TF2.cxx

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// @(#)root/hist:$Id$
// Author: Rene Brun   23/08/95
 
/*************************************************************************
 * 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 "TF2.h"
#include "TMath.h"
#include "TRandom.h"
#include "TBuffer.h"
#include "TH2.h"
#include "TVirtualPad.h"
#include <iostream>
#include "TColor.h"
#include "TVirtualFitter.h"
#include "Math/IntegratorOptions.h"
#include "snprintf.h"
 
 
/** \class TF2
    \ingroup Functions
    \brief A 2-Dim function with parameters.
 
The following types of functions can be created:
 
1.  [Expression using variables x and y](\ref TF2a)
2.  [Expression using a user defined function](\ref TF2b)
3.  [Lambda Expression with x and y variables and parameters](\ref TF2c)
 
\anchor TF2a
### Expression using variables x and y
 
Begin_Macro (source)
{
    auto f2 = new TF2("f2","sin(x)*sin(y)/(x*y)",0,5,0,5);
    f2->Draw();
}
End_Macro
 
\anchor TF2b
### Expression using a user defined function
 
~~~~{.cpp}
Double_t func(Double_t *val, Double_t *par)
{
   Float_t x = val[0];
   Float_t y = val[1];
   Double_t f = x*x-y*y;
   return f;
}
 
void fplot()
{
   auto f = new TF2("f",func,-1,1,-1,1);
   f->Draw("surf1");
}
~~~~
 
\anchor TF2c
### Lambda Expression with x and y variables and parameters
 
~~~~{.cpp}
root [0] TF2 f2("f2", [](double* x, double*p) { return x[0] + x[1] * p[0]; }, 0., 1., 0., 1., 1)
(TF2 &) Name: f2 Title: f2
root [1] f2.SetParameter(0, 1.)
root [2] f2.Eval(1., 2.)
(double) 3.0000000
~~~~
 
See TF1 class for the list of functions formats
*/
 
////////////////////////////////////////////////////////////////////////////////
/// TF2 default constructor
 
TF2::TF2(): fYmin(0),fYmax(0),fNpy(100)
{
}
 
////////////////////////////////////////////////////////////////////////////////
/// TF2 constructor using a formula definition and string option args
///
/// See TFormula constructor for explanation of the formula syntax.
///
/// If formula has the form "fffffff;xxxx;yyyy", it is assumed that
/// the formula string is "fffffff" and "xxxx" and "yyyy" are the
/// titles for the X and Y axis respectively.
 
TF2::TF2(const char *name, const char *formula, Double_t xmin, Double_t xmax, Double_t ymin, Double_t ymax,
         Option_t *opt)
   : TF1(name, formula, xmax, xmin, opt) // purposely swapped xmax, xmin to signal that TFormula may be 1D or 2D
{
   if (ymin < ymax) {
      fYmin   = ymin;
      fYmax   = ymax;
   } else {
      fYmin = ymax;
      fYmax = ymin;
   }
   fNpx    = 30;
   fNpy    = 30;
   fContour.Set(0);
   // accept 1-d formula
   if (GetNdim() < 2) fNdim = 2;
   // dimension is obtained by TFormula
   // accept cases where formula dim is less than 2
   if (GetNdim() > 2 && xmin < xmax && ymin < ymax) {
      Error("TF2","function: %s/%s has dimension %d instead of 2",name,formula,GetNdim());
      MakeZombie();
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// TF2 constructor using a formula definition and explicit option args
///
/// See TFormula constructor for explanation of the formula syntax.
///
/// If formula has the form "fffffff;xxxx;yyyy", it is assumed that
/// the formula string is "fffffff" and "xxxx" and "yyyy" are the
/// titles for the X and Y axis respectively.
 
TF2::TF2(const char *name, const char *formula, Double_t xmin, Double_t xmax, Double_t ymin, Double_t ymax,
         EAddToList addToGlobList, bool vectorize)
   : TF1(name, formula, xmax, xmin, addToGlobList,
         vectorize) // purposely swapped xmax, xmin to signal that TFormula may be 1D or 2D
{
   if (ymin < ymax) {
      fYmin = ymin;
      fYmax = ymax;
   } else {
      fYmin = ymax;
      fYmax = ymin;
   }
   fNpx = 30;
   fNpy = 30;
   fContour.Set(0);
   // accept 1-d formula
   if (GetNdim() < 2)
      fNdim = 2;
   // dimension is obtained by TFormula
   // accept cases where formula dim is less than 2
   if (GetNdim() > 2 && xmin < xmax && ymin < ymax) {
      Error("TF2", "function: %s/%s has dimension %d instead of 2", name, formula, GetNdim());
      MakeZombie();
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// TF2 constructor using a pointer to a compiled function
///
/// npar is the number of free parameters used by the function
///
/// This constructor creates a function of type C when invoked
/// with the normal C++ compiler.
///
/// WARNING! A function created with this constructor cannot be Cloned.
 
TF2::TF2(const char *name, Double_t (*fcn)(Double_t *, Double_t *), Double_t xmin, Double_t xmax, Double_t ymin, Double_t ymax, Int_t npar, Int_t ndim, EAddToList addToGlobList)
      : TF1(name, fcn, xmin, xmax, npar, ndim, addToGlobList)
{
   fYmin   = ymin;
   fYmax   = ymax;
   fNpx    = 30;
   fNpy    = 30;
   fContour.Set(0);
}
 
TF2::TF2(const char *name, Double_t xmin, Double_t xmax, Double_t ymin, Double_t ymax, Int_t npar, Int_t ndim, EAddToList addToGlobList):
   TF1(name, xmin, xmax, npar, ndim, addToGlobList)
{
   fYmin   = ymin;
   fYmax   = ymax;
   fNpx    = 30;
   fNpy    = 30;
   fContour.Set(0);
}
 
////////////////////////////////////////////////////////////////////////////////
/// TF2 constructor using a pointer to a compiled function
///
/// npar is the number of free parameters used by the function
///
/// This constructor creates a function of type C when invoked
/// with the normal C++ compiler.
///
/// WARNING! A function created with this constructor cannot be Cloned.
 
TF2::TF2(const char *name, Double_t (*fcn)(const Double_t *, const Double_t *), Double_t xmin, Double_t xmax, Double_t ymin, Double_t ymax, Int_t npar, Int_t ndim, EAddToList addToGlobList)
      : TF1(name, fcn, xmin, xmax, npar, ndim, addToGlobList)
{
   fYmin   = ymin;
   fYmax   = ymax;
   fNpx    = 30;
   fNpy    = 30;
   fContour.Set(0);
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// TF2 constructor using a ParamFunctor,
///          a functor class implementing operator() (double *, double *)
///
/// npar is the number of free parameters used by the function
///
/// WARNING! A function created with this constructor cannot be Cloned.
 
TF2::TF2(const char *name, ROOT::Math::ParamFunctor f, Double_t xmin, Double_t xmax, Double_t ymin, Double_t ymax, Int_t npar, Int_t ndim, EAddToList addToGlobList)
      : TF1(name, f, xmin, xmax, npar, ndim, addToGlobList)
{
   fYmin   = ymin;
   fYmax   = ymax;
   fNpx    = 30;
   fNpy    = 30;
   fContour.Set(0);
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// Operator =
 
TF2& TF2::operator=(const TF2 &rhs)
{
   if (this != &rhs)
      rhs.TF2::Copy(*this);
   return *this;
}
 
////////////////////////////////////////////////////////////////////////////////
/// F2 default destructor
 
TF2::~TF2()
{
}
 
////////////////////////////////////////////////////////////////////////////////
/// Copy constructor.
 
TF2::TF2(const TF2 &f2) : TF1()
{
   f2.TF2::Copy(*this);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Copy this F2 to a new F2
 
void TF2::Copy(TObject &obj) const
{
   TF1::Copy(obj);
   ((TF2&)obj).fYmin    = fYmin;
   ((TF2&)obj).fYmax    = fYmax;
   ((TF2&)obj).fNpy     = fNpy;
   fContour.Copy(((TF2&)obj).fContour);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Compute distance from point px,py to a function
///
/// \param[in] px x position
/// \param[in] py y position
///
/// Compute the closest distance of approach from point px,py to this function.
/// The distance is computed in pixels units.
 
Int_t TF2::DistancetoPrimitive(Int_t px, Int_t py)
{
   if (!fHistogram) return 9999;
   Int_t distance = fHistogram->DistancetoPrimitive(px,py);
   if (distance <= 1) return distance;
 
   Double_t x = gPad->PadtoX(gPad->AbsPixeltoX(px));
   Double_t y = gPad->PadtoY(gPad->AbsPixeltoY(py));
   const char *drawOption = GetDrawOption();
   Double_t uxmin,uxmax;
   Double_t uymin,uymax;
   if (gPad->GetView() || strncmp(drawOption,"cont",4) == 0
                       || strncmp(drawOption,"CONT",4) == 0) {
      uxmin=gPad->GetUxmin();
      uxmax=gPad->GetUxmax();
      x = fXmin +(fXmax-fXmin)*(x-uxmin)/(uxmax-uxmin);
      uymin=gPad->GetUymin();
      uymax=gPad->GetUymax();
      y = fYmin +(fYmax-fYmin)*(y-uymin)/(uymax-uymin);
   }
   if (x < fXmin || x > fXmax) return distance;
   if (y < fYmin || y > fYmax) return distance;
   return 0;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Draw this function with its current attributes
///
/// NB. You must use DrawCopy if you want to draw several times the same
///     function in the current canvas.
 
void TF2::Draw(Option_t *option)
{
   TString opt = option;
   opt.ToLower();
   if (gPad && !opt.Contains("same")) gPad->Clear();
 
   AppendPad(option);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Draw a copy of this function with its current attributes-*
///
/// This function MUST be used instead of Draw when you want to draw
/// the same function with different parameters settings in the same canvas.
///
/// Possible option values are:
///
/// option   | description
/// ---------|------------
///   "SAME" | superimpose on top of existing picture
///   "L"    | connect all computed points with a straight line
///   "C"    | connect all computed points with a smooth curve.
///
/// Note that the default value is "F". Therefore to draw on top
/// of an existing picture, specify option "SL"
 
 
TF1 *TF2::DrawCopy(Option_t *option) const
{
   TF2 *newf2 = new TF2();
   Copy(*newf2);
   newf2->AppendPad(option);
   newf2->SetBit(kCanDelete);
   return newf2;
}
 
// remove this function
//______________________________________________________________________________
// void TF2::DrawF2(const char *formula, Double_t xmin, Double_t ymin, Double_t xmax, Double_t ymax, Option_t *option)
// {
// //----Draw formula between xmin,ymin and xmax,ymax---
// //                ============================================
// //
 
//    //if (Compile((char*)formula)) return;
 
//    SetRange(xmin, ymin, xmax, ymax);
 
//    Draw(option);
 
// }
 
////////////////////////////////////////////////////////////////////////////////
/// Execute action corresponding to one event
///
/// This member function is called when a F2 is clicked with the locator
 
void TF2::ExecuteEvent(Int_t event, Int_t px, Int_t py)
{
   TF1::ExecuteEvent(event, px, py);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return contour values into array levels
///
/// The number of contour levels can be returned by getContourLevel
 
Int_t TF2::GetContour(Double_t *levels)
{
   Int_t nlevels = fContour.fN;
   if (levels) {
      for (Int_t level=0; level<nlevels; level++) levels[level] = GetContourLevel(level);
   }
   return nlevels;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return the number of contour levels
 
Double_t TF2::GetContourLevel(Int_t level) const
{
   if (level <0 || level >= fContour.fN) return 0;
   if (fContour.fArray[0] != -9999) return fContour.fArray[level];
   if (fHistogram == nullptr) return 0;
   return fHistogram->GetContourLevel(level);
}
 
////////////////////////////////////////////////////////////////////////////////
/// 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 infinity 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 TF2::FindMinMax(Double_t *x, Bool_t findmax) const
{
   //First do a grid search with step size fNpx and fNpy
 
   Double_t xx[2];
   Double_t rsign = (findmax) ? -1. : 1.;
   TF2 & function = const_cast<TF2&>(*this); // needed since EvalPar is not const
   Double_t xxmin = 0, yymin = 0, zzmin = 0;
   if (x == nullptr || ( (x!= nullptr) && ( !TMath::Finite(x[0]) || !TMath::Finite(x[1]) ) ) ){
      Double_t dx = (fXmax - fXmin)/fNpx;
      Double_t dy = (fYmax - fYmin)/fNpy;
      xxmin = fXmin;
      yymin = fYmin;
      zzmin = 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;
            Double_t zz = function(xx);
            if (rsign*zz < rsign*zzmin) {xxmin = xx[0], yymin = xx[1]; zzmin = zz;}
         }
      }
 
      xxmin = TMath::Min(fXmax, xxmin);
      yymin = TMath::Min(fYmax, yymin);
   }
   else {
      xxmin = x[0];
      yymin = x[1];
      zzmin = function(x);
   }
   xx[0] = xxmin;
   xx[1] = yymin;
 
   double fmin = GetMinMaxNDim(xx,findmax);
   if (rsign*fmin < rsign*zzmin) {
      if (x) {x[0] = xx[0]; x[1] = xx[1]; }
      return fmin;
   }
   // here if minimization failed
   if (x) { x[0] = xxmin; x[1] = yymin; }
   return zzmin;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Compute the X and Y values corresponding to the minimum value of the function
///
/// Return the minimum value of the function
/// To find the minimum on a range, first set this range via the SetRange function
///
/// 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 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
///
///   Note that this method will always do first a grid search in contrast to GetMinimum
 
Double_t TF2::GetMinimumXY(Double_t &x, Double_t &y) const
{
   double xx[2] = { 0,0 };
   xx[0] = TMath::QuietNaN();  // to force to do grid search in TF2::FindMinMax
   double fmin = FindMinMax(xx, false);
   x = xx[0]; y = xx[1];
   return fmin;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Compute the X and Y values corresponding to the maximum value of the function
///
/// Return the maximum value of the function
///  See TF2::GetMinimumXY
 
Double_t TF2::GetMaximumXY(Double_t &x, Double_t &y) const
{
   double xx[2] = { 0,0 };
   xx[0] = TMath::QuietNaN();  // to force to do grid search in TF2::FindMinMax
   double fmax = FindMinMax(xx, true);
   x = xx[0]; y = xx[1];
   return fmax;
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// 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 infinity 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 TF2::GetMinimum(Double_t *x) const
{
   return FindMinMax(x, false);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return maximum value of the function
/// See TF2::GetMinimum
 
Double_t TF2::GetMaximum(Double_t *x) const
{
   return FindMinMax(x, true);
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Redefines TObject::GetObjectInfo.
///
/// Displays the function value
/// corresponding to cursor position px,py
 
char *TF2::GetObjectInfo(Int_t px, Int_t py) const
{
   const char *snull = "";
   if (!gPad) return (char*)snull;
   static char info[64];
   Double_t x = gPad->PadtoX(gPad->AbsPixeltoX(px));
   Double_t y = gPad->PadtoY(gPad->AbsPixeltoY(py));
   const char *drawOption = GetDrawOption();
   Double_t uxmin,uxmax;
   Double_t uymin,uymax;
   if (gPad->GetView() || strncmp(drawOption,"cont",4) == 0
                       || strncmp(drawOption,"CONT",4) == 0) {
      uxmin=gPad->GetUxmin();
      uxmax=gPad->GetUxmax();
      x = fXmin +(fXmax-fXmin)*(x-uxmin)/(uxmax-uxmin);
      uymin=gPad->GetUymin();
      uymax=gPad->GetUymax();
      y = fYmin +(fYmax-fYmin)*(y-uymin)/(uymax-uymin);
   }
   snprintf(info,64,"(x=%g, y=%g, f=%.18g)",x,y,((TF2*)this)->Eval(x,y));
   return info;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return a random number following this function shape
 
Double_t TF2::GetRandom(TRandom *, Option_t *)
{
   Error("GetRandom","cannot be called for TF2/3, use GetRandom2/3 instead");
   return 0;  // not yet implemented
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return a random number following this function shape
 
 
Double_t TF2::GetRandom(Double_t, Double_t, TRandom *, Option_t *)
{
   Error("GetRandom","cannot be called for TF2/3, use GetRandom2/3 instead");
   return 0;  // not yet implemented
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return 2 random numbers following this function shape
///
///   The distribution contained in this TF2 function is integrated
///   over the cell contents.
///   It is normalized to 1.
///   Getting the two 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 points.
///  If the function has sharp peaks, you should increase the number of
///  points (SetNpx, SetNpy) such that the peak is correctly tabulated
///  at several points.
 
void TF2::GetRandom2(Double_t &xrandom, Double_t &yrandom, TRandom * rng)
{
   //  Check if integral array must be built
   Int_t i,j,cell;
   Double_t dx   = (fXmax-fXmin)/fNpx;
   Double_t dy   = (fYmax-fYmin)/fNpy;
   Int_t ncells = fNpx*fNpy;
   if (fIntegral.empty()) {
      fIntegral.resize(ncells+1);
      fIntegral[0] = 0;
      Double_t integ;
      Int_t intNegative = 0;
      cell = 0;
      for (j=0;j<fNpy;j++) {
         for (i=0;i<fNpx;i++) {
            integ = Integral(fXmin+i*dx,fXmin+i*dx+dx,fYmin+j*dy,fYmin+j*dy+dy);
            if (integ < 0) {intNegative++; integ = -integ;}
            fIntegral[cell+1] = fIntegral[cell] + integ;
            cell++;
         }
      }
      if (intNegative > 0) {
         Warning("GetRandom2","function:%s has %d negative values: abs assumed",GetName(),intNegative);
      }
      if (fIntegral[ncells] == 0) {
         Error("GetRandom2","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,ddx,ddy,dxint;
   if (!rng) rng = gRandom;
   r     = rng->Rndm();
   cell  = TMath::BinarySearch(ncells,fIntegral.data(),r);
   dxint = fIntegral[cell+1] - fIntegral[cell];
   if (dxint > 0) ddx = dx*(r - fIntegral[cell])/dxint;
   else           ddx = 0;
   ddy = dy*rng->Rndm();
   j   = cell/fNpx;
   i   = cell%fNpx;
   xrandom = fXmin +dx*i +ddx;
   yrandom = fYmin +dy*j +ddy;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return range of a 2-D function
 
void TF2::GetRange(Double_t &xmin, Double_t &ymin,  Double_t &xmax, Double_t &ymax) const
{
   xmin = fXmin;
   xmax = fXmax;
   ymin = fYmin;
   ymax = fYmax;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return range of function
 
void TF2::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 = 0;
   zmax = 0;
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Get value corresponding to X in array of fSave values
 
Double_t TF2::GetSave(const Double_t *xx)
{
   if (fSave.size() < 6) return 0;
   Int_t nsave = fSave.size() - 6;
   Double_t xmin = fSave[nsave+0];
   Double_t xmax = fSave[nsave+1];
   Double_t ymin = fSave[nsave+2];
   Double_t ymax = fSave[nsave+3];
   Int_t npx     = Int_t(fSave[nsave+4]);
   Int_t npy     = Int_t(fSave[nsave+5]);
   Double_t x    = Double_t(xx[0]);
   Double_t dx   = (xmax-xmin)/npx;
   if (x < xmin || x > xmax) return 0;
   if (dx <= 0) return 0;
   Double_t y    = Double_t(xx[1]);
   Double_t dy   = (ymax-ymin)/npy;
   if (y < ymin || y > ymax) return 0;
   if (dy <= 0) return 0;
 
   //we make a bilinear interpolation using the 4 points surrounding x,y
   Int_t ibin    = TMath::Min(npx-1, Int_t((x-xmin)/dx));
   Int_t jbin    = TMath::Min(npy-1, Int_t((y-ymin)/dy));
   Double_t xlow = xmin + ibin*dx;
   Double_t ylow = ymin + jbin*dy;
   Double_t t    = (x-xlow)/dx;
   Double_t u    = (y-ylow)/dy;
   Int_t k1      = jbin*(npx+1) + ibin;
   Int_t k2      = jbin*(npx+1) + ibin +1;
   Int_t k3      = (jbin+1)*(npx+1) + ibin +1;
   Int_t k4      = (jbin+1)*(npx+1) + ibin;
   Double_t z    = (1-t)*(1-u)*fSave[k1] +t*(1-u)*fSave[k2] +t*u*fSave[k3] + (1-t)*u*fSave[k4];
   return z;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Create the basic function objects
 
void TF2::InitStandardFunctions()
{
   TF2 *f2;
   R__LOCKGUARD(gROOTMutex);
   if (!gROOT->GetListOfFunctions()->FindObject("xygaus")) {
      f2 = new TF2("xygaus", "xygaus", -1, 1, -1, 1);
      f2->SetParameters(1, 0, 1, 0, 1);
      f2 = new TF2("bigaus", "bigaus", -1, 1, -1, 1);
      f2->SetParameters(1, 0, 1, 0, 1, 0);
      f2 = new TF2("xyexpo", "xyexpo", -1, 1, -1, 1);
      f2->SetParameters(1, 0, 1, 1, 0, 1);
      f2 = new TF2("xylandau", "xylandau", -1, 1, -1, 1);
      f2->SetParameters(1, 0, 1, 1, 0, 1);
      f2 = new TF2("xylandaun", "xylandaun", -1, 1, -1, 1);
      f2->SetParameters(1, 0, 1, 1, 0, 1);
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return Integral of a 2d function in range [ax,bx],[ay,by]
/// with desired relative accuracy (defined by eps)
 
Double_t TF2::Integral(Double_t ax, Double_t bx, Double_t ay, Double_t by, Double_t epsrel)
{
   Double_t a[2], b[2];
   a[0] = ax;
   b[0] = bx;
   a[1] = ay;
   b[1] = by;
   Double_t relerr  = 0;
   Int_t n = 2;
   UInt_t maxpts = TMath::Max( UInt_t( 20*fNpx*fNpy), 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 TF2::IsInside(const Double_t *x) const
{
   if (x[0] < fXmin || x[0] > fXmax) return kFALSE;
   if (x[1] < fYmin || x[1] > fYmax) return kFALSE;
   return kTRUE;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Create a histogram from function.
///
/// always created it, even if it is already existing
 
TH1* TF2::CreateHistogram()
{
   Int_t i,j,bin;
   Double_t dx, dy;
   Double_t xv[2];
 
 
   Double_t *parameters = GetParameters();
   TH2F* h = new TH2F("Func",(char*)GetTitle(),fNpx,fXmin,fXmax,fNpy,fYmin,fYmax);
   h->SetDirectory(nullptr);
 
   InitArgs(xv,parameters);
   dx = (fXmax - fXmin)/Double_t(fNpx);
   dy = (fYmax - fYmin)/Double_t(fNpy);
   for (i=1;i<=fNpx;i++) {
      xv[0] = fXmin + (Double_t(i) - 0.5)*dx;
      for (j=1;j<=fNpy;j++) {
         xv[1] = fYmin + (Double_t(j) - 0.5)*dy;
         bin   = j*(fNpx + 2) + i;
         h->SetBinContent(bin,EvalPar(xv,parameters));
      }
   }
   h->Fill(fXmin-1,fYmin-1,0);  //This call to force fNentries non zero
 
   Double_t *levels = fContour.GetArray();
   if (levels && levels[0] == -9999) levels = nullptr;
   h->SetMinimum(fMinimum);
   h->SetMaximum(fMaximum);
   h->SetContour(fContour.fN, levels);
   h->SetLineColor(GetLineColor());
   h->SetLineStyle(GetLineStyle());
   h->SetLineWidth(GetLineWidth());
   h->SetFillColor(GetFillColor());
   h->SetFillStyle(GetFillStyle());
   h->SetMarkerColor(GetMarkerColor());
   h->SetMarkerStyle(GetMarkerStyle());
   h->SetMarkerSize(GetMarkerSize());
   h->SetStats(false);
 
   return h;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Paint this 2-D function with its current attributes
 
void TF2::Paint(Option_t *option)
{
   Int_t i,j,bin;
   Double_t dx, dy;
   Double_t xv[2];
   Double_t *parameters = GetParameters();
   TString opt = option;
   opt.ToLower();
 
//-  Create a temporary histogram and fill each channel with the function value
   if (!fHistogram) {
      fHistogram = new TH2F("Func",(char*)GetTitle(),fNpx,fXmin,fXmax,fNpy,fYmin,fYmax);
      if (!fHistogram) return;
      fHistogram->SetDirectory(nullptr);
   }
   InitArgs(xv,parameters);
   dx = (fXmax - fXmin)/Double_t(fNpx);
   dy = (fYmax - fYmin)/Double_t(fNpy);
   for (i=1;i<=fNpx;i++) {
      xv[0] = fXmin + (Double_t(i) - 0.5)*dx;
      for (j=1;j<=fNpy;j++) {
         xv[1] = fYmin + (Double_t(j) - 0.5)*dy;
         bin   = j*(fNpx + 2) + i;
         fHistogram->SetBinContent(bin,EvalPar(xv,parameters));
      }
   }
   ((TH2F*)fHistogram)->Fill(fXmin-1,fYmin-1,0);  //This call to force fNentries non zero
 
//- Copy Function attributes to histogram attributes
   Double_t *levels = fContour.GetArray();
   if (levels && levels[0] == -9999) levels = nullptr;
   fHistogram->SetMinimum(fMinimum);
   fHistogram->SetMaximum(fMaximum);
   fHistogram->SetContour(fContour.fN, levels);
   fHistogram->SetLineColor(GetLineColor());
   fHistogram->SetLineStyle(GetLineStyle());
   fHistogram->SetLineWidth(GetLineWidth());
   fHistogram->SetFillColor(GetFillColor());
   fHistogram->SetFillStyle(GetFillStyle());
   fHistogram->SetMarkerColor(GetMarkerColor());
   fHistogram->SetMarkerStyle(GetMarkerStyle());
   fHistogram->SetMarkerSize(GetMarkerSize());
   fHistogram->SetStats(false);
   fHistogram->Sumw2(kFALSE);
 
//-  Draw the histogram
   if (!gPad) return;
   if (opt.Length() == 0)  fHistogram->Paint("cont3");
   else if (opt == "same") fHistogram->Paint("cont2same");
   else                    fHistogram->Paint(option);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Save values of function in array fSave
 
void TF2::Save(Double_t xmin, Double_t xmax, Double_t ymin, Double_t ymax, Double_t, Double_t)
{
   if (!fSave.empty())
      fSave.clear();
   Int_t npx = fNpx, npy = fNpy;
   if ((npx < 2) || (npy < 2))
      return;
   Double_t dx = (xmax-xmin)/fNpx;
   Double_t dy = (ymax-ymin)/fNpy;
   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;
   }
 
   Int_t nsave = (npx + 1) * (npy + 1);
   fSave.resize(nsave + 6);
   Double_t xv[2];
   Double_t *parameters = GetParameters();
   InitArgs(xv, parameters);
   for (Int_t j = 0, k = 0; j <= npy; j++) {
      xv[1]    = ymin + dy*j;
      for (Int_t i = 0; i <= npx; i++) {
         xv[0]    = xmin + dx*i;
         fSave[k++] = EvalPar(xv, parameters);
      }
   }
   fSave[nsave+0] = xmin;
   fSave[nsave+1] = xmax;
   fSave[nsave+2] = ymin;
   fSave[nsave+3] = ymax;
   fSave[nsave+4] = npx;
   fSave[nsave+5] = npy;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Restore value of function saved at point
 
void TF2::SetSavedPoint(Int_t point, Double_t value)
{
   if (fSave.empty())
      fSave.resize((fNpx + 1) * (fNpy + 1) + 6);
   if (point >= 0 && point < (Int_t)fSave.size())
      fSave[point] = value;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Save primitive as a C++ statement(s) on output stream out
 
void TF2::SavePrimitive(std::ostream &out, Option_t *option /*= ""*/)
{
   TString f2Name = ProvideSaveName(option);
   out << "   \n";
   if (!fType)
      out << "   TF2 *" << f2Name << " = new TF2(\"" << GetName() << "\", \""
          << TString(GetTitle()).ReplaceSpecialCppChars() << "\", " << fXmin << "," << fXmax << "," << fYmin << ","
          << fYmax << ");\n";
   else {
      out << "   TF2 *" << f2Name << " = new TF2(\"" << "*" << GetName() << "\", " << fXmin << "," << fXmax << ","
          << fYmin << "," << fYmax << "," << GetNpar() << ");\n";
      SavePrimitiveNameTitle(out, f2Name);
   }
 
   if (GetNpx() != 30)
      out << "   " << f2Name << "->SetNpx(" << GetNpx() << ");\n";
   if (GetNpy() != 30)
      out << "   " << f2Name << "->SetNpy(" << GetNpy() << ");\n";
 
   if (GetChisquare() != 0)
      out << "   " << f2Name << "->SetChisquare(" << GetChisquare() << ");\n";
 
   Double_t parmin, parmax;
   for (Int_t i = 0; i < GetNpar(); i++) {
      out << "   " << f2Name << "->SetParameter(" << i << "," << GetParameter(i) << ");\n";
      out << "   " << f2Name << "->SetParError(" << i << "," << GetParError(i) << ");\n";
      GetParLimits(i, parmin, parmax);
      out << "   " << f2Name << "->SetParLimits(" << i << "," << parmin << "," << parmax << ");\n";
   }
 
   Bool_t saved = kFALSE;
 
   if ((fType != EFType::kFormula) && ((Int_t) fSave.size() != ((GetNpx() + 1) * (GetNpy() + 1) + 6))) {
      saved = kTRUE;
      Save(fXmin, fXmax, fYmin, fYmax, 0, 0);
   }
 
   if (!fSave.empty()) {
      TString vect = SavePrimitiveVector(out, f2Name, fSave.size(), fSave.data());
      out << "   for (int n = 0; n < " << fSave.size() << "; n++)\n";
      out << "      " << f2Name << "->SetSavedPoint(n, " << vect << "[n]);\n";
   }
 
   if (saved)
      fSave.clear();
 
   if (fContour.fN > 0) {
      TString vectname;
      if (fContour.fArray[0] != -9999)
         vectname = SavePrimitiveVector(out, f2Name, fContour.fN, fContour.GetArray());
      out << "   " << f2Name << "->SetContour(" << fContour.fN;
      if (!vectname.IsNull())
         out << ", " << vectname << ".data()";
      out << ");\n";
   }
 
   SaveFillAttributes(out, f2Name, -1, 0);
   SaveMarkerAttributes(out, f2Name, -1, -1, -1);
   SaveLineAttributes(out, f2Name, -1, -1, -1);
 
   if (fHistogram && !strstr(option, "same")) {
      GetXaxis()->SaveAttributes(out, f2Name, "->GetXaxis()");
      GetYaxis()->SaveAttributes(out, f2Name, "->GetYaxis()");
      GetZaxis()->SaveAttributes(out, f2Name, "->GetZaxis()");
   }
 
   SavePrimitiveDraw(out, f2Name, option);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set the number and values of contour levels
///
///  By default the number of contour levels is set to 20.
///
///  if argument levels = 0 or missing, equidistant contours are computed
 
void TF2::SetContour(Int_t  nlevels, const Double_t *levels)
{
   Int_t level;
   if (nlevels <=0 ) {
      fContour.Set(0);
      return;
   }
   fContour.Set(nlevels);
 
   //-  Contour levels are specified
   if (levels) {
      for (level=0; level<nlevels; level++) fContour.fArray[level] = levels[level];
   } else {
      fContour.fArray[0] = -9999; // means not defined at this point
   }
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Set value for one contour level
 
void TF2::SetContourLevel(Int_t level, Double_t value)
{
   if (level <0 || level >= fContour.fN) return;
   fContour.fArray[level] = value;
}
 
////////////////////////////////////////////////////////////////////////////////
/// 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 TF2::GetRandom2
/// the minimum number of points is 4, the maximum is 10000 for 2-d/3-d functions
 
void TF2::SetNpy(Int_t npy)
{
   if (npy < 4) {
      Warning("SetNpy","Number of points must be >=4 && <= 10000, fNpy set to 4");
      fNpy = 4;
   } else if(npy > 10000) {
      Warning("SetNpy","Number of points must be >=4 && <= 10000, fNpy set to 10000");
      fNpy = 10000;
   } else {
      fNpy = npy;
   }
   Update();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Initialize the upper and lower bounds to draw the function-
 
void TF2::SetRange(Double_t xmin, Double_t ymin, Double_t xmax, Double_t ymax)
{
   fXmin = xmin;
   fXmax = xmax;
   fYmin = ymin;
   fYmax = ymax;
   Update();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Stream an object of class TF2.
 
void TF2::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 > 3) {
         R__b.ReadClassBuffer(TF2::Class(), this, R__v, R__s, R__c);
         return;
      }
      //====process old versions before automatic schema evolution
      Int_t nlevels;
      TF1::Streamer(R__b);
      if (R__v < 3) {
         Float_t ymin,ymax;
         R__b >> ymin; fYmin = ymin;
         R__b >> ymax; fYmax = ymax;
      } else {
         R__b >> fYmin;
         R__b >> fYmax;
      }
      R__b >> fNpy;
      R__b >> nlevels;
      if (R__v < 3) {
         Float_t *contour = nullptr;
         Int_t n = R__b.ReadArray(contour);
         fContour.Set(n);
         for (Int_t i=0;i<n;i++) fContour.fArray[i] = contour[i];
         delete [] contour;
      } else {
         fContour.Streamer(R__b);
      }
      R__b.CheckByteCount(R__s, R__c, TF2::IsA());
      //====end of old versions
 
   } else {
      Int_t saved = 0;
      if (fType != EFType::kFormula && fSave.empty()) { saved = 1; Save(fXmin,fXmax,fYmin,fYmax,0,0);}
 
      R__b.WriteClassBuffer(TF2::Class(),this);
 
      if (saved) {fSave.clear(); }
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return x^nx * y^ny moment of a 2d function in range [ax,bx],[ay,by]
/// \author Gene Van Buren <gene@bnl.gov>
 
Double_t TF2::Moment2(Double_t nx, Double_t ax, Double_t bx, Double_t ny, Double_t ay, Double_t by, Double_t epsilon)
{
   Double_t norm = Integral(ax,bx,ay,by,epsilon);
   if (norm == 0) {
      Error("Moment2", "Integral zero over range");
      return 0;
   }
 
   // define  integrand function as a lambda : g(x,y)=  x^(nx) * y^(ny) * f(x,y)
   auto integrand = [&](double *x, double *) {
      return std::pow(x[0], nx) * std::pow(x[1], ny) * this->EvalPar(x, nullptr);
   };
   // compute integral of g(x,y)
   TF2 fnc("TF2_ExpValHelper",integrand,ax,bx,ay,by,0);
   // set same points as current function to get correct max points when computing the integral
   fnc.fNpx = fNpx;
   fnc.fNpy = fNpy;
   return fnc.Integral(ax,bx,ay,by,epsilon)/norm;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return x^nx * y^ny central moment of a 2d function in range [ax,bx],[ay,by]
/// \author Gene Van Buren <gene@bnl.gov>
 
Double_t TF2::CentralMoment2(Double_t nx, Double_t ax, Double_t bx, Double_t ny, Double_t ay, Double_t by, Double_t epsilon)
{
   Double_t norm = Integral(ax,bx,ay,by,epsilon);
   if (norm == 0) {
      Error("CentralMoment2", "Integral zero over range");
      return 0;
   }
 
   Double_t xbar = 0;
   Double_t ybar = 0;
   if (nx!=0) {
      // compute first momentum in x
      auto integrandX = [&](double *x, double *) { return x[0] * this->EvalPar(x, nullptr); };
      TF2 fncx("TF2_ExpValHelperx",integrandX, ax, bx, ay, by, 0);
      fncx.fNpx = fNpx;
      fncx.fNpy = fNpy;
      xbar = fncx.Integral(ax,bx,ay,by,epsilon)/norm;
   }
   if (ny!=0) {
      // compute first momentum in y
      auto integrandY = [&](double *x, double *) { return x[1] * this->EvalPar(x, nullptr); };
      TF2 fncy("TF2_ExpValHelperx", integrandY, ax, bx, ay, by, 0);
      fncy.fNpx = fNpx;
      fncy.fNpy = fNpy;
      ybar = fncy.Integral(ax,bx,ay,by,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.;
      return xxx * yyy * this->EvalPar(x, nullptr);
   };
   // compute integral of g(x,y)
   TF2 fnc("TF2_ExpValHelper", integrand, ax, bx, ay, by, 0);
   fnc.fNpx = fNpx;
   fnc.fNpy = fNpy;
   return fnc.Integral(ax, bx, ay, by, epsilon) / norm;
}