if(fVerbose > 0) {
      G4cout << "G4PolynomialPDF::HasNegativeMinimum() WARNING: Invalid range " 
         << x1 << " - " << x2 << G4endl;
    }
    return false;
  }
 
  // If flat, then check anywhere.
  if(GetNCoefficients() == 1) return (Evaluate(x1) < -fTolerance);
 
  // If linear, or if quadratic with negative second derivative, 
  // just check the endpoints
  if(GetNCoefficients() == 2 || 
     (GetNCoefficients() == 3 && GetCoefficient(2) <= 0)) {
    return (Evaluate(x1) < -fTolerance) || (Evaluate(x2) < -fTolerance);
  }
 
  // If quadratic and second dervative is positive, check at the mininum
  if(GetNCoefficients() == 3) {
    G4double xMin = -GetCoefficient(1)*0.5/GetCoefficient(2);
    if(xMin < x1) xMin = x1;
    if(xMin > x2) xMin = x2;
    return Evaluate(xMin) < -fTolerance;
  }
 
  // Higher-order polynomials: consider any extremum between x1 and x2. If none
  // are found, check the endpoints.
  G4double extremum = GetX(0, x1, x2, 1);
  if(Evaluate(extremum) < -fTolerance) return true;
  else if(extremum <= x1+(x2-x1)*fTolerance || 
      extremum >= x2-(x2-x1)*fTolerance) return false;
  else return 
     HasNegativeMinimum(x1, extremum) || HasNegativeMinimum(extremum, x2);
}
 
G4double G4PolynomialPDF::GetRandomX()
{
  if(fChanged) {
    Normalize(); 
    if(HasNegativeMinimum(fX1, fX2)) {
      if(fVerbose > 0) {
    G4cout << "G4PolynomialPDF::GetRandomX() WARNING: PDF has negative values, returning 0..." 
           << G4endl;
      }
      return 0.0;
    }
    fChanged = false;
  }
  return EvalInverseCDF(G4UniformRand());
}
 
G4double G4PolynomialPDF::GetX(G4double p, G4double x1, G4double x2, 
                   G4int ddxPower, G4double guess, G4bool bisect)
{
  /// Find a value of X between x1 and x2 at which f(x) = p.
  /// ddxPower = -1: f = CDF
  /// ddxPower = 0: f = PDF
  /// ddxPower = 1: f = (d/dx) PDF
  /// Uses the Newton-Raphson method to find the zero of f(x) - p.
  /// If not found in range, returns the nearest boundary
 
  // input range checking
  if(GetNCoefficients() == 0) {
    if(fVerbose > 0) {
      G4cout << "G4PolynomialPDF::GetX() WARNING: no PDF defined!" << G4endl;
    }
    return x2;
  }
  if(ddxPower < -1 || ddxPower > 1) {
    if(fVerbose > 0) {
      G4cout << "G4PolynomialPDF::GetX() WARNING: ddxPower " << ddxPower 
         << " not implemented" << G4endl;
    }
    return x2;
  }
  if(ddxPower == -1 && (p<0 || p>1)) {
    if(fVerbose > 0) {
      G4cout << "G4PolynomialPDF::GetX() WARNING: p is out of range" << G4endl;
    }
    return fX2;
  }
 
  // check limits
  if(x2 <= x1 || x1 < fX1 || x2 > fX2) {
    if(fVerbose > 0) {
      G4cout << "G4PolynomialPDF::GetX() WARNING: domain must have fX1 <= x1 < x2 <= fX2. "
         << "You sent x1 = " << x1 << ", x2 = " << x2 << "." << G4endl;
    }
    return x2;
  }
 
  // Return x2 for flat lines
  if((ddxPower == 0 && GetNCoefficients() == 1) ||
     (ddxPower == 1 && GetNCoefficients() == 2)) return x2;
 
  // Solve p = mx + b -> x = (p-b)/m for linear functions
  if((ddxPower == -1 && GetNCoefficients() == 1) ||
     (ddxPower ==  0 && GetNCoefficients() == 2) ||
     (ddxPower ==  1 && GetNCoefficients() == 3)) {
    G4double b = (ddxPower > -1) ? GetCoefficient(ddxPower) : -GetCoefficient(0)*fX1;
    G4double slope = GetCoefficient(ddxPower+1); // the highest-order coefficient
    if(slope == 0) { // the highest-order coefficient should never be zero if simplified
      if(fVerbose > 0) {
        G4cout << "G4PolynomialPDF::GetX() WARNING: Got slope = 0. "
               << "Did you forget to Simplify()?" << G4endl;
      }
      return x2;
    }
    if(ddxPower == 1) slope *= 2.;
    G4double value = (p-b)/slope;
    if(value < x1) {
      return x1;
    }
    else if(value > x2) {
      return x2;
    }
    else {
      return value;
    }
  }
 
  // Solve quadratic equation for f-p=0 when f is quadratic
  if((ddxPower == -1 && GetNCoefficients() == 2) ||
     (ddxPower ==  0 && GetNCoefficients() == 3) ||
     (ddxPower ==  1 && GetNCoefficients() == 4)) {
    G4double c = -p + ((ddxPower > -1) ? GetCoefficient(ddxPower) : 0);
    if(ddxPower == -1) c -= (GetCoefficient(0) + GetCoefficient(1)/2.*fX1)*fX1;
    G4double b = GetCoefficient(ddxPower+1);
    if(ddxPower == 1) b *= 2.;
    G4double a = GetCoefficient(ddxPower+2); // the highest-order coefficient
    if(a == 0) { // the highest-order coefficient should never be 0 if simplified
      if(fVerbose > 0) {
        G4cout << "G4PolynomialPDF::GetX() WARNING: Got a = 0. "
               << "Did you forget to Simplify()?" << G4endl;
      }
      return x2;
    }
    if(ddxPower == 1) a *= 3;
    else if(ddxPower == -1) a *= 0.5;
    double sqrtFactor = b*b - 4.*a*c;
    if(sqrtFactor < 0) return x2; // quadratic equation has no solution (p not in range of f)
    sqrtFactor = sqrt(sqrtFactor)/2./fabs(a);
    G4double valueMinus = -b/2./a - sqrtFactor;
    if(valueMinus >= x1 && valueMinus <= x2) return valueMinus;
    else if(valueMinus > x2) return x2;
    G4double valuePlus = -b/2./a + sqrtFactor;
    if(valuePlus >= x1 && valuePlus <= x2) return valuePlus;
    else if(valuePlus < x1) return x2;
    return (x1-valueMinus <= valuePlus-x2) ? x1 : x2;
  }
 
  // f is non-trivial, so use Newton-Raphson
  // start in the middle if no good guess is provided
  if(guess < x1 || guess > x2) guess = (x2+x1)*0.5; 
  G4double lastChange = 1;
  size_t iterations = 0;
  while(fabs(lastChange) > fTolerance) {  /* Loop checking, 02.11.2015, A.Ribon */
    // calculate f and f' simultaneously
    G4double f = -p;
    G4double dfdx = 0;
    G4double xN = 1;
    G4double x1N = 1; // only used by CDF
    for(size_t i=0; i<=GetNCoefficients(); ++i) {
      if(ddxPower == -1) { // CDF
        if(i>0) f += GetCoefficient(i-1)*(xN - x1N)/G4double(i);
        if(i<GetNCoefficients()) dfdx += GetCoefficient(i)*xN;
        x1N *= fX1;
      }
      else if(ddxPower == 0) { // PDF
        if(i<GetNCoefficients()) f += GetCoefficient(i)*xN;
        if(i+1<GetNCoefficients()) dfdx += GetCoefficient(i+1)*xN*G4double(i+1);
      }
      else { // ddxPower == 1: (d/dx) PDF
        if(i+1<GetNCoefficients()) f += GetCoefficient(i+1)*xN*G4double(i+1);
        if(i+2<GetNCoefficients()) dfdx += GetCoefficient(i+2)*xN*G4double(i+2);
      }
      xN *= guess;
    }
    if(f == 0) return guess;
    if(dfdx == 0) {
      if(fVerbose > 0) {
    G4cout << "G4PolynomialPDF::GetX() WARNING: got f != 0 but slope = 0 for ddxPower = " 
           << ddxPower << G4endl;
      }
      return x2;
    }
    lastChange = - f/dfdx;
 
    if(guess + lastChange < x1) {
      lastChange = x1 - guess;
    } else if(guess + lastChange > x2) {
      lastChange = x2 - guess;
    }
 
    guess += lastChange;
    lastChange /= (fX2-fX1);
     
    ++iterations;
    if(iterations > 50) {
      if(p!=0) {
    if(fVerbose > 0) {
      G4cout << "G4PolynomialPDF::GetX() WARNING: got stuck searching for " << p 
         << " between " << x1 << " and " << x2 << " with ddxPower = " 
         << ddxPower 
         << ". Last guess was " << guess << "." << G4endl;
    }
      }
      if(ddxPower==-1 && bisect) {
    if(fVerbose > 0) {
      G4cout << "G4PolynomialPDF::GetX() WARNING: Biseting and trying again..." 
         << G4endl;
    }
        return Bisect(p, x1, x2);
      }
      else return guess;
    }
  }
  return guess;
}
 
G4double G4PolynomialPDF::Bisect( G4double p, G4double x1, G4double x2 ) {
  // Bisect to get 1% precision, then use Newton-Raphson
  G4double z = (x2 + x1)/2.0; // [x1 z x2]
  if((x2 - x1)/(fX2 - fX1) < 0.01) return GetX(p, fX1, fX2, -1, z, false);
  G4double fz = Evaluate(z, -1) - p;
  if(fz < 0) return Bisect(p, z, x2); // [z x2]
  return Bisect(p, x1, z); // [x1 z]
}
 
void G4PolynomialPDF::Dump()
{
  G4cout << "G4PolynomialPDF::Dump() - PDF(x) = ";
  for(size_t i=0; i<GetNCoefficients(); i++) {
    if(i>0) G4cout << " + ";
    G4cout << GetCoefficient(i);
    if(i>0) G4cout << "*x";
    if(i>1) G4cout << "^" << i;
  }
  G4cout << G4endl;
  G4cout << "G4PolynomialPDF::Dump() - Interval: " << fX1 << " <= x < " 
     << fX2 << G4endl;
}