TF3.cxx

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// @(#)root/hist:$Id$
// Author: Rene Brun   27/10/95
 
/*************************************************************************
 * Copyright (C) 1995-2000, Rene Brun and Fons Rademakers.               *
 * All rights reserved.                                                  *
 *                                                                       *
 * For the licensing terms see $ROOTSYS/LICENSE.                         *
 * For the list of contributors see $ROOTSYS/README/CREDITS.             *
 *************************************************************************/
 
#include "TROOT.h"
#include "TF3.h"
#include "TBuffer.h"
#include "TMath.h"
#include "TH3.h"
#include "TVirtualPad.h"
#include "TRandom.h"
#include "TVectorD.h"
#include "Riostream.h"
#include "TColor.h"
#include "TVirtualFitter.h"
#include "TVirtualHistPainter.h"
#include "Math/IntegratorOptions.h"
#include <cassert>
 
 
/** \class TF3
    \ingroup Functions
TF3 defines a 3D Function with Parameters.
 
3D implicit functions can be visualized as iso-surfaces. An implicit surface is defined 
by the equation f(x,y,z) = 0 and is rendered in Cartesian coordinates.
 
In the example below, the drawing options "FB" and "BB" are used to remove the 
front box and back box of the 3D frame keeping only the three axes.
 
Begin_Macro(source)
{
   auto C = new TCanvas("C","C",500,500);
   auto f3 = new TF3("gyroid",
      "sin(x)*cos(y) + sin(y)*cos(z) + sin(z)*cos(x)",
      -4, 4, -4, 4, -4, 4);
   f3->SetFillColor(50);
   f3->SetLineColor(15);
   f3->Draw("FBBB");
}
End_Macro
 
*/
 
////////////////////////////////////////////////////////////////////////////////
/// F3 default constructor
 
TF3::TF3()
{
   fNpz  = 0;
   fZmin = 0;
   fZmax = 1;
}
 
////////////////////////////////////////////////////////////////////////////////
/// TF3 constructor using a formula definition and string option args
///
/// See TFormula constructor for explanation of the formula syntax.
 
TF3::TF3(const char *name, const char *formula, Double_t xmin, Double_t xmax, Double_t ymin, Double_t ymax,
         Double_t zmin, Double_t zmax, Option_t *opt)
   : TF2(name, formula, xmin, xmax, ymax, ymin,
         opt) // purposely swapped ymax, ymin to signal that TFormula may be 1D or 2D or 3D
{
   fZmin   = zmin;
   fZmax   = zmax;
   fNpz    = 30;
   Int_t ndim = GetNdim();
   // accept 1-d or 2-d formula
   if (ndim < 3) fNdim = 3;
   if (ndim > 3 && xmin < xmax && ymin < ymax && zmin < zmax) {
      Error("TF3","function: %s/%s has dimension %d instead of 3",name,formula,ndim);
      MakeZombie();
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// TF3 constructor using a formula definition and explicit option args
///
/// See TFormula constructor for explanation of the formula syntax.
 
TF3::TF3(const char *name, const char *formula, Double_t xmin, Double_t xmax, Double_t ymin, Double_t ymax,
         Double_t zmin, Double_t zmax, EAddToList addToGlobList, bool vectorize)
   : TF2(name, formula, xmin, xmax, ymax, ymin, addToGlobList,
         vectorize) // purposely swapped ymax, ymin to signal that TFormula may be 1D or 2D or 3D
{
   fZmin = zmin;
   fZmax = zmax;
   fNpz = 30;
   Int_t ndim = GetNdim();
   // accept 1-d or 2-d formula
   if (ndim < 3)
      fNdim = 3;
   if (ndim > 3 && xmin < xmax && ymin < ymax && zmin < zmax) {
      Error("TF3", "function: %s/%s has dimension %d instead of 3", name, formula, ndim);
      MakeZombie();
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// TF3 constructor using a pointer to real function
///
/// \param[in] name object name
/// \param[in] fcn pointer to real function
/// \param[in] xmin,xmax x axis limits
/// \param[in] ymin,ymax y axis limits
/// \param[in] zmin,zmax z axis limits
/// \param[in] npar is the number of free parameters used by the function
/// \param[in] ndim number of dimensions
///
/// For example, for a 3-dim function with 3 parameters, the user function
/// looks like:
///
///     Double_t fun1(Double_t *x, Double_t *par)
///     return par[0]*x[2] + par[1]*exp(par[2]*x[0]*x[1]);
///
/// \warning A function created with this constructor cannot be Cloned.
 
TF3::TF3(const char *name,Double_t (*fcn)(Double_t *, Double_t *), Double_t xmin, Double_t xmax, Double_t ymin, Double_t ymax, Double_t zmin, Double_t zmax, Int_t npar, Int_t ndim, EAddToList addToGlobList)
      :TF2(name,fcn,xmin,xmax,ymin,ymax,npar,ndim,addToGlobList)
{
   fZmin   = zmin;
   fZmax   = zmax;
   fNpz    = 30;
}
 
////////////////////////////////////////////////////////////////////////////////
/// TF3 constructor using a pointer to real function---
///
/// \param[in] name object name
/// \param[in] fcn pointer to real function
/// \param[in] xmin,xmax x axis limits
/// \param[in] ymin,ymax y axis limits
/// \param[in] zmin,zmax z axis limits
/// \param[in] npar is the number of free parameters used by the function
/// \param[in] ndim number of dimensions
///
/// For example, for a 3-dim function with 3 parameters, the user function
/// looks like:
///
///     Double_t fun1(Double_t *x, Double_t *par)
///     return par[0]*x[2] + par[1]*exp(par[2]*x[0]*x[1]);
///
/// WARNING! A function created with this constructor cannot be Cloned.
 
TF3::TF3(const char *name,Double_t (*fcn)(const Double_t *, const Double_t *), Double_t xmin, Double_t xmax, Double_t ymin, Double_t ymax, Double_t zmin, Double_t zmax, Int_t npar, Int_t ndim, EAddToList addToGlobList)
   : TF2(name,fcn,xmin,xmax,ymin,ymax,npar,ndim,addToGlobList),
   fZmin(zmin),
   fZmax(zmax),
   fNpz(30)
{
}
 
////////////////////////////////////////////////////////////////////////////////
/// TF3 constructor using a ParamFunctor
///
/// a functor class implementing operator() (double *, double *)
///
/// \param[in] name object name
/// \param[in] f parameter functor
/// \param[in] xmin,xmax x axis limits
/// \param[in] ymin,ymax y axis limits
/// \param[in] zmin,zmax z axis limits
/// \param[in] npar is the number of free parameters used by the function
/// \param[in] ndim number of dimensions
///
/// \warning A function created with this constructor cannot be Cloned.
 
TF3::TF3(const char *name, ROOT::Math::ParamFunctor f, Double_t xmin, Double_t xmax, Double_t ymin, Double_t ymax, Double_t zmin, Double_t zmax, Int_t npar, Int_t ndim, EAddToList addToGlobList)
   : TF2(name, f, xmin, xmax, ymin, ymax,  npar, ndim, addToGlobList),
   fZmin(zmin),
   fZmax(zmax),
   fNpz(30)
{
}
 
////////////////////////////////////////////////////////////////////////////////
/// Operator =
 
TF3& TF3::operator=(const TF3 &rhs)
{
   if (this != &rhs)
      rhs.TF3::Copy(*this);
   return *this;
}
 
////////////////////////////////////////////////////////////////////////////////
/// F3 default destructor
 
TF3::~TF3()
{
}
 
////////////////////////////////////////////////////////////////////////////////
/// Copy constructor.
 
TF3::TF3(const TF3 &f3) : TF2()
{
   f3.TF3::Copy(*this);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Copy this F3 to a new F3
 
void TF3::Copy(TObject &obj) const
{
   TF2::Copy(obj);
   ((TF3&)obj).fZmin = fZmin;
   ((TF3&)obj).fZmax = fZmax;
   ((TF3&)obj).fNpz  = fNpz;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Compute distance from point px,py to a function
///
///  Compute the closest distance of approach from point px,py to this function.
///  The distance is computed in pixels units.
 
 
Int_t TF3::DistancetoPrimitive(Int_t px, Int_t py)
{
   return TF1::DistancetoPrimitive(px, py);
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// Draw this function with iso-surfaces.
 
void TF3::Draw(Option_t *option)
{
   TString opt = option;
   opt.ToLower();
   if (gPad && !opt.Contains("same")) gPad->Clear();
 
   AppendPad(option);
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// Execute action corresponding to one event
///
///  This member function is called when a F3 is clicked with the locator
 
void TF3::ExecuteEvent(Int_t event, Int_t px, Int_t py)
{
   TF1::ExecuteEvent(event, px, py);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return minimum/maximum value of the function
///
/// To find the minimum on a range, first set this range via the SetRange function
/// If a vector x of coordinate is passed it will be used as starting point for the minimum.
/// In addition on exit x will contain the coordinate values at the minimuma
/// If x is NULL or x is inifinity or NaN, first, a grid search is performed to find the initial estimate of the
/// minimum location. The range of the function is divided into fNpx and fNpy
/// sub-ranges. If the function is "good" (or "bad"), these values can be changed
/// by SetNpx and SetNpy functions
///
/// Then, a minimization is used with starting values found by the grid search
/// The minimizer algorithm used (by default Minuit) can be changed by callinga
///  ROOT::Math::Minimizer::SetDefaultMinimizerType("..")
/// Other option for the minimizer can be set using the static method of the MinimizerOptions class
 
Double_t TF3::FindMinMax(Double_t *x, Bool_t findmax) const
{
   //First do a grid search with step size fNpx and fNpy
 
   Double_t xx[3];
   Double_t rsign = (findmax) ? -1. : 1.;
   TF3 & function = const_cast<TF3&>(*this); // needed since EvalPar is not const
   Double_t xxmin = 0, yymin = 0, zzmin = 0, ttmin = 0;
   if (x == nullptr || ( (x!= nullptr) && ( !TMath::Finite(x[0]) || !TMath::Finite(x[1]) || !TMath::Finite(x[2]) ) ) ){
      Double_t dx = (fXmax - fXmin)/fNpx;
      Double_t dy = (fYmax - fYmin)/fNpy;
      Double_t dz = (fZmax - fZmin)/fNpz;
      xxmin = fXmin;
      yymin = fYmin;
      zzmin = fZmin;
      ttmin = rsign * TMath::Infinity();
      for (Int_t i=0; i<fNpx; i++){
         xx[0]=fXmin + (i+0.5)*dx;
         for (Int_t j=0; j<fNpy; j++){
            xx[1]=fYmin+(j+0.5)*dy;
            for (Int_t k=0; k<fNpz; k++){
               xx[2] = fZmin+(k+0.5)*dz;
               Double_t tt = function(xx);
               if (rsign*tt < rsign*ttmin) {xxmin = xx[0], yymin = xx[1]; zzmin = xx[2]; ttmin=tt;}
            }
         }
      }
 
      xxmin = TMath::Min(fXmax, xxmin);
      yymin = TMath::Min(fYmax, yymin);
      zzmin = TMath::Min(fZmax, zzmin);
   }
   else {
      xxmin = x[0];
      yymin = x[1];
      zzmin = x[2];
      zzmin = function(x);
   }
   xx[0] = xxmin;
   xx[1] = yymin;
   xx[2] = zzmin;
 
   double fmin = GetMinMaxNDim(xx,findmax);
   if (rsign*fmin < rsign*zzmin) {
      if (x) {x[0] = xx[0]; x[1] = xx[1]; x[2] = xx[2];}
      return fmin;
   }
   // here if minimization failed
   if (x) { x[0] = xxmin; x[1] = yymin; x[2] = zzmin; }
   return ttmin;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Compute the X, Y and Z values corresponding to the minimum value of the function
/// on its range.
///
/// Returns the function value at the minimum.
/// To find the minimum on a subrange, use the SetRange() function first.
///
/// Method:
///   First, a grid search is performed to find the initial estimate of the
///   minimum location. The range of the function is divided
///   into fNpx,fNpy and fNpz sub-ranges. If the function is "good" (or "bad"),
///   these values can be changed by SetNpx(), SetNpy() and SetNpz() functions.
///   Then, Minuit minimization is used with starting values found by the grid search
///
///   Note that this method will always do first a grid search in contrast to GetMinimum
 
Double_t TF3::GetMinimumXYZ(Double_t &x, Double_t &y, Double_t &z)
{
   double xx[3] = { 0,0,0 };
   xx[0] = TMath::QuietNaN();  // to force to do grid search in TF3::FindMinMax
   double fmin = FindMinMax(xx, false);
   x = xx[0]; y = xx[1]; z = xx[2];
   return fmin;
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// Compute the X, Y and Z values corresponding to the maximum value of the function
/// on its range.
///
/// Return the function value at the maximum. See TF3::GetMinimumXYZ
 
Double_t TF3::GetMaximumXYZ(Double_t &x, Double_t &y, Double_t &z)
{
   double xx[3] = { 0,0,0 };
   xx[0] = TMath::QuietNaN();  // to force to do grid search in TF3::FindMinMax
   double fmax = FindMinMax(xx, true);
   x = xx[0]; y = xx[1]; z = xx[2];
   return fmax;
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return 3 random numbers following this function shape
///
/// The distribution contained in this TF3 function is integrated
/// over the cell contents.
/// It is normalized to 1.
/// Getting the three random numbers implies:
///   - Generating a random number between 0 and 1 (say r1)
///   - Look in which cell in the normalized integral r1 corresponds to
///   - make a linear interpolation in the returned cell
///
///  IMPORTANT NOTE
///
///  The integral of the function is computed at fNpx * fNpy * fNpz points.
///  If the function has sharp peaks, you should increase the number of
///  points (SetNpx, SetNpy, SetNpz) such that the peak is correctly tabulated
///  at several points.
 
void TF3::GetRandom3(Double_t &xrandom, Double_t &yrandom, Double_t &zrandom, TRandom * rng)
{
   //  Check if integral array must be built
   Int_t i,j,k,cell;
   Double_t dx   = (fXmax-fXmin)/fNpx;
   Double_t dy   = (fYmax-fYmin)/fNpy;
   Double_t dz   = (fZmax-fZmin)/fNpz;
   Int_t ncells = fNpx*fNpy*fNpz;
   Double_t xx[3];
   Double_t *parameters = GetParameters();
   InitArgs(xx,parameters);
   if (fIntegral.empty() ) {
      fIntegral.resize(ncells+1);
      //fIntegral = new Double_t[ncells+1];
      fIntegral[0] = 0;
      Double_t integ;
      Int_t intNegative = 0;
      cell = 0;
      for (k=0;k<fNpz;k++) {
         xx[2] = fZmin+(k+0.5)*dz;
         for (j=0;j<fNpy;j++) {
            xx[1] = fYmin+(j+0.5)*dy;
            for (i=0;i<fNpx;i++) {
               xx[0] = fXmin+(i+0.5)*dx;
               integ = EvalPar(xx,parameters);
               if (integ < 0) {intNegative++; integ = -integ;}
               fIntegral[cell+1] = fIntegral[cell] + integ;
               cell++;
            }
         }
      }
      if (intNegative > 0) {
         Warning("GetRandom3","function:%s has %d negative values: abs assumed",GetName(),intNegative);
      }
      if (fIntegral[ncells] == 0) {
         Error("GetRandom3","Integral of function is zero");
         return;
      }
      for (i=1;i<=ncells;i++) {  // normalize integral to 1
         fIntegral[i] /= fIntegral[ncells];
      }
   }
 
// return random numbers
   Double_t r;
   if (!rng) rng = gRandom;
   r    = rng->Rndm();
   cell = TMath::BinarySearch(ncells,fIntegral.data(),r);
   k    = cell/(fNpx*fNpy);
   j    = (cell -k*fNpx*fNpy)/fNpx;
   i    = cell -fNpx*(j +fNpy*k);
   xrandom = fXmin +dx*i +dx*rng->Rndm();
   yrandom = fYmin +dy*j +dy*rng->Rndm();
   zrandom = fZmin +dz*k +dz*rng->Rndm();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return range of function
 
void TF3::GetRange(Double_t &xmin, Double_t &ymin, Double_t &zmin, Double_t &xmax, Double_t &ymax, Double_t &zmax) const
{
   xmin = fXmin;
   xmax = fXmax;
   ymin = fYmin;
   ymax = fYmax;
   zmin = fZmin;
   zmax = fZmax;
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Get value corresponding to X in array of fSave values
 
Double_t TF3::GetSave(const Double_t *xx)
{
   if (fSave.size() < 9) return 0;
   Int_t nsave = fSave.size() - 9;
   Double_t xmin = fSave[nsave+0];
   Double_t xmax = fSave[nsave+1];
   Double_t ymin = fSave[nsave+2];
   Double_t ymax = fSave[nsave+3];
   Double_t zmin = fSave[nsave+4];
   Double_t zmax = fSave[nsave+5];
   Int_t npx     = Int_t(fSave[nsave+6]);
   Int_t npy     = Int_t(fSave[nsave+7]);
   Int_t npz     = Int_t(fSave[nsave+8]);
   Double_t x    = xx[0];
   Double_t dx   = (xmax-xmin)/npx;
   if (x < xmin || x > xmax) return 0;
   if (dx <= 0) return 0;
   Double_t y    = xx[1];
   Double_t dy   = (ymax-ymin)/npy;
   if (y < ymin || y > ymax) return 0;
   if (dy <= 0) return 0;
   Double_t z    = xx[2];
   Double_t dz   = (zmax-zmin)/npz;
   if (z < zmin || z > zmax) return 0;
   if (dz <= 0) return 0;
 
   //we make a trilinear interpolation using the 8 points surrounding x,y,z
   Int_t ibin    = TMath::Min(npx-1, Int_t((x-xmin)/dx));
   Int_t jbin    = TMath::Min(npy-1, Int_t((y-ymin)/dy));
   Int_t kbin    = TMath::Min(npz-1, Int_t((z-zmin)/dz));
   Double_t xlow = xmin + ibin*dx;
   Double_t ylow = ymin + jbin*dy;
   Double_t zlow = zmin + kbin*dz;
   Double_t t    = (x-xlow)/dx;
   Double_t u    = (y-ylow)/dy;
   Double_t v    = (z-zlow)/dz;
   Int_t k1      = (ibin  ) + (npx+1)*((jbin  ) + (npy+1)*(kbin  ));
   Int_t k2      = (ibin+1) + (npx+1)*((jbin  ) + (npy+1)*(kbin  ));
   Int_t k3      = (ibin+1) + (npx+1)*((jbin+1) + (npy+1)*(kbin  ));
   Int_t k4      = (ibin  ) + (npx+1)*((jbin+1) + (npy+1)*(kbin  ));
   Int_t k5      = (ibin  ) + (npx+1)*((jbin  ) + (npy+1)*(kbin+1));
   Int_t k6      = (ibin+1) + (npx+1)*((jbin  ) + (npy+1)*(kbin+1));
   Int_t k7      = (ibin+1) + (npx+1)*((jbin+1) + (npy+1)*(kbin+1));
   Int_t k8      = (ibin  ) + (npx+1)*((jbin+1) + (npy+1)*(kbin+1));
   Double_t r    = (1-t)*(1-u)*(1-v)*fSave[k1] + t*(1-u)*(1-v)*fSave[k2] + t*u*(1-v)*fSave[k3] + (1-t)*u*(1-v)*fSave[k4] +
                   (1-t)*(1-u)*v*fSave[k5] + t*(1-u)*v*fSave[k6] + t*u*v*fSave[k7] + (1-t)*u*v*fSave[k8];
   return r;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Create the basic function objects
 
void TF3::InitStandardFunctions()
{
   TF3 *f3;
   R__LOCKGUARD(gROOTMutex);
   if (!gROOT->GetListOfFunctions()->FindObject("xyzgaus")) {
      f3 = new TF3("xyzgaus", "xyzgaus", -1, 1, -1, 1, -1, 1);
      f3->SetParameters(1, 0, 1, 0, 1, 0, 1);
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return Integral of a 3d function in range [ax,bx],[ay,by],[az,bz]
/// with a desired relative accuracy.
 
Double_t TF3::Integral(Double_t ax, Double_t bx, Double_t ay, Double_t by, Double_t az, Double_t bz, Double_t epsrel)
{
   Double_t a[3], b[3];
   a[0] = ax;
   b[0] = bx;
   a[1] = ay;
   b[1] = by;
   a[2] = az;
   b[2] = bz;
   Double_t relerr  = 0;
   Int_t n = 3;
   Int_t maxpts = TMath::Max(UInt_t(fNpx * fNpy * fNpz), ROOT::Math::IntegratorMultiDimOptions::DefaultNCalls());
   Int_t nfnevl, ifail;
   Double_t result = IntegralMultiple(n,a,b,maxpts,epsrel,epsrel, relerr,nfnevl,ifail);
   if (ifail > 0) {
      Warning("Integral","failed for %s code=%d, maxpts=%d, epsrel=%g, nfnevl=%d, relerr=%g ",GetName(),ifail,maxpts,epsrel,nfnevl,relerr);
   }
   if (gDebug) {
      Info("Integral","Integral of %s using %d and tol=%f is %f , relerr=%f nfcn=%d",GetName(),maxpts,epsrel,result,relerr,nfnevl);
   }
   return result;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return kTRUE is the point is inside the function range
 
Bool_t TF3::IsInside(const Double_t *x) const
{
   if (x[0] < fXmin || x[0] > fXmax) return kFALSE;
   if (x[1] < fYmin || x[1] > fYmax) return kFALSE;
   if (x[2] < fZmin || x[2] > fZmax) return kFALSE;
   return kTRUE;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Create a histogram for axis range.
 
TH1* TF3::CreateHistogram()
{
   TH1* h = new TH3F("R__TF3",(char*)GetTitle(),fNpx,fXmin,fXmax
                         ,fNpy,fYmin,fYmax
                         ,fNpz,fZmin,fZmax);
   h->SetDirectory(nullptr);
   return h;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Paint this 3-D function with its current attributes
 
void TF3::Paint(Option_t *option)
{
 
   TString opt = option;
   opt.ToLower();
 
//-  Create a temporary histogram and fill each channel with the function value
   if (!fHistogram) {
      fHistogram = new TH3F("R__TF3",(char*)GetTitle(),fNpx,fXmin,fXmax
                                                      ,fNpy,fYmin,fYmax
                                                      ,fNpz,fZmin,fZmax);
      fHistogram->SetDirectory(nullptr);
   }
 
   fHistogram->GetPainter(option)->ProcessMessage("SetF3",this);
 
   if (opt.Index("tf3") == kNPOS)
      opt.Append("tf3");
 
   fHistogram->Paint(opt.Data());
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set the function clipping box (for drawing) "off".
 
void TF3::SetClippingBoxOff()
{
   fClipBoxOn = kFALSE;
   fClipBox[0] = fClipBox[1] = fClipBox[2] = 0;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Save values of function in array fSave
 
void TF3::Save(Double_t xmin, Double_t xmax, Double_t ymin, Double_t ymax, Double_t zmin, Double_t zmax)
{
   if (!fSave.empty()) fSave.clear();
   Int_t npx = fNpx, npy = fNpy, npz = fNpz;
   if ((npx < 2) || (npy < 2) || (npz < 2))
      return;
 
   Double_t dx = (xmax-xmin)/fNpx;
   Double_t dy = (ymax-ymin)/fNpy;
   Double_t dz = (zmax-zmin)/fNpz;
   if (dx <= 0) {
      dx = (fXmax-fXmin)/fNpx;
      npx--;
      xmin = fXmin + 0.5*dx;
      xmax = fXmax - 0.5*dx;
   }
   if (dy <= 0) {
      dy = (fYmax-fYmin)/fNpy;
      npy--;
      ymin = fYmin + 0.5*dy;
      ymax = fYmax - 0.5*dy;
   }
   if (dz <= 0) {
      dz = (fZmax-fZmin)/fNpz;
      npz--;
      zmin = fZmin + 0.5*dz;
      zmax = fZmax - 0.5*dz;
   }
   Int_t nsave = (npx + 1)*(npy + 1)*(npz + 1);
   fSave.resize(nsave + 9);
   Double_t xv[3];
   Double_t *pp = GetParameters();
   InitArgs(xv,pp);
   for (Int_t k = 0, l = 0; k <= npz; k++) {
      xv[2]    = zmin + dz*k;
      for (Int_t j = 0; j <= npy; j++) {
         xv[1]    = ymin + dy*j;
         for (Int_t i = 0; i <= npx; i++) {
            xv[0]    = xmin + dx*i;
            fSave[l++] = EvalPar(xv, pp);
         }
      }
   }
   fSave[nsave+0] = xmin;
   fSave[nsave+1] = xmax;
   fSave[nsave+2] = ymin;
   fSave[nsave+3] = ymax;
   fSave[nsave+4] = zmin;
   fSave[nsave+5] = zmax;
   fSave[nsave+6] = npx;
   fSave[nsave+7] = npy;
   fSave[nsave+8] = npz;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Save primitive as a C++ statement(s) on output stream out
 
void TF3::SavePrimitive(std::ostream &out, Option_t *option /*= ""*/)
{
   TString f3Name = ProvideSaveName(option);
   out << "   \n";
   out << "   TF3 *";
 
   if (!fMethodCall)
      out << f3Name << " = new TF3(\"" << GetName() << "\", \"" << TString(GetTitle()).ReplaceSpecialCppChars() << "\","
          << fXmin << "," << fXmax << "," << fYmin << "," << fYmax << "," << fZmin << "," << fZmax << ");\n";
   else
      out << f3Name << " = new TF3(\"" << GetName() << "\", " << GetTitle() << "," << fXmin << "," << fXmax << ","
          << fYmin << "," << fYmax << "," << fZmin << "," << fZmax << "," << GetNpar() << ");\n";
 
   SaveFillAttributes(out, f3Name, -1, 0);
   SaveMarkerAttributes(out, f3Name, 1, 1, 1);
   SaveLineAttributes(out, f3Name, 1, 1, 4);
 
   if (GetNpx() != 30)
      out << "   " << f3Name << "->SetNpx(" << GetNpx() << ");\n";
   if (GetNpy() != 30)
      out << "   " << f3Name << "->SetNpy(" << GetNpy() << ");\n";
   if (GetNpz() != 30)
      out << "   " << f3Name << "->SetNpz(" << GetNpz() << ");\n";
 
   if (GetChisquare() != 0)
      out << "   " << f3Name << "->SetChisquare(" << GetChisquare() << ");\n";
 
   Double_t parmin, parmax;
   for (Int_t i = 0; i < GetNpar(); i++) {
      out << "   " << f3Name << "->SetParameter(" << i << "," << GetParameter(i) << ");\n";
      out << "   " << f3Name << "->SetParError(" << i << "," << GetParError(i) << ");\n";
      GetParLimits(i, parmin, parmax);
      out << "   " << f3Name << "->SetParLimits(" << i << "," << parmin << "," << parmax << ");\n";
   }
 
   if (GetXaxis())
      GetXaxis()->SaveAttributes(out, f3Name, "->GetXaxis()");
   if (GetYaxis())
      GetYaxis()->SaveAttributes(out, f3Name, "->GetYaxis()");
   if (GetZaxis())
      GetZaxis()->SaveAttributes(out, f3Name, "->GetZaxis()");
 
   SavePrimitiveDraw(out, f3Name, option);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set the function clipping box (for drawing) "on" and define the clipping box.
/// xclip, yclip and zclip is a point within the function range. All the
/// function values having x<=xclip and y<=yclip and z>=zclip are clipped.
 
void TF3::SetClippingBoxOn(Double_t xclip, Double_t yclip, Double_t zclip)
{
   fClipBoxOn = kTRUE;
   fClipBox[0] = xclip;
   fClipBox[1] = yclip;
   fClipBox[2] = zclip;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set the number of points used to draw the function
///
/// The default number of points along x is 30 for 2-d/3-d functions.
/// You can increase this value to get a better resolution when drawing
/// pictures with sharp peaks or to get a better result when using TF3::GetRandom2
/// the minimum number of points is 4, the maximum is 10000 for 2-d/3-d functions
 
void TF3::SetNpz(Int_t npz)
{
   if (npz < 4) {
      Warning("SetNpz","Number of points must be >=4 && <= 10000, fNpz set to 4");
      fNpz = 4;
   } else if(npz > 10000) {
      Warning("SetNpz","Number of points must be >=4 && <= 10000, fNpz set to 10000");
      fNpz = 10000;
   } else {
      fNpz = npz;
   }
   Update();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Initialize the upper and lower bounds to draw the function
 
void TF3::SetRange(Double_t xmin, Double_t ymin, Double_t zmin, Double_t xmax, Double_t ymax, Double_t zmax)
{
   fXmin = xmin;
   fXmax = xmax;
   fYmin = ymin;
   fYmax = ymax;
   fZmin = zmin;
   fZmax = zmax;
   Update();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Stream an object of class TF3.
 
void TF3::Streamer(TBuffer &R__b)
{
   if (R__b.IsReading()) {
      UInt_t R__s, R__c;
      Version_t R__v = R__b.ReadVersion(&R__s, &R__c);
      if (R__v > 0) {
         R__b.ReadClassBuffer(TF3::Class(), this, R__v, R__s, R__c);
         return;
      }
 
   } else {
      Int_t saved = 0;
      if (fType != EFType::kFormula && fSave.empty() ) { saved = 1; Save(fXmin,fXmax,fYmin,fYmax,fZmin,fZmax);}
 
      R__b.WriteClassBuffer(TF3::Class(),this);
 
      if (saved) { fSave.clear(); }
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return x^nx * y^ny * z^nz moment of a 3d function in range [ax,bx],[ay,by],[az,bz]
/// \author Gene Van Buren <gene@bnl.gov>
 
Double_t TF3::Moment3(Double_t nx, Double_t ax, Double_t bx, Double_t ny, Double_t ay, Double_t by, Double_t nz, Double_t az, Double_t bz, Double_t epsilon)
{
   Double_t norm = Integral(ax,bx,ay,by,az,bz,epsilon);
   if (norm == 0) {
      Error("Moment3", "Integral zero over range");
      return 0;
   }
 
   // define  integrand function as a lambda : g(x,y,z)=  x^(nx) * y^(ny) * z^(nz) * f(x,y,z)
   auto integrand = [&](double *x, double *) {
      return std::pow(x[0], nx) * std::pow(x[1], ny) * std::pow(x[2], nz) * this->EvalPar(x, nullptr);
   };
   // compute integral of g(x,y,z)
   TF3 fnc("TF3_ExpValHelper", integrand, ax, bx, ay, by, az, bz, 0);
   // set same points as current function to get correct max points when computing the integral
   fnc.fNpx = fNpx;
   fnc.fNpy = fNpy;
   fnc.fNpz = fNpz;
   return fnc.Integral(ax, bx, ay, by, az, bz, epsilon) / norm;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return x^nx * y^ny * z^nz central moment of a 3d function in range [ax,bx],[ay,by],[az,bz]
/// \author Gene Van Buren <gene@bnl.gov>
 
Double_t TF3::CentralMoment3(Double_t nx, Double_t ax, Double_t bx, Double_t ny, Double_t ay, Double_t by, Double_t nz, Double_t az, Double_t bz, Double_t epsilon)
{
   Double_t norm = Integral(ax,bx,ay,by,az,bz,epsilon);
   if (norm == 0) {
      Error("CentralMoment3", "Integral zero over range");
      return 0;
   }
 
   Double_t xbar = 0;
   Double_t ybar = 0;
   Double_t zbar = 0;
   if (nx!=0) {
      // compute first momentum in x
      auto integrandX = [&](double *x, double *) { return x[0] * this->EvalPar(x, nullptr); };
      TF3 fncx("TF3_ExpValHelperx", integrandX, ax, bx, ay, by, az, bz, 0);
      fncx.fNpx = fNpx;
      fncx.fNpy = fNpy;
      fncx.fNpz = fNpz;
      xbar = fncx.Integral(ax, bx, ay, by, az, bz, epsilon) / norm;
   }
   if (ny!=0) {
      auto integrandY = [&](double *x, double *) { return x[1] * this->EvalPar(x, nullptr); };
      TF3 fncy("TF3_ExpValHelpery", integrandY, ax, bx, ay, by, az, bz, 0);
      fncy.fNpx = fNpx;
      fncy.fNpy = fNpy;
      fncy.fNpz = fNpz;
      ybar = fncy.Integral(ax,bx,ay,by,az,bz,epsilon)/norm;
   }
   if (nz!=0) {
      auto integrandZ = [&](double *x, double *) { return x[2] * this->EvalPar(x, nullptr); };
      TF3 fncz("TF3_ExpValHelperz", integrandZ, ax, bx, ay, by, az, bz, 0);
      fncz.fNpx = fNpx;
      fncz.fNpy = fNpy;
      fncz.fNpz = fNpz;
      zbar = fncz.Integral(ax,bx,ay,by,az,bz,epsilon)/norm;
   }
   // define  integrand function as a lambda : g(x,y)=  (x-xbar)^(nx) * (y-ybar)^(ny) * f(x,y)
   auto integrand = [&](double *x, double *) {
      double xxx = (nx != 0) ? std::pow(x[0] - xbar, nx) : 1.;
      double yyy = (ny != 0) ? std::pow(x[1] - ybar, ny) : 1.;
      double zzz = (nz != 0) ? std::pow(x[2] - zbar, nz) : 1.;
      return xxx * yyy * zzz * this->EvalPar(x, nullptr);
   };
   // compute integral of g(x,y, z)
   TF3 fnc("TF3_ExpValHelper",integrand,ax,bx,ay,by,az,bz,0) ;
   fnc.fNpx = fNpx;
   fnc.fNpy = fNpy;
   fnc.fNpz = fNpz;
   return fnc.Integral(ax,bx,ay,by,az,bz,epsilon)/norm;
}