AdaptiveIntegratorMultiDim.cxx

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// Implementation file for class
// AdaptiveIntegratorMultiDim
//
#include "Math/IFunction.h"
#include "Math/AdaptiveIntegratorMultiDim.h"
#include "Math/IntegratorOptions.h"
#include "Math/Error.h"
 
#include <cmath>
#include <algorithm>
 
namespace ROOT {
namespace Math {
 
 
 
AdaptiveIntegratorMultiDim::AdaptiveIntegratorMultiDim(double absTol, double relTol, unsigned int maxpts, unsigned int size):
   fDim(0),
   fMinPts(0),
   fMaxPts(maxpts),
   fSize(size),
   fAbsTol(absTol),
   fRelTol(relTol),
   fResult(0),
   fError(0), fRelError(0),
   fNEval(0),
   fStatus(-1),
   fFun(nullptr)
{
   // constructor - without passing a function
   if (fAbsTol < 0) fAbsTol = ROOT::Math::IntegratorMultiDimOptions::DefaultAbsTolerance();
   if (fRelTol < 0) fRelTol = ROOT::Math::IntegratorMultiDimOptions::DefaultRelTolerance();
   if (fMaxPts == 0) fMaxPts = ROOT::Math::IntegratorMultiDimOptions::DefaultNCalls();
   if (fSize   == 0) fSize = ROOT::Math::IntegratorMultiDimOptions::DefaultWKSize();
}
 
AdaptiveIntegratorMultiDim::AdaptiveIntegratorMultiDim( const IMultiGenFunction &f, double absTol, double relTol, unsigned int maxpts, unsigned int size):
   fDim(f.NDim()),
   fMinPts(0),
   fMaxPts(maxpts),
   fSize(size),
   fAbsTol(absTol),
   fRelTol(relTol),
   fResult(0),
   fError(0), fRelError(0),
   fNEval(0),
   fStatus(-1),
   fFun(&f)
{
   // constructor passing a multi-dimensional function interface
   // constructor - without passing a function
   if (fAbsTol < 0) fAbsTol = ROOT::Math::IntegratorMultiDimOptions::DefaultAbsTolerance();
   if (fRelTol < 0) fRelTol = ROOT::Math::IntegratorMultiDimOptions::DefaultRelTolerance();
   if (fMaxPts == 0) fMaxPts = ROOT::Math::IntegratorMultiDimOptions::DefaultNCalls();
   if (fSize   == 0) fSize = ROOT::Math::IntegratorMultiDimOptions::DefaultWKSize();
}
 
 
 
//double AdaptiveIntegratorMultiDim::Result() const { return fIntegrator->Result(); }
//double AdaptiveIntegratorMultiDim::Error() const { return fIntegrator->Error(); }
 
void AdaptiveIntegratorMultiDim::SetFunction(const IMultiGenFunction &f)
{
   // set the integration function
   fFun = &f;
   fDim = f.NDim();
}
 
void AdaptiveIntegratorMultiDim::SetRelTolerance(double relTol){ this->fRelTol = relTol; }
 
 
void AdaptiveIntegratorMultiDim::SetAbsTolerance(double absTol){ this->fAbsTol = absTol; }
 
 
double AdaptiveIntegratorMultiDim::DoIntegral(const double* xmin, const double * xmax, bool absValue)
{
   // References:
   //
   //   1.A.C. Genz and A.A. Malik, Remarks on algorithm 006:
   //     An adaptive algorithm for numerical integration over
   //     an N-dimensional rectangular region, J. Comput. Appl. Math. 6 (1980) 295-302.
   //   2.A. van Doren and L. de Ridder, An adaptive algorithm for numerical
   //     integration over an n-dimensional cube, J.Comput. Appl. Math. 2 (1976) 207-217.
 
   //to be changed later
   unsigned int n=fDim;
   bool kFALSE = false;
   bool kTRUE = true;
 
   double epsrel = fRelTol; //specified relative accuracy
   double epsabs = fAbsTol; //specified absolute accuracy
   //output parameters
   fStatus = 0; //report status
   unsigned int nfnevl; //nr of function evaluations
   double relerr; //an estimation of the relative accuracy of the result
 
 
   double ctr[15], width[15], widthl[15], z[15];
 
   static const double xl2 = 0.358568582800318073;//lambda_2
   static const double xl4 = 0.948683298050513796;//lambda_4
   static const double xl5 = 0.688247201611685289;//lambda_5
   static const double w2  = 980./6561; //weights/2^n
   static const double w4  = 200./19683;
   static const double wp2 = 245./486;//error weights/2^n
   static const double wp4 = 25./729;
 
   static const double wn1[14] = {     -0.193872885230909911, -0.555606360818980835,
                                       -0.876695625666819078, -1.15714067977442459,  -1.39694152314179743,
                                       -1.59609815576893754,  -1.75461057765584494,  -1.87247878880251983,
                                       -1.94970278920896201,  -1.98628257887517146,  -1.98221815780114818,
                                       -1.93750952598689219,  -1.85215668343240347,  -1.72615963013768225};
 
   static const double wn3[14] = {     0.0518213686937966768,  0.0314992633236803330,
                                       0.0111771579535639891,-0.00914494741655235473,-0.0294670527866686986,
                                       -0.0497891581567850424,-0.0701112635269013768, -0.0904333688970177241,
                                       -0.110755474267134071, -0.131077579637250419,  -0.151399685007366752,
                                       -0.171721790377483099, -0.192043895747599447,  -0.212366001117715794};
 
   static const double wn5[14] = {         0.871183254585174982e-01,  0.435591627292587508e-01,
                                           0.217795813646293754e-01,  0.108897906823146873e-01,  0.544489534115734364e-02,
                                           0.272244767057867193e-02,  0.136122383528933596e-02,  0.680611917644667955e-03,
                                           0.340305958822333977e-03,  0.170152979411166995e-03,  0.850764897055834977e-04,
                                           0.425382448527917472e-04,  0.212691224263958736e-04,  0.106345612131979372e-04};
 
   static const double wpn1[14] = {   -1.33196159122085045, -2.29218106995884763,
                                      -3.11522633744855959, -3.80109739368998611, -4.34979423868312742,
                                      -4.76131687242798352, -5.03566529492455417, -5.17283950617283939,
                                      -5.17283950617283939, -5.03566529492455417, -4.76131687242798352,
                                      -4.34979423868312742, -3.80109739368998611, -3.11522633744855959};
 
   static const double wpn3[14] = {     0.0445816186556927292, -0.0240054869684499309,
                                        -0.0925925925925925875, -0.161179698216735251,  -0.229766803840877915,
                                        -0.298353909465020564,  -0.366941015089163228,  -0.435528120713305891,
                                        -0.504115226337448555,  -0.572702331961591218,  -0.641289437585733882,
                                        -0.709876543209876532,  -0.778463648834019195,  -0.847050754458161859};
 
   double result = 0;
   double abserr = 0;
   fStatus  = 3;
   nfnevl = 0;
   relerr = 0;
   // does not work for 1D functions
   if (n < 2 || n > 15) {
      MATH_WARN_MSGVAL("AdaptiveIntegratorMultiDim::Integral","Wrong function dimension",n);
      return 0;
   }
 
   double twondm = std::pow(2.0,static_cast<int>(n));
   //unsigned int minpts = Int_t(twondm)+ 2*n*(n+1)+1;
 
   unsigned int ifncls = 0;
   bool  ldv   = kFALSE;
   unsigned int irgnst = 2*n+3;
   unsigned int  irlcls = (unsigned int)(twondm) +2*n*(n+1)+1;//minimal number of nodes in n dim
   unsigned int isbrgn = irgnst;
   unsigned int isbrgs = irgnst;
 
 
   unsigned int minpts = fMinPts;
   unsigned int maxpts = std::max(fMaxPts, irlcls) ;//specified maximal number of function evaluations
 
   if (minpts < 1)      minpts = irlcls;
   if (maxpts < minpts) maxpts = 10*minpts;
 
   // The original algorithm expected a working space array WK of length IWK
   // with IWK Length ( >= (2N + 3) * (1 + MAXPTS/(2**N + 2N(N + 1) + 1))/2).
   // Here, this array is allocated dynamically
 
   unsigned int iwk = std::max( fSize, irgnst*(1 +maxpts/irlcls)/2 );
   double *wk = new double[iwk+10];
 
   unsigned int j;
   for (j=0; j<n; j++) {
      ctr[j] = (xmax[j] + xmin[j])*0.5;//center of a hypercube
      width[j] = (xmax[j] - xmin[j])*0.5;//its width
   }
 
   double rgnvol, sum1, sum2, sum3, sum4, sum5, difmax, f2, f3, dif, aresult;
   double rgncmp=0, rgnval, rgnerr;
 
   unsigned int j1, k, l, m, idvaxn=0, idvax0=0, isbtmp, isbtpp;
 
   //InitArgs(z,fParams);
 
L20:
   rgnvol = twondm;//=2^n
   for (j=0; j<n; j++) {
      rgnvol *= width[j]; //region volume
      z[j]    = ctr[j]; //temporary node
   }
   sum1 = (*fFun)((const double*)z);//EvalPar(z,fParams); //evaluate function
 
   difmax = 0;
   sum2   = 0;
   sum3   = 0;
 
   //loop over coordinates
   for (j=0; j<n; j++) {
      z[j]    = ctr[j] - xl2*width[j];
      if (absValue) f2 = std::abs((*fFun)(z));
      else          f2 = (*fFun)(z);
      z[j]    = ctr[j] + xl2*width[j];
      if (absValue) f2 += std::abs((*fFun)(z));
      else          f2 += (*fFun)(z);
      widthl[j] = xl4*width[j];
      z[j]    = ctr[j] - widthl[j];
      if (absValue) f3 = std::abs((*fFun)(z));
      else          f3 = (*fFun)(z);
      z[j]    = ctr[j] + widthl[j];
      if (absValue) f3 += std::abs((*fFun)(z));
      else          f3 += (*fFun)(z);
      sum2   += f2;//sum func eval with different weights separately
      sum3   += f3;//for a given region
      dif     = std::abs(7*f2-f3-12*sum1);
      //storing dimension with biggest error/difference (?)
      if (dif >= difmax) {
         difmax=dif;
         idvaxn=j+1;
      }
      z[j]    = ctr[j];
   }
 
   sum4 = 0;
   for (j=1;j<n;j++) {
      j1 = j-1;
      for (k=j;k<n;k++) {
         for (l=0;l<2;l++) {
            widthl[j1] = -widthl[j1];
            z[j1]    = ctr[j1] + widthl[j1];
            for (m=0;m<2;m++) {
               widthl[k] = -widthl[k];
               z[k]    = ctr[k] + widthl[k];
               if (absValue) sum4 += std::abs((*fFun)(z));
               else            sum4 += (*fFun)(z);
            }
         }
         z[k] = ctr[k];
      }
      z[j1] = ctr[j1];
   }
 
   sum5 = 0;
 
   for (j=0;j<n;j++) {
      widthl[j] = -xl5*width[j];
      z[j] = ctr[j] + widthl[j];
   }
L90: //sum over end nodes ~gray codes
   if (absValue) sum5 += std::abs((*fFun)(z));
   else          sum5 += (*fFun)(z);
   for (j=0;j<n;j++) {
      widthl[j] = -widthl[j];
      z[j] = ctr[j] + widthl[j];
      if (widthl[j] > 0) goto L90;
   }
 
   rgncmp  = rgnvol*(wpn1[n-2]*sum1+wp2*sum2+wpn3[n-2]*sum3+wp4*sum4);
   rgnval  = wn1[n-2]*sum1+w2*sum2+wn3[n-2]*sum3+w4*sum4+wn5[n-2]*sum5;
   rgnval *= rgnvol;
   // avoid difference of too small numbers
   //rgnval = 1.0E-30;
   //rgnerr  = TMath::Max( std::abs(rgnval-rgncmp), TMath::Max(std::abs(rgncmp), std::abs(rgnval) )*4.0E-16 );
   rgnerr  = std::abs(rgnval-rgncmp);//compares estim error with expected error
 
   result += rgnval;
   abserr += rgnerr;
   ifncls += irlcls;
   aresult = std::abs(result);
   //if (result > 0 && aresult< 1e-100) {
   //   delete [] wk;
   //   fStatus = 0;  //function is probably symmetric ==> integral is null: not an error
   //   return result;
   //}
 
   //if division
   if (ldv) {
   L110:
      isbtmp = 2*isbrgn;
      if (isbtmp > isbrgs) goto L160;
      if (isbtmp < isbrgs) {
         isbtpp = isbtmp + irgnst;
         if (wk[isbtmp-1] < wk[isbtpp-1]) isbtmp = isbtpp;
      }
      if (rgnerr >= wk[isbtmp-1]) goto L160;
      for (k=0;k<irgnst;k++) {
         wk[isbrgn-k-1] = wk[isbtmp-k-1];
      }
      isbrgn = isbtmp;
      goto L110;
   }
L140:
   isbtmp = (isbrgn/(2*irgnst))*irgnst;
   if (isbtmp >= irgnst && rgnerr > wk[isbtmp-1]) {
      for (k=0;k<irgnst;k++) {
         wk[isbrgn-k-1] = wk[isbtmp-k-1];
      }
      isbrgn = isbtmp;
      goto L140;
   }
 
L160: //to divide or not
   wk[isbrgn-1] = rgnerr;//storing value & error in last
   wk[isbrgn-2] = rgnval;//table records
   wk[isbrgn-3] = double(idvaxn);//coordinate with biggest error
   for (j=0;j<n;j++) {
      isbtmp = isbrgn-2*j-4;
      wk[isbtmp]   = ctr[j];
      wk[isbtmp-1] = width[j];
   }
   if (ldv) { // division along chosen coordinate
      ldv = kFALSE;
      ctr[idvax0-1] += 2*width[idvax0-1];
      isbrgs += irgnst;//updating the number of nodes/regions(?)
      isbrgn  = isbrgs;
      goto L20;
   }
   //if no divisions to be made..
   relerr = abserr;
   if (aresult != 0)  relerr = abserr/aresult;
 
   if (isbrgs+irgnst > iwk) fStatus = 2;
   if (ifncls+2*irlcls > maxpts) {
      if (sum1==0 && sum2==0 && sum3==0 && sum4==0 && sum5==0){
         fStatus = 0;
         result = 0;
      }
      else
         fStatus = 1;
   }
   //..and accuracy appropriare
   // should not use absolute tolerance especially for sharp peaks
   if ( ( relerr < epsrel || abserr < epsabs ) && ifncls >= minpts) fStatus = 0;  
 
#ifdef DEBUG   
   if (ifncls >= minpts) {
      if (relerr < epsrel ) {
         printf("relative tol reached for value %20.10g an rel error %20.10g  \n",aresult, relerr);
         fStatus = 0;  // We do not use the absolute error.
      }
      if (abserr < epsabs ) { 
         printf("Absolute tol reached for value %20.10g and abs error %20.10g  \n",aresult, abserr);
         fStatus = 0;  // We do not use the absolute error.
      }
   }
#endif
   
   if (fStatus == 3) {
      ldv = kTRUE;
      isbrgn  = irgnst;
      abserr -= wk[isbrgn-1];
      result -= wk[isbrgn-2];
      idvax0  = (unsigned int)(wk[isbrgn-3]);
      for (j=0;j<n;j++) {
         isbtmp = isbrgn-2*j-4;
         ctr[j] = wk[isbtmp];
         width[j] = wk[isbtmp-1];
      }
      if (idvax0 < 1) {
         // Can happen for overflows / degenerate floats.
         idvax0 = 1;
         MATH_ERROR_MSG("AdaptiveIntegratorMultiDim::DoIntegral()", "Logic error: idvax0 < 1!");
         //::Error("AdaptiveIntegratorMultiDim::DoIntegral()", "Logic error: idvax0 < 1!");
      }
      width[idvax0-1]  = 0.5*width[idvax0-1];
      ctr[idvax0-1] -= width[idvax0-1];
      goto L20;
   }
   nfnevl = ifncls;       //number of function evaluations performed.
   fResult = result;
   fError = abserr;//wk[isbrgn-1];
   fRelError = relerr;
   fNEval = nfnevl;
   delete [] wk;
 
   return result;         //an approximate value of the integral
}
 
 
 
double AdaptiveIntegratorMultiDim::Integral(const IMultiGenFunction &f, const double* xmin, const double * xmax)
{
   // calculate integral passing a function object
   fFun = &f;
   return Integral(xmin, xmax);
 
}
 
ROOT::Math::IntegratorMultiDimOptions  AdaptiveIntegratorMultiDim::Options() const {
   // return the used options
   ROOT::Math::IntegratorMultiDimOptions opt;
   opt.SetAbsTolerance(fAbsTol);
   opt.SetRelTolerance(fRelTol);
   opt.SetNCalls(fMaxPts);
   opt.SetWKSize(fSize);
   opt.SetIntegrator("ADAPTIVE");
   return opt;
}
 
void AdaptiveIntegratorMultiDim::SetOptions(const ROOT::Math::IntegratorMultiDimOptions & opt)
{
   //   set integration options
   if (opt.IntegratorType() != IntegrationMultiDim::kADAPTIVE) {
      MATH_ERROR_MSG("AdaptiveIntegratorMultiDim::SetOptions","Invalid options");
      return;
   }
   SetAbsTolerance( opt.AbsTolerance() );
   SetRelTolerance( opt.RelTolerance() );
   SetMaxPts( opt.NCalls() );
   SetSize( opt.WKSize() );
}
 
} // namespace Math
} // namespace ROOT