TGraph.cxx

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// @(#)root/hist:$Id$
// Author: Rene Brun, Olivier Couet   12/12/94
 
/*************************************************************************
 * 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 "TBuffer.h"
#include "TEnv.h"
#include "TGraph.h"
#include "TGraphErrors.h"
#include "TGraphAsymmErrors.h"
#include "TGraphBentErrors.h"
#include "TH1.h"
#include "TF1.h"
#include "TStyle.h"
#include "TMath.h"
#include "TVectorD.h"
#include "Foption.h"
#include "TRandom.h"
#include "TSpline.h"
#include "TVirtualFitter.h"
#include "TVirtualPad.h"
#include "TVirtualGraphPainter.h"
#include "TBrowser.h"
#include "TSystem.h"
#include "TPluginManager.h"
#include "strtok.h"
 
#include <cstdlib>
#include <string>
#include <cassert>
#include <iostream>
#include <fstream>
#include <cstring>
 
#include "HFitInterface.h"
#include "Fit/DataRange.h"
#include "Math/MinimizerOptions.h"
 
extern void H1LeastSquareSeqnd(Int_t n, Double_t *a, Int_t idim, Int_t &ifail, Int_t k, Double_t *b);
 
ClassImp(TGraph);
 
////////////////////////////////////////////////////////////////////////////////
 
/** \class TGraph
    \ingroup Graphs
A TGraph is an object made of two arrays X and Y with npoints each.
The TGraph painting is performed thanks to the TGraphPainter
class. All details about the various painting options are given in this class.
 
#### Notes
 
  - Unlike histogram or tree (or even TGraph2D), TGraph objects
    are not automatically attached to the current TFile, in order to keep the
    management and size of the TGraph as small as possible.
  - The TGraph constructors do not have the TGraph title and name as parameters.
    A TGraph has the default title and name "Graph". To change the default title
    and name `SetTitle` and `SetName` should be called on the TGraph after its creation.
    TGraph was a light weight object to start with, like TPolyline or TPolyMarker.
    That’s why it did not have any title and name parameters in the constructors.
 
#### Example
 
The picture below gives an example:
 
Begin_Macro(source)
{
   double x[100], y[100];
   int n = 20;
   for (int i=0;i<n;i++) {
     x[i] = i*0.1;
     y[i] = 10*sin(x[i]+0.2);
   }
   auto g = new TGraph(n,x,y);
   g->SetTitle("Graph title;X title;Y title");
   g->Draw("AC*");
}
End_Macro
 
#### Default X-Points
 
If one doesn't specify the points in the x-axis, they will get the default values 0, 1, 2, 3, (etc. depending
on the length of the y-points):
 
Begin_Macro(source)
{
   double y[6] = {3, 8, 1, 10, 5, 7};
   auto g = new TGraph(6,y);
   g->SetTitle("A Graph with default X points");
   g->Draw();
}
End_Macro
 
*/
 
////////////////////////////////////////////////////////////////////////////////
/// Graph default constructor.
 
TGraph::TGraph(): TNamed(), TAttLine(), TAttFill(0, 1000), TAttMarker()
{
   fNpoints = -1;  //will be reset to 0 in CtorAllocate
   if (!CtorAllocate()) return;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Constructor with only the number of points set
/// the arrays x and y will be set later
 
TGraph::TGraph(Int_t n)
   : TNamed("Graph", "Graph"), TAttLine(), TAttFill(0, 1000), TAttMarker()
{
   fNpoints = n;
   if (!CtorAllocate()) return;
   FillZero(0, fNpoints);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Graph normal constructor with ints.
 
TGraph::TGraph(Int_t n, const Int_t *x, const Int_t *y)
   : TNamed("Graph", "Graph"), TAttLine(), TAttFill(0, 1000), TAttMarker()
{
   if (!x || !y) {
      fNpoints = 0;
   } else {
      fNpoints = n;
   }
   if (!CtorAllocate()) return;
   for (Int_t i = 0; i < n; i++) {
      fX[i] = (Double_t)x[i];
      fY[i] = (Double_t)y[i];
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Graph normal constructor with floats.
 
TGraph::TGraph(Int_t n, const Float_t *x, const Float_t *y)
   : TNamed("Graph", "Graph"), TAttLine(), TAttFill(0, 1000), TAttMarker()
{
   if (!x || !y) {
      fNpoints = 0;
   } else {
      fNpoints = n;
   }
   if (!CtorAllocate()) return;
   for (Int_t i = 0; i < n; i++) {
      fX[i] = x[i];
      fY[i] = y[i];
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Default X-Points constructor. The points along the x-axis get the default
/// values `start`, `start+step`, `start+2*step`, `start+3*step`, etc ...
 
TGraph::TGraph(Int_t n, const Double_t *y, Double_t start, Double_t step)
   : TNamed("Graph", "Graph"), TAttLine(), TAttFill(0, 1000), TAttMarker()
{
   if (!y) {
      fNpoints = 0;
   } else {
      fNpoints = n;
   }
   if (!CtorAllocate()) return;
   for (Int_t i = 0; i < n; i++) {
      fX[i] = start+i*step;
      fY[i] = y[i];
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Graph normal constructor with doubles.
 
TGraph::TGraph(Int_t n, const Double_t *x, const Double_t *y)
   : TNamed("Graph", "Graph"), TAttLine(), TAttFill(0, 1000), TAttMarker()
{
   if (!x || !y) {
      fNpoints = 0;
   } else {
      fNpoints = n;
   }
   if (!CtorAllocate()) return;
   n = fNpoints * sizeof(Double_t);
   memcpy(fX, x, n);
   memcpy(fY, y, n);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Copy constructor for this graph
 
TGraph::TGraph(const TGraph &gr)
   : TNamed(gr), TAttLine(gr), TAttFill(gr), TAttMarker(gr)
{
   fNpoints = gr.fNpoints;
   fMaxSize = gr.fMaxSize;
   if (gr.fFunctions) fFunctions = (TList*)gr.fFunctions->Clone();
   else fFunctions = new TList;
   if (gr.fHistogram) {
      fHistogram = (TH1F*)gr.fHistogram->Clone();
      fHistogram->SetDirectory(nullptr);
   } else {
      fHistogram = nullptr;
   }
   fMinimum = gr.fMinimum;
   fMaximum = gr.fMaximum;
   if (!fMaxSize) {
      fX = fY = nullptr;
      return;
   } else {
      fX = new Double_t[fMaxSize];
      fY = new Double_t[fMaxSize];
   }
 
   Int_t n = gr.GetN() * sizeof(Double_t);
   memcpy(fX, gr.fX, n);
   memcpy(fY, gr.fY, n);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Equal operator for this graph
 
TGraph& TGraph::operator=(const TGraph &gr)
{
   if (this != &gr) {
      TNamed::operator=(gr);
      TAttLine::operator=(gr);
      TAttFill::operator=(gr);
      TAttMarker::operator=(gr);
 
      fNpoints = gr.fNpoints;
      fMaxSize = gr.fMaxSize;
 
      // delete list of functions and their contents before copying it
      if (fFunctions) {
         // delete previous lists of functions
         if (!fFunctions->IsEmpty()) {
            fFunctions->SetBit(kInvalidObject);
            // use TList::Remove to take into account the case the same object is
            // added multiple times in the list
            TObject *obj;
            while ((obj  = fFunctions->First())) {
               while (fFunctions->Remove(obj)) { }
               delete obj;
            }
         }
         delete fFunctions;
      }
 
      if (gr.fFunctions) fFunctions = (TList*)gr.fFunctions->Clone();
      else fFunctions = new TList;
 
      if (fHistogram) delete fHistogram;
      if (gr.fHistogram) {
         fHistogram = new TH1F(*(gr.fHistogram));
         fHistogram->SetDirectory(nullptr);
      } else {
         fHistogram = nullptr;
      }
 
      fMinimum = gr.fMinimum;
      fMaximum = gr.fMaximum;
      if (fX) delete [] fX;
      if (fY) delete [] fY;
      if (!fMaxSize) {
         fX = fY = nullptr;
         return *this;
      } else {
         fX = new Double_t[fMaxSize];
         fY = new Double_t[fMaxSize];
      }
 
      Int_t n = gr.GetN() * sizeof(Double_t);
      if (n > 0) {
         memcpy(fX, gr.fX, n);
         memcpy(fY, gr.fY, n);
      }
   }
   return *this;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Graph constructor with two vectors of floats in input
/// A graph is build with the X coordinates taken from vx and Y coord from vy
/// The number of points in the graph is the minimum of number of points
/// in vx and vy.
 
TGraph::TGraph(const TVectorF &vx, const TVectorF &vy)
   : TNamed("Graph", "Graph"), TAttLine(), TAttFill(0, 1000), TAttMarker()
{
   fNpoints = TMath::Min(vx.GetNrows(), vy.GetNrows());
   if (!CtorAllocate()) return;
   Int_t ivxlow  = vx.GetLwb();
   Int_t ivylow  = vy.GetLwb();
   for (Int_t i = 0; i < fNpoints; i++) {
      fX[i]  = vx(i + ivxlow);
      fY[i]  = vy(i + ivylow);
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Graph constructor with two vectors of doubles in input
/// A graph is build with the X coordinates taken from vx and Y coord from vy
/// The number of points in the graph is the minimum of number of points
/// in vx and vy.
 
TGraph::TGraph(const TVectorD &vx, const TVectorD &vy)
   : TNamed("Graph", "Graph"), TAttLine(), TAttFill(0, 1000), TAttMarker()
{
   fNpoints = TMath::Min(vx.GetNrows(), vy.GetNrows());
   if (!CtorAllocate()) return;
   Int_t ivxlow  = vx.GetLwb();
   Int_t ivylow  = vy.GetLwb();
   for (Int_t i = 0; i < fNpoints; i++) {
      fX[i]  = vx(i + ivxlow);
      fY[i]  = vy(i + ivylow);
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Graph constructor importing its parameters from the TH1 object passed as argument
 
TGraph::TGraph(const TH1 *h)
   : TNamed("Graph", "Graph"), TAttLine(), TAttFill(0, 1000), TAttMarker()
{
   if (!h) {
      Error("TGraph", "Pointer to histogram is null");
      fNpoints = 0;
      return;
   }
   if (h->GetDimension() != 1) {
      Error("TGraph", "Histogram must be 1-D; h %s is %d-D", h->GetName(), h->GetDimension());
      fNpoints = 0;
   } else {
      fNpoints = ((TH1*)h)->GetXaxis()->GetNbins();
   }
 
   if (!CtorAllocate()) return;
 
   TAxis *xaxis = ((TH1*)h)->GetXaxis();
   for (Int_t i = 0; i < fNpoints; i++) {
      fX[i] = xaxis->GetBinCenter(i + 1);
      fY[i] = h->GetBinContent(i + 1);
   }
   h->TAttLine::Copy(*this);
   h->TAttFill::Copy(*this);
   h->TAttMarker::Copy(*this);
 
   std::string gname = "Graph_from_" + std::string(h->GetName());
   SetName(gname.c_str());
   SetTitle(h->GetTitle());
}
 
////////////////////////////////////////////////////////////////////////////////
/// Graph constructor importing its parameters from the TF1 object passed as argument
/// - if option =="" (default), a TGraph is created with points computed
///                at the fNpx points of f.
/// - if option =="d", a TGraph is created with points computed with the derivatives
///                at the fNpx points of f.
/// - if option =="i", a TGraph is created with points computed with the integral
///                at the fNpx points of f.
/// - if option =="I", a TGraph is created with points computed with the integral
///                at the fNpx+1 points of f and the integral is normalized to 1.
 
TGraph::TGraph(const TF1 *f, Option_t *option)
   : TNamed("Graph", "Graph"), TAttLine(), TAttFill(0, 1000), TAttMarker()
{
   char coption = ' ';
   if (!f) {
      Error("TGraph", "Pointer to function is null");
      fNpoints = 0;
   } else {
      fNpoints   = f->GetNpx();
      if (option) coption = *option;
      if (coption == 'i' || coption == 'I') fNpoints++;
   }
   if (!CtorAllocate()) return;
 
   Double_t xmin = f->GetXmin();
   Double_t xmax = f->GetXmax();
   Double_t dx   = (xmax - xmin) / fNpoints;
   Double_t integ = 0;
   Int_t i;
   for (i = 0; i < fNpoints; i++) {
      if (coption == 'i' || coption == 'I') {
         fX[i] = xmin + i * dx;
         if (i == 0) fY[i] = 0;
         else        fY[i] = integ + ((TF1*)f)->Integral(fX[i] - dx, fX[i]);
         integ = fY[i];
      } else if (coption == 'd' || coption == 'D') {
         fX[i] = xmin + (i + 0.5) * dx;
         fY[i] = ((TF1*)f)->Derivative(fX[i]);
      } else {
         fX[i] = xmin + (i + 0.5) * dx;
         fY[i] = ((TF1*)f)->Eval(fX[i]);
      }
   }
   if (integ != 0 && coption == 'I') {
      for (i = 1; i < fNpoints; i++) fY[i] /= integ;
   }
 
   f->TAttLine::Copy(*this);
   f->TAttFill::Copy(*this);
   f->TAttMarker::Copy(*this);
 
   SetName(f->GetName());
   SetTitle(f->GetTitle());
}
 
////////////////////////////////////////////////////////////////////////////////
/// Graph constructor reading input from filename.
///
/// `filename` is assumed to contain at least two columns of numbers.
/// The string format is by default `"%lg %lg"`.
/// This is a standard c formatting for `scanf()`.
///
/// If columns of numbers should be skipped, a `"%*lg"` or `"%*s"` for each column
/// can be added,  e.g. `"%lg %*lg %lg"` would read x-values from the first and
/// y-values from the third column.
///
/// For files separated by a specific delimiter different from ' ' and '\\t' (e.g.
/// ';' in csv files) you can avoid using `%*s` to bypass this delimiter by explicitly
/// specify the `option` argument,
/// e.g. option=`" \\t,;"` for columns of figures separated by any of these characters
/// (' ', '\\t', ',', ';')
/// used once (e.g. `"1;1"`) or in a combined way (`" 1;,;;  1"`).
/// Note in that case, the instantiation is about two times slower.
 
TGraph::TGraph(const char *filename, const char *format, Option_t *option)
   : TNamed("Graph", filename), TAttLine(), TAttFill(0, 1000), TAttMarker()
{
   Double_t x, y;
   TString fname = filename;
   gSystem->ExpandPathName(fname);
 
   std::ifstream infile(fname.Data());
   if (!infile.good()) {
      MakeZombie();
      Error("TGraph", "Cannot open file: %s, TGraph is Zombie", filename);
      fNpoints = 0;
      return;
   } else {
      fNpoints = 100;  //initial number of points
   }
   if (!CtorAllocate()) return;
   std::string line;
   Int_t np = 0;
 
   // No delimiters specified (standard constructor).
   if (strcmp(option, "") == 0) {
 
      while (std::getline(infile, line, '\n')) {
         if (2 != sscanf(line.c_str(), format, &x, &y)) {
            continue; //skip empty and ill-formed lines
         }
         SetPoint(np, x, y);
         np++;
      }
      Set(np);
 
      // A delimiter has been specified in "option"
   } else {
 
      // Checking format and creating its boolean counterpart
      TString format_ = TString(format) ;
      format_.ReplaceAll(" ", "") ;
      format_.ReplaceAll("\t", "") ;
      format_.ReplaceAll("lg", "") ;
      format_.ReplaceAll("s", "") ;
      format_.ReplaceAll("%*", "0") ;
      format_.ReplaceAll("%", "1") ;
      if (!format_.IsDigit()) {
         Error("TGraph", "Incorrect input format! Allowed formats are {\"%%lg\",\"%%*lg\" or \"%%*s\"}");
         return;
      }
      Int_t ntokens = format_.Length() ;
      if (ntokens < 2) {
         Error("TGraph", "Incorrect input format! Only %d tag(s) in format whereas 2 \"%%lg\" tags are expected!", ntokens);
         return;
      }
      Int_t ntokensToBeSaved = 0 ;
      Bool_t * isTokenToBeSaved = new Bool_t [ntokens] ;
      for (Int_t idx = 0; idx < ntokens; idx++) {
         isTokenToBeSaved[idx] = TString::Format("%c", format_[idx]).Atoi() ; //atoi(&format_[idx]) does not work for some reason...
         if (isTokenToBeSaved[idx] == 1) {
            ntokensToBeSaved++ ;
         }
      }
      if (ntokens >= 2 && ntokensToBeSaved != 2) { //first condition not to repeat the previous error message
         Error("TGraph", "Incorrect input format! There are %d \"%%lg\" tag(s) in format whereas 2 and only 2 are expected!", ntokensToBeSaved);
         delete [] isTokenToBeSaved ;
         return;
      }
 
      // Initializing loop variables
      Bool_t isLineToBeSkipped = kFALSE ; //empty and ill-formed lines
      char * token = NULL ;
      TString token_str = "" ;
      Int_t token_idx = 0 ;
      Double_t * value = new Double_t [2] ; //x,y buffers
      Int_t value_idx = 0 ;
 
      // Looping
      char *rest;
      while (std::getline(infile, line, '\n')) {
         if (line != "") {
            if (line[line.size() - 1] == char(13)) {  // removing DOS CR character
               line.erase(line.end() - 1, line.end()) ;
            }
            //token = R__STRTOK_R(const_cast<char *>(line.c_str()), option, rest);
            token = R__STRTOK_R(const_cast<char *>(line.c_str()), option, &rest);
            while (token != NULL && value_idx < 2) {
               if (isTokenToBeSaved[token_idx]) {
                  token_str = TString(token) ;
                  token_str.ReplaceAll("\t", "") ;
                  if (!token_str.IsFloat()) {
                     isLineToBeSkipped = kTRUE ;
                     break ;
                  } else {
                     value[value_idx] = token_str.Atof() ;
                     value_idx++ ;
                  }
               }
               token = R__STRTOK_R(NULL, option, &rest); // next token
               token_idx++ ;
            }
            if (!isLineToBeSkipped && value_idx == 2) {
               x = value[0] ;
               y = value[1] ;
               SetPoint(np, x, y) ;
               np++ ;
            }
         }
         isLineToBeSkipped = kFALSE ;
         token = NULL ;
         token_idx = 0 ;
         value_idx = 0 ;
      }
      Set(np) ;
 
      // Cleaning
      delete [] isTokenToBeSaved ;
      delete [] value ;
      delete token ;
   }
   infile.close();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Graph default destructor.
 
TGraph::~TGraph()
{
   delete [] fX;
   delete [] fY;
   if (fFunctions) {
      fFunctions->SetBit(kInvalidObject);
      //special logic to support the case where the same object is
      //added multiple times in fFunctions.
      //This case happens when the same object is added with different
      //drawing modes
      TObject *obj;
      while ((obj  = fFunctions->First())) {
         while (fFunctions->Remove(obj)) { }
         delete obj;
      }
      delete fFunctions;
      fFunctions = nullptr; //to avoid accessing a deleted object in RecursiveRemove
   }
   delete fHistogram;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Allocate internal data structures for `newsize` points.
 
Double_t **TGraph::Allocate(Int_t newsize)
{
   return AllocateArrays(2, newsize);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Allocate arrays.
 
Double_t** TGraph::AllocateArrays(Int_t Narrays, Int_t arraySize)
{
   if (arraySize < 0) {
      arraySize = 0;
   }
   Double_t **newarrays = new Double_t*[Narrays];
   if (!arraySize) {
      for (Int_t i = 0; i < Narrays; ++i)
         newarrays[i] = nullptr;
   } else {
      for (Int_t i = 0; i < Narrays; ++i)
         newarrays[i] = new Double_t[arraySize];
   }
   fMaxSize = arraySize;
   return newarrays;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Apply function f to all the data points
/// f may be a 1-D function TF1 or 2-d function TF2
/// The Y values of the graph are replaced by the new values computed
/// using the function
 
void TGraph::Apply(TF1 *f)
{
   if (fHistogram) SetBit(kResetHisto);
 
   for (Int_t i = 0; i < fNpoints; i++) {
      fY[i] = f->Eval(fX[i], fY[i]);
   }
   if (gPad) gPad->Modified();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Browse
 
void TGraph::Browse(TBrowser *b)
{
   TString opt = gEnv->GetValue("TGraph.BrowseOption", "");
   if (opt.IsNull()) {
      opt = b ? b->GetDrawOption() : "alp";
      opt = (opt == "") ? "alp" : opt.Data();
   }
   Draw(opt.Data());
   gPad->Update();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return the chisquare of this graph with respect to f1.
/// The chisquare is computed as the sum of the quantity below at each point:
/// \f[
///   \frac{(y-f1(x))^{2}}{ey^{2}+(\frac{1}{2}(exl+exh)f1'(x))^{2}}
/// \f]
/// where x and y are the graph point coordinates and f1'(x) is the derivative of function f1(x).
/// This method to approximate the uncertainty in y because of the errors in x, is called
/// "effective variance" method.
/// In case of a pure TGraph, the denominator is 1.
/// In case of a TGraphErrors or TGraphAsymmErrors the errors are taken
/// into account.
/// By default the range of the graph is used whatever function range.
/// Use option "R" to use the function range
 
Double_t TGraph::Chisquare(TF1 *func, Option_t * option) const
{
   if (!func) {
      Error("Chisquare","Function pointer is Null - return -1");
      return -1;
   }
 
   TString opt(option); opt.ToUpper();
   bool useRange = opt.Contains("R");
 
   return ROOT::Fit::Chisquare(*this, *func,useRange);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return kTRUE if point number "left"'s argument (angle with respect to positive
/// x-axis) is bigger than that of point number "right". Can be used by Sort.
 
Bool_t TGraph::CompareArg(const TGraph* gr, Int_t left, Int_t right)
{
   Double_t xl = 0, yl = 0, xr = 0, yr = 0;
   gr->GetPoint(left, xl, yl);
   gr->GetPoint(right, xr, yr);
   return (TMath::ATan2(yl, xl) > TMath::ATan2(yr, xr));
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return kTRUE if fX[left] > fX[right]. Can be used by Sort.
 
Bool_t TGraph::CompareX(const TGraph* gr, Int_t left, Int_t right)
{
   return gr->fX[left] > gr->fX[right];
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return kTRUE if fY[left] > fY[right]. Can be used by Sort.
 
Bool_t TGraph::CompareY(const TGraph* gr, Int_t left, Int_t right)
{
   return gr->fY[left] > gr->fY[right];
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return kTRUE if point number "left"'s distance to origin is bigger than
/// that of point number "right". Can be used by Sort.
 
Bool_t TGraph::CompareRadius(const TGraph* gr, Int_t left, Int_t right)
{
   return gr->fX[left] * gr->fX[left] + gr->fY[left] * gr->fY[left]
          > gr->fX[right] * gr->fX[right] + gr->fY[right] * gr->fY[right];
}
 
////////////////////////////////////////////////////////////////////////////////
/// Compute the x/y range of the points in this graph
 
void TGraph::ComputeRange(Double_t &xmin, Double_t &ymin, Double_t &xmax, Double_t &ymax) const
{
   if (fNpoints <= 0) {
      xmin = xmax = ymin = ymax = 0;
      return;
   }
   xmin = xmax = fX[0];
   ymin = ymax = fY[0];
 
   Double_t xminl = 0; // Positive minimum. Used in case of log scale along X axis.
   Double_t yminl = 0; // Positive minimum. Used in case of log scale along Y axis.
 
   for (Int_t i = 1; i < fNpoints; i++) {
      if (fX[i] < xmin) xmin = fX[i];
      if (fX[i] > xmax) xmax = fX[i];
      if (fY[i] < ymin) ymin = fY[i];
      if (fY[i] > ymax) ymax = fY[i];
      if (ymin>0 && (yminl==0 || ymin<yminl)) yminl = ymin;
      if (xmin>0 && (xminl==0 || xmin<xminl)) xminl = xmin;
   }
 
   if (gPad && gPad->GetLogy() && yminl>0) ymin = yminl;
   if (gPad && gPad->GetLogx() && xminl>0) xmin = xminl;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Copy points from fX and fY to arrays[0] and arrays[1]
/// or to fX and fY if arrays == 0 and ibegin != iend.
/// If newarrays is non null, replace fX, fY with pointers from newarrays[0,1].
/// Delete newarrays, old fX and fY
 
void TGraph::CopyAndRelease(Double_t **newarrays, Int_t ibegin, Int_t iend,
                            Int_t obegin)
{
   CopyPoints(newarrays, ibegin, iend, obegin);
   if (newarrays) {
      delete[] fX;
      fX = newarrays[0];
      delete[] fY;
      fY = newarrays[1];
      delete[] newarrays;
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Copy points from fX and fY to arrays[0] and arrays[1]
/// or to fX and fY if arrays == 0 and ibegin != iend.
 
Bool_t TGraph::CopyPoints(Double_t **arrays, Int_t ibegin, Int_t iend,
                          Int_t obegin)
{
   if (ibegin < 0 || iend <= ibegin || obegin < 0) { // Error;
      return kFALSE;
   }
   if (!arrays && ibegin == obegin) { // No copying is needed
      return kFALSE;
   }
   Int_t n = (iend - ibegin) * sizeof(Double_t);
   if (arrays) {
      memmove(&arrays[0][obegin], &fX[ibegin], n);
      memmove(&arrays[1][obegin], &fY[ibegin], n);
   } else {
      memmove(&fX[obegin], &fX[ibegin], n);
      memmove(&fY[obegin], &fY[ibegin], n);
   }
   return kTRUE;
}
 
////////////////////////////////////////////////////////////////////////////////
/// In constructors set fNpoints than call this method.
/// Return kFALSE if the graph will contain no points.
///Note: This function should be called only from the constructor
/// since it does not delete previously existing arrays
 
Bool_t TGraph::CtorAllocate()
{
   fHistogram = nullptr;
   fMaximum = -1111;
   fMinimum = -1111;
   SetBit(kClipFrame);
   fFunctions = new TList;
   if (fNpoints <= 0) {
      fNpoints = 0;
      fMaxSize   = 0;
      fX         = nullptr;
      fY         = nullptr;
      return kFALSE;
   } else {
      fMaxSize   = fNpoints;
      fX = new Double_t[fMaxSize];
      fY = new Double_t[fMaxSize];
   }
   return kTRUE;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Draw this graph with its current attributes.
///
/// The options to draw a graph are described in TGraphPainter class.
 
void TGraph::Draw(Option_t *option)
{
   TString opt = option;
   opt.ToLower();
 
   if (opt.Contains("same")) {
      opt.ReplaceAll("same", "");
   }
 
   // in case of option *, set marker style to 3 (star) and replace
   // * option by option P.
   Ssiz_t pos;
   if ((pos = opt.Index("*")) != kNPOS) {
      SetMarkerStyle(3);
      opt.Replace(pos, 1, "p");
   }
 
   // If no option is specified, it is defined as "alp" in case there
   // no current pad or if the current pad as no axis defined.
   if (!option || !strlen(option)) {
      if (gPad) {
         if (!gPad->GetListOfPrimitives()->FindObject("TFrame")) opt = "alp";
      } else {
         opt = "alp";
      }
   }
 
   if (gPad) {
      if (!gPad->IsEditable()) gROOT->MakeDefCanvas();
      if (opt.Contains("a")) gPad->Clear();
   }
 
   AppendPad(opt);
 
   gPad->IncrementPaletteColor(1, opt);
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// Compute distance from point px,py to a graph.
///
///  Compute the closest distance of approach from point px,py to this line.
///  The distance is computed in pixels units.
 
Int_t TGraph::DistancetoPrimitive(Int_t px, Int_t py)
{
   TVirtualGraphPainter *painter = TVirtualGraphPainter::GetPainter();
   if (painter) return painter->DistancetoPrimitiveHelper(this, px, py);
   else return 0;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Draw this graph with new attributes.
 
void TGraph::DrawGraph(Int_t n, const Int_t *x, const Int_t *y, Option_t *option)
{
   TGraph *newgraph = new TGraph(n, x, y);
   TAttLine::Copy(*newgraph);
   TAttFill::Copy(*newgraph);
   TAttMarker::Copy(*newgraph);
   newgraph->SetBit(kCanDelete);
   newgraph->AppendPad(option);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Draw this graph with new attributes.
 
void TGraph::DrawGraph(Int_t n, const Float_t *x, const Float_t *y, Option_t *option)
{
   TGraph *newgraph = new TGraph(n, x, y);
   TAttLine::Copy(*newgraph);
   TAttFill::Copy(*newgraph);
   TAttMarker::Copy(*newgraph);
   newgraph->SetBit(kCanDelete);
   newgraph->AppendPad(option);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Draw this graph with new attributes.
 
void TGraph::DrawGraph(Int_t n, const Double_t *x, const Double_t *y, Option_t *option)
{
   const Double_t *xx = x;
   const Double_t *yy = y;
   if (!xx) xx = fX;
   if (!yy) yy = fY;
   TGraph *newgraph = new TGraph(n, xx, yy);
   TAttLine::Copy(*newgraph);
   TAttFill::Copy(*newgraph);
   TAttMarker::Copy(*newgraph);
   newgraph->SetBit(kCanDelete);
   newgraph->AppendPad(option);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Display a panel with all graph drawing options.
 
void TGraph::DrawPanel()
{
   TVirtualGraphPainter *painter = TVirtualGraphPainter::GetPainter();
   if (painter) painter->DrawPanelHelper(this);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Interpolate points in this graph at x using a TSpline.
///
///  - if spline==0 and option="" a linear interpolation between the two points
///    close to x is computed. If x is outside the graph range, a linear
///    extrapolation is computed.
///  - if spline==0 and option="S" a TSpline3 object is created using this graph
///    and the interpolated value from the spline is returned.
///    the internally created spline is deleted on return.
///  - if spline is specified, it is used to return the interpolated value.
///
///   If the points are sorted in X a binary search is used (significantly faster)
///   One needs to set the bit  TGraph::SetBit(TGraph::kIsSortedX) before calling
///   TGraph::Eval to indicate that the graph is sorted in X.
 
Double_t TGraph::Eval(Double_t x, TSpline *spline, Option_t *option) const
{
 
   if (spline) {
      //spline interpolation using the input spline
      return spline->Eval(x);
   }
 
   if (fNpoints == 0) return 0;
   if (fNpoints == 1) return fY[0];
 
   if (option && *option) {
      TString opt = option;
      opt.ToLower();
      // create a TSpline every time when using option "s" and no spline pointer is given
      if (opt.Contains("s")) {
 
         // points must be sorted before using a TSpline
         std::vector<Double_t> xsort(fNpoints);
         std::vector<Double_t> ysort(fNpoints);
         std::vector<Int_t> indxsort(fNpoints);
         TMath::Sort(fNpoints, fX, &indxsort[0], false);
         for (Int_t i = 0; i < fNpoints; ++i) {
            xsort[i] = fX[ indxsort[i] ];
            ysort[i] = fY[ indxsort[i] ];
         }
 
         // spline interpolation creating a new spline
         TSpline3 s("", &xsort[0], &ysort[0], fNpoints);
         Double_t result = s.Eval(x);
         return result;
      }
   }
   //linear interpolation
   //In case x is < fX[0] or > fX[fNpoints-1] return the extrapolated point
 
   //find points in graph around x assuming points are not sorted
   // (if point are sorted use a binary search)
   Int_t low  = -1;
   Int_t up  = -1;
   if (TestBit(TGraph::kIsSortedX) ) {
      low = TMath::BinarySearch(fNpoints, fX, x);
      if (low == -1)  {
         // use first two points for doing an extrapolation
         low = 0;
      }
      if (fX[low] == x) return fY[low];
      if (low == fNpoints-1) low--; // for extrapolating
      up = low+1;
   }
   else {
      // case TGraph is not sorted
 
   // find neighbours simply looping  all points
   // and find also the 2 adjacent points: (low2 < low < x < up < up2 )
   // needed in case x is outside the graph ascissa interval
      Int_t low2 = -1;
      Int_t up2 = -1;
 
      for (Int_t i = 0; i < fNpoints; ++i) {
         if (fX[i] < x) {
            if (low == -1 || fX[i] > fX[low])  {
               low2 = low;
               low = i;
            } else if (low2 == -1) low2 = i;
         } else if (fX[i] > x) {
            if (up  == -1 || fX[i] < fX[up])  {
               up2 = up;
               up = i;
            } else if (up2 == -1) up2 = i;
         } else // case x == fX[i]
            return fY[i]; // no interpolation needed
      }
 
      // treat cases when x is outside graph min max abscissa
      if (up == -1)  {
         up  = low;
         low = low2;
      }
      if (low == -1) {
         low = up;
         up  = up2;
      }
   }
   // do now the linear interpolation
   assert(low != -1 && up != -1);
 
   if (fX[low] == fX[up]) return fY[low];
   Double_t yn = fY[up] + (x - fX[up]) * (fY[low] - fY[up]) / (fX[low] - fX[up]);
   return yn;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Execute action corresponding to one event.
///
///  This member function is called when a graph is clicked with the locator
///
///  If Left button clicked on one of the line end points, this point
///     follows the cursor until button is released.
///
///  if Middle button clicked, the line is moved parallel to itself
///     until the button is released.
 
void TGraph::ExecuteEvent(Int_t event, Int_t px, Int_t py)
{
   TVirtualGraphPainter *painter = TVirtualGraphPainter::GetPainter();
   if (painter) painter->ExecuteEventHelper(this, event, px, py);
}
 
////////////////////////////////////////////////////////////////////////////////
/// If array sizes <= newsize, expand storage to 2*newsize.
 
void TGraph::Expand(Int_t newsize)
{
   Double_t **ps = ExpandAndCopy(newsize, fNpoints);
   CopyAndRelease(ps, 0, 0, 0);
}
 
////////////////////////////////////////////////////////////////////////////////
/// If graph capacity is less than newsize points then make array sizes
/// equal to least multiple of step to contain newsize points.
 
void TGraph::Expand(Int_t newsize, Int_t step)
{
   if (newsize <= fMaxSize) {
      return;
   }
   Double_t **ps = Allocate(step * (newsize / step + (newsize % step ? 1 : 0)));
   CopyAndRelease(ps, 0, fNpoints, 0);
}
 
////////////////////////////////////////////////////////////////////////////////
/// if size > fMaxSize allocate new arrays of 2*size points and copy iend first
/// points.
/// Return pointer to new arrays.
 
Double_t **TGraph::ExpandAndCopy(Int_t size, Int_t iend)
{
   if (size <= fMaxSize)
      return nullptr;
   Double_t **newarrays = Allocate(2 * size);
   CopyPoints(newarrays, 0, iend, 0);
   return newarrays;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set zero values for point arrays in the range [begin, end)
/// Should be redefined in descendant classes
 
void TGraph::FillZero(Int_t begin, Int_t end, Bool_t)
{
   memset(fX + begin, 0, (end - begin)*sizeof(Double_t));
   memset(fY + begin, 0, (end - begin)*sizeof(Double_t));
}
 
////////////////////////////////////////////////////////////////////////////////
/// Search object named name in the list of functions
 
TObject *TGraph::FindObject(const char *name) const
{
   return fFunctions ? fFunctions->FindObject(name) : nullptr;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Search object obj in the list of functions
 
TObject *TGraph::FindObject(const TObject *obj) const
{
   return fFunctions ? fFunctions->FindObject(obj) : nullptr;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Fit this graph with function f1.
///
/// \param[in] f1 pointer to the function object
/// \param[in] option string defining the fit options (see table below).
/// \param[in] goption specify a list of graphics options. See TGraph::Draw and TGraphPainter for a complete list of these possible options.
/// \param[in] rxmin lower fitting range
/// \param[in] rxmax upper fitting range
///
/// \anchor GFitOpt
/// ### Graph Fitting Options
/// The list of fit options is given in parameter option.
///
/// option | description
/// -------|------------
/// "S"  | The full result of the fit is returned in the `TFitResultPtr`. This is needed to get the covariance matrix of the fit. See `TFitResult` and the base class `ROOT::Math::FitResult`.
/// "W"  | Ignore all point errors when fitting a TGraphErrors or TGraphAsymmErrors
/// "F"  | Uses the default minimizer (e.g. Minuit) when fitting a linear function (e.g. polN) instead of the linear fitter.
/// "U"  | Uses a user specified objective function (e.g. user providedlikelihood function) defined using `TVirtualFitter::SetFCN`
/// "E"  | Performs a better parameter errors estimation using the Minos technique for all fit parameters.
/// "M"  | Uses the IMPROVE algorithm (available only in TMinuit). This algorithm attempts improve the found local minimum by searching for a better one.
/// "Q"  | Quiet mode (minimum printing)
/// "V"  | Verbose mode (default is between Q and V)
/// "+"  | Adds this new fitted function to the list of fitted functions. By default, the previous function is deleted and only the last one is kept.
/// "N"  | Does not store the graphics function, does not draw the histogram with the function after fitting.
/// "0"  | Does not draw the histogram and the fitted function after fitting, but in contrast to option "N", it stores the fitted function in the histogram list of functions.
/// "R"  | Fit using a fitting range specified in the function range with `TF1::SetRange`.
/// "B"  | Use this option when you want to fix one or more parameters and the fitting function is a predefined one (e.g gaus, expo,..), otherwise in case of pre-defined functions, some default initial values and limits are set.
/// "C"  | In case of linear fitting, do no calculate the chisquare (saves CPU time).
/// "G"  | Uses the gradient implemented in `TF1::GradientPar` for the minimization. This allows to use Automatic Differentiation when it is supported by the provided TF1 function.
/// "EX0" | When fitting a TGraphErrors or TGraphAsymErrors do not consider errors in the X coordinates
/// "ROB" | In case of linear fitting, compute the LTS regression coefficients (robust (resistant) regression), using the default fraction of good points "ROB=0.x" - compute the LTS regression coefficients, using 0.x as a fraction of good points
///
///
/// This function is used for fitting also the derived TGraph classes such as TGraphErrors or TGraphAsymmErrors.
/// See the note below on how the errors are used when fitting a TGraphErrors or TGraphAsymmErrors.
///
/// The fitting of the TGraph, i.e simple data points without any error associated, is performed using the
/// un-weighted least-square (chi-square) method.
///
///
///\anchor GFitErrors
/// ### TGraphErrors fit:
///
///   In case of a TGraphErrors or TGraphAsymmErrors object, when `x` errors are present, the error along x,
///   is projected along the y-direction by calculating the function at the points `x-ex_low` and
///   `x+ex_high`, where `ex_low` and `ex_high` are the corresponding lower and upper error in x.
///   The chi-square is then computed as the sum of the quantity below at each data point:
///
/// \f[
///   \frac{(y-f(x))^{2}}{ey^{2}+(\frac{1}{2}(exl+exh)f'(x))^{2}}
/// \f]
///
///   where `x` and `y` are the point coordinates, and `f'(x)` is the derivative of the
///   function `f(x)`.
///
///   In case of asymmetric errors, if the function lies below (above) the data point, `ey` is `ey_low` (`ey_high`).
///
///   The approach used to approximate the uncertainty in y because of the
///   errors in x is to make it equal the error in x times the slope of the line.
///   This approach is called "effective variance method" and
///   the implementation is provided in the function FitUtil::EvaluateChi2Effective
///
/// \anchor GFitLinear
/// ### Linear fitting:
///   When the fitting function is linear (contains the `++` sign) or the fitting
///   function is a polynomial, a linear fitter is initialised.
///   To create a linear function, use the following syntax: linear parts
///   separated by `++` sign.
///   Example: to fit the parameters of the function `p0*x + p1*sin(x)`, you can create a
///   TF1 object as
///
///       TF1 *f1 = new TF1("f1", "x++sin(x)", xmin, xmax);
///
///   For such a TF1 you don't have to set the initial conditions and the linear fitter is used.
///   Going via the linear fitter for functions, linear in parameters, gives a
///   considerable advantage in speed.
///   When using the linear fitting it is also possible to perform a robust fitting with the
///   Least Trimmed Square (LTS) regression algorithm, by using the fit option `ROB`.
///   See the tutorial `fitLinearRobust.C`.
///
/// ### Notes on TGraph/TGraphErrors Fitting:
///
/// 1. By using the "effective variance" method a simple linear regression
///    becomes a non-linear case, which takes several iterations
///    instead of 0 as in the linear case.
/// 2. The effective variance technique assumes that there is no correlation
///    between the x and y coordinate.
/// 3. The standard chi2 (least square) method without error in the coordinates (x) can
///    be forced by using option "EX0"
/// 4. The linear fitter doesn't take into account the errors in x. When fitting a
///    TGraphErrors with a linear functions the errors in x will not be considered.
///    If errors in x are important, use option "F" for linear function fitting.
/// 5. When fitting a TGraph (i.e. no errors associated with each point),
///    a correction is applied to the errors on the parameters with the following
///    formula:
///    `parameter_error *= sqrt(chisquare/(ndf-1))`
///
/// ### General Fitting documentation
///
/// See in TH1::Fit for the documentation of
///  - [Fit Result](\ref HFitRes)
///  - [Fit Status](\ref HFitStatus)
///  - [Fit Statistics Box](\ref HFitStatBox)
///  - [Fitting in a Range](\ref HFitRange)
///  - [Setting Initial Conditions](\ref HFitInitial)
 
TFitResultPtr TGraph::Fit(TF1 *f1, Option_t *option, Option_t *goption, Axis_t rxmin, Axis_t rxmax)
{
   Foption_t fitOption;
   ROOT::Fit::FitOptionsMake(ROOT::Fit::kGraph, option, fitOption);
   // create range and minimizer options with default values
   ROOT::Fit::DataRange range(rxmin, rxmax);
   ROOT::Math::MinimizerOptions minOption;
   return ROOT::Fit::FitObject(this, f1 , fitOption , minOption, goption, range);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Fit this graph with function with name `fname`.
///
/// This is a different interface to TGraph fitting using TGraph::Fit(TF1 *f1,Option_t *, Option_t *, Axis_t, Axis_t)
/// See there for the details about fitting a TGraph.
///
/// The parameter `fname` is the name of an already predefined function created by TF1 or TF2
/// Predefined functions such as gaus, expo and poln are automatically
/// created by ROOT.
///
/// The parameter `fname` can also be a formula, accepted by the linear fitter (linear parts divided
/// by "++" sign), for example "x++sin(x)" for fitting "[0]*x+[1]*sin(x)"
 
TFitResultPtr TGraph::Fit(const char *fname, Option_t *option, Option_t *, Axis_t xmin, Axis_t xmax)
{
   const char *linear = fname ? strstr(fname, "++") : nullptr;
   if (linear) {
      TF1 f1(fname, fname, xmin, xmax);
      return Fit(&f1, option, "", xmin, xmax);
   }
   TF1 * f1 = (TF1*)gROOT->GetFunction(fname);
   if (!f1) {
      Printf("Unknown function: %s", fname);
      return -1;
   }
   return Fit(f1, option, "", xmin, xmax);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Display a GUI panel with all graph fit options.
///
/// See class TFitEditor for example
 
void TGraph::FitPanel()
{
   if (!gPad)
      gROOT->MakeDefCanvas();
 
   if (!gPad) {
      Error("FitPanel", "Unable to create a default canvas");
      return;
   }
 
   // use plugin manager to create instance of TFitEditor
   TPluginHandler *handler = gROOT->GetPluginManager()->FindHandler("TFitEditor");
   if (handler && handler->LoadPlugin() != -1) {
      if (handler->ExecPlugin(2, gPad, this) == 0)
         Error("FitPanel", "Unable to crate the FitPanel");
   } else
      Error("FitPanel", "Unable to find the FitPanel plug-in");
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return graph correlation factor
 
Double_t TGraph::GetCorrelationFactor() const
{
   Double_t rms1 = GetRMS(1);
   if (rms1 == 0) return 0;
   Double_t rms2 = GetRMS(2);
   if (rms2 == 0) return 0;
   return GetCovariance() / rms1 / rms2;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return covariance of vectors x,y
 
Double_t TGraph::GetCovariance() const
{
   if (fNpoints <= 0) return 0;
   Double_t sum = fNpoints, sumx = 0, sumy = 0, sumxy = 0;
 
   for (Int_t i = 0; i < fNpoints; i++) {
      sumx  += fX[i];
      sumy  += fY[i];
      sumxy += fX[i] * fY[i];
   }
   return sumxy / sum - sumx / sum * sumy / sum;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return mean value of X (axis=1)  or Y (axis=2)
 
Double_t TGraph::GetMean(Int_t axis) const
{
   if (axis < 1 || axis > 2) return 0;
   if (fNpoints <= 0) return 0;
   Double_t sumx = 0;
   for (Int_t i = 0; i < fNpoints; i++) {
      if (axis == 1) sumx += fX[i];
      else           sumx += fY[i];
   }
   return sumx / fNpoints;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return RMS of X (axis=1)  or Y (axis=2)
 
Double_t TGraph::GetRMS(Int_t axis) const
{
   if (axis < 1 || axis > 2) return 0;
   if (fNpoints <= 0) return 0;
   Double_t sumx = 0, sumx2 = 0;
   for (Int_t i = 0; i < fNpoints; i++) {
      if (axis == 1) {
         sumx += fX[i];
         sumx2 += fX[i] * fX[i];
      } else           {
         sumx += fY[i];
         sumx2 += fY[i] * fY[i];
      }
   }
   Double_t x = sumx / fNpoints;
   Double_t rms2 = TMath::Abs(sumx2 / fNpoints - x * x);
   return TMath::Sqrt(rms2);
}
 
////////////////////////////////////////////////////////////////////////////////
/// It always returns a negative value. Real implementation in TGraphErrors
 
Double_t TGraph::GetErrorX(Int_t) const
{
   return -1;
}
 
////////////////////////////////////////////////////////////////////////////////
/// It always returns a negative value. Real implementation in TGraphErrors
 
Double_t TGraph::GetErrorY(Int_t) const
{
   return -1;
}
 
////////////////////////////////////////////////////////////////////////////////
/// It always returns a negative value. Real implementation in TGraphErrors
/// and TGraphAsymmErrors
 
Double_t TGraph::GetErrorXhigh(Int_t) const
{
   return -1;
}
 
////////////////////////////////////////////////////////////////////////////////
/// It always returns a negative value. Real implementation in TGraphErrors
/// and TGraphAsymmErrors
 
Double_t TGraph::GetErrorXlow(Int_t) const
{
   return -1;
}
 
////////////////////////////////////////////////////////////////////////////////
/// It always returns a negative value. Real implementation in TGraphErrors
/// and TGraphAsymmErrors
 
Double_t TGraph::GetErrorYhigh(Int_t) const
{
   return -1;
}
 
////////////////////////////////////////////////////////////////////////////////
/// It always returns a negative value. Real implementation in TGraphErrors
/// and TGraphAsymmErrors
 
Double_t TGraph::GetErrorYlow(Int_t) const
{
   return -1;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return pointer to function with name.
///
/// Functions such as TGraph::Fit store the fitted function in the list of
/// functions of this graph.
 
TF1 *TGraph::GetFunction(const char *name) const
{
   if (!fFunctions) return nullptr;
   return (TF1*)fFunctions->FindObject(name);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns a pointer to the histogram used to draw the axis
/// Takes into account the two following cases.
///  1. option 'A' was specified in TGraph::Draw. Return fHistogram
///  2. user had called TPad::DrawFrame. return pointer to hframe histogram
 
TH1F *TGraph::GetHistogram() const
{
   Double_t rwxmin, rwxmax, rwymin, rwymax, maximum, minimum, dx, dy;
   Double_t uxmin, uxmax;
 
   ComputeRange(rwxmin, rwymin, rwxmax, rwymax);  //this is redefined in TGraphErrors
 
   // (if fHistogram exist) && (if the log scale is on) &&
   // (if the computed range minimum is > 0) && (if the fHistogram minimum is zero)
   // then it means fHistogram limits have been computed in linear scale
   // therefore they might be too strict and cut some points. In that case the
   // fHistogram limits should be recomputed ie: the existing fHistogram
   // should not be returned.
   TH1F *historg = nullptr;
   if (fHistogram) {
      if (!TestBit(kResetHisto)) {
         if (gPad && gPad->GetLogx()) {
            if (rwxmin <= 0 || fHistogram->GetXaxis()->GetXmin() != 0) return fHistogram;
         } else if (gPad && gPad->GetLogy()) {
            if (rwymin <= 0 || fHistogram->GetMinimum() != 0) return fHistogram;
         } else {
            return fHistogram;
         }
      } else {
         const_cast <TGraph*>(this)->ResetBit(kResetHisto);
      }
      historg = fHistogram;
   }
 
   if (rwxmin == rwxmax) rwxmax += 1.;
   if (rwymin == rwymax) rwymax += 1.;
   dx = 0.1 * (rwxmax - rwxmin);
   dy = 0.1 * (rwymax - rwymin);
   uxmin    = rwxmin - dx;
   uxmax    = rwxmax + dx;
   minimum  = rwymin - dy;
   maximum  = rwymax + dy;
 
   if (fMinimum != -1111) minimum = fMinimum;
   if (fMaximum != -1111) maximum = fMaximum;
 
   // the graph is created with at least as many channels as there are points
   // to permit zooming on the full range
   if (uxmin < 0 && rwxmin >= 0) {
      if (gPad && gPad->GetLogx()) uxmin = 0.9 * rwxmin;
      else                         uxmin = 0;
   }
   if (uxmax > 0 && rwxmax <= 0) {
      if (gPad && gPad->GetLogx()) uxmax = 1.1 * rwxmax;
      else                         uxmax = 0;
   }
 
   if (minimum < 0 && rwymin >= 0) minimum = 0.9 * rwymin;
 
   if (minimum <= 0 && gPad && gPad->GetLogy()) minimum = 0.001 * maximum;
   if (uxmin <= 0 && gPad && gPad->GetLogx()) {
      if (uxmax > 1000) uxmin = 1;
      else              uxmin = 0.001 * uxmax;
   }
 
   rwxmin = uxmin;
   rwxmax = uxmax;
   Int_t npt = 100;
   if (fNpoints > npt) npt = fNpoints;
   const char *gname = GetName();
   if (!gname[0]) gname = "Graph";
   // do not add the histogram to gDirectory
   // use local TDirectory::TContect that will set temporarly gDirectory to a nullptr and
   // will avoid that histogram is added in the global directory
   {
      TDirectory::TContext ctx(nullptr);
      ((TGraph*)this)->fHistogram = new TH1F(gname, GetTitle(), npt, rwxmin, rwxmax);
   }
   if (!fHistogram) return nullptr;
   fHistogram->SetMinimum(minimum);
   fHistogram->SetBit(TH1::kNoStats);
   fHistogram->SetMaximum(maximum);
   fHistogram->GetYaxis()->SetLimits(minimum, maximum);
   // Restore the axis attributes if needed
   if (historg) {
      fHistogram->GetXaxis()->SetTitle(historg->GetXaxis()->GetTitle());
      fHistogram->GetXaxis()->CenterTitle(historg->GetXaxis()->GetCenterTitle());
      fHistogram->GetXaxis()->RotateTitle(historg->GetXaxis()->GetRotateTitle());
      fHistogram->GetXaxis()->SetNoExponent(historg->GetXaxis()->GetNoExponent());
      fHistogram->GetXaxis()->SetTimeDisplay(historg->GetXaxis()->GetTimeDisplay());
      fHistogram->GetXaxis()->SetTimeFormat(historg->GetXaxis()->GetTimeFormat());
      historg->GetXaxis()->TAttAxis::Copy(*(fHistogram->GetXaxis()));
 
      fHistogram->GetYaxis()->SetTitle(historg->GetYaxis()->GetTitle());
      fHistogram->GetYaxis()->CenterTitle(historg->GetYaxis()->GetCenterTitle());
      fHistogram->GetYaxis()->RotateTitle(historg->GetYaxis()->GetRotateTitle());
      fHistogram->GetYaxis()->SetNoExponent(historg->GetYaxis()->GetNoExponent());
      fHistogram->GetYaxis()->SetTimeDisplay(historg->GetYaxis()->GetTimeDisplay());
      fHistogram->GetYaxis()->SetTimeFormat(historg->GetYaxis()->GetTimeFormat());
      historg->GetYaxis()->TAttAxis::Copy(*(fHistogram->GetYaxis()));
 
      delete historg;
   }
   return fHistogram;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Get x and y values for point number i.
/// The function returns -1 in case of an invalid request or the point number otherwise
 
Int_t TGraph::GetPoint(Int_t i, Double_t &x, Double_t &y) const
{
   if (i < 0 || i >= fNpoints || !fX || !fY) return -1;
   x = fX[i];
   y = fY[i];
   return i;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Get x value for point i.
 
Double_t TGraph::GetPointX(Int_t i) const
{
   if (i < 0 || i >= fNpoints || !fX)
      return -1.;
 
   return fX[i];
}
 
////////////////////////////////////////////////////////////////////////////////
/// Get y value for point i.
 
Double_t TGraph::GetPointY(Int_t i) const
{
   if (i < 0 || i >= fNpoints || !fY)
      return -1.;
 
   return fY[i];
}
 
////////////////////////////////////////////////////////////////////////////////
/// Get x axis of the graph.
 
TAxis *TGraph::GetXaxis() const
{
   auto h = GetHistogram();
   return h ? h->GetXaxis() : nullptr;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Get y axis of the graph.
 
TAxis *TGraph::GetYaxis() const
{
   auto h = GetHistogram();
   return h ? h->GetYaxis() : nullptr;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Implementation to get information on point of graph at cursor position
/// Adapted from class TH1
 
char *TGraph::GetObjectInfo(Int_t px, Int_t py) const
{
   if (!gPad) {
      Error("GetObjectInfo", "Cannot be used without gPad");
      return nullptr;
   }
 
   // localize point
   Int_t ipoint = -2;
   // start with a small window (in case the mouse is very close to one point)
   for (Int_t i = 0; i < fNpoints; i++) {
      Int_t dpx = px - gPad->XtoAbsPixel(gPad->XtoPad(fX[i]));
      Int_t dpy = py - gPad->YtoAbsPixel(gPad->YtoPad(fY[i]));
 
      if (dpx * dpx + dpy * dpy < 25) {
         ipoint = i;
         break;
      }
   }
 
   Double_t x = gPad->PadtoX(gPad->AbsPixeltoX(px));
   Double_t y = gPad->PadtoY(gPad->AbsPixeltoY(py));
 
   if (ipoint == -2)
      return Form("x=%g, y=%g", x, y);
 
   Double_t xval = fX[ipoint];
   Double_t yval = fY[ipoint];
 
   return Form("x=%g, y=%g, point=%d, xval=%g, yval=%g", x, y, ipoint, xval, yval);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Compute Initial values of parameters for a gaussian.
 
void TGraph::InitGaus(Double_t xmin, Double_t xmax)
{
   Double_t allcha, sumx, sumx2, x, val, rms, mean;
   Int_t bin;
   const Double_t sqrtpi = 2.506628;
 
   // Compute mean value and RMS of the graph in the given range
   if (xmax <= xmin) {
      xmin = fX[0];
      xmax = fX[fNpoints-1];
   }
   Int_t np = 0;
   allcha = sumx = sumx2 = 0;
   for (bin = 0; bin < fNpoints; bin++) {
      x       = fX[bin];
      if (x < xmin || x > xmax) continue;
      np++;
      val     = fY[bin];
      sumx   += val * x;
      sumx2  += val * x * x;
      allcha += val;
   }
   if (np == 0 || allcha == 0) return;
   mean = sumx / allcha;
   rms  = TMath::Sqrt(sumx2 / allcha - mean * mean);
   Double_t binwidx = TMath::Abs((xmax - xmin) / np);
   if (rms == 0) rms = 1;
   TVirtualFitter *grFitter = TVirtualFitter::GetFitter();
   TF1 *f1 = (TF1*)grFitter->GetUserFunc();
   f1->SetParameter(0, binwidx * allcha / (sqrtpi * rms));
   f1->SetParameter(1, mean);
   f1->SetParameter(2, rms);
   f1->SetParLimits(2, 0, 10 * rms);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Compute Initial values of parameters for an exponential.
 
void TGraph::InitExpo(Double_t xmin, Double_t xmax)
{
   Double_t constant, slope;
   Int_t ifail;
   if (xmax <= xmin) {
      xmin = fX[0];
      xmax = fX[fNpoints-1];
   }
   Int_t nchanx = fNpoints;
 
   LeastSquareLinearFit(-nchanx, constant, slope, ifail, xmin, xmax);
 
   TVirtualFitter *grFitter = TVirtualFitter::GetFitter();
   TF1 *f1 = (TF1*)grFitter->GetUserFunc();
   f1->SetParameter(0, constant);
   f1->SetParameter(1, slope);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Compute Initial values of parameters for a polynom.
 
void TGraph::InitPolynom(Double_t xmin, Double_t xmax)
{
   Double_t fitpar[25];
 
   TVirtualFitter *grFitter = TVirtualFitter::GetFitter();
   TF1 *f1 = (TF1*)grFitter->GetUserFunc();
   Int_t npar   = f1->GetNpar();
   if (xmax <= xmin) {
      xmin = fX[0];
      xmax = fX[fNpoints-1];
   }
 
   LeastSquareFit(npar, fitpar, xmin, xmax);
 
   for (Int_t i = 0; i < npar; i++) f1->SetParameter(i, fitpar[i]);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Insert a new point at the mouse position
 
Int_t TGraph::InsertPoint()
{
   if (!gPad) {
      Error("InsertPoint", "Cannot be used without gPad, requires last mouse position");
      return -1;
   }
 
   Int_t px = gPad->GetEventX();
   Int_t py = gPad->GetEventY();
 
   //localize point where to insert
   Int_t ipoint = -2;
   Int_t i, d = 0;
   // start with a small window (in case the mouse is very close to one point)
   for (i = 0; i < fNpoints - 1; i++) {
      d = DistancetoLine(px, py, gPad->XtoPad(fX[i]), gPad->YtoPad(fY[i]), gPad->XtoPad(fX[i+1]), gPad->YtoPad(fY[i+1]));
      if (d < 5) {
         ipoint = i + 1;
         break;
      }
   }
   if (ipoint == -2) {
      //may be we are far from one point, try again with a larger window
      for (i = 0; i < fNpoints - 1; i++) {
         d = DistancetoLine(px, py, gPad->XtoPad(fX[i]), gPad->YtoPad(fY[i]), gPad->XtoPad(fX[i+1]), gPad->YtoPad(fY[i+1]));
         if (d < 10) {
            ipoint = i + 1;
            break;
         }
      }
   }
   if (ipoint == -2) {
      //distinguish between first and last point
      Int_t dpx = px - gPad->XtoAbsPixel(gPad->XtoPad(fX[0]));
      Int_t dpy = py - gPad->YtoAbsPixel(gPad->XtoPad(fY[0]));
      if (dpx * dpx + dpy * dpy < 25) ipoint = 0;
      else                      ipoint = fNpoints;
   }
 
 
   InsertPointBefore(ipoint, gPad->AbsPixeltoX(px), gPad->AbsPixeltoY(py));
 
   gPad->Modified();
   return ipoint;
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Insert a new point with coordinates (x,y) before the point number `ipoint`.
 
void TGraph::InsertPointBefore(Int_t ipoint, Double_t x, Double_t y)
{
   if (ipoint < 0) {
      Error("TGraph", "Inserted point index should be >= 0");
      return;
   }
 
   if (ipoint > fNpoints) {
      Error("TGraph", "Inserted point index should be <= %d", fNpoints);
      return;
   }
 
   if (ipoint == fNpoints) {
       SetPoint(ipoint, x, y);
       return;
   }
 
   Double_t **ps = ExpandAndCopy(fNpoints + 1, ipoint);
   CopyAndRelease(ps, ipoint, fNpoints++, ipoint + 1);
 
   // To avoid redefinitions in descendant classes
   FillZero(ipoint, ipoint + 1);
 
   fX[ipoint] = x;
   fY[ipoint] = y;
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Integrate the TGraph data within a given (index) range.
/// Note that this function computes the area of the polygon enclosed by the points of the TGraph.
/// The polygon segments, which are defined by the points of the TGraph, do not need to form a closed polygon,
/// since the last polygon segment, which closes the polygon, is taken as the line connecting the last TGraph point
/// with the first one. It is clear that the order of the point is essential in defining the polygon.
/// Also note that the segments should not intersect.
///
/// NB:
///  - if last=-1 (default) last is set to the last point.
///  - if (first <0) the first point (0) is taken.
///
/// ### Method:
///
/// There are many ways to calculate the surface of a polygon. It all depends on what kind of data
/// you have to deal with. The most evident solution would be to divide the polygon in triangles and
/// calculate the surface of them. But this can quickly become complicated as you will have to test
/// every segments of every triangles and check if they are intersecting with a current polygon's
/// segment or if it goes outside the polygon. Many calculations that would lead to many problems...
///
/// ### The solution (implemented by R.Brun)
/// Fortunately for us, there is a simple way to solve this problem, as long as the polygon's
/// segments don't intersect.
/// It takes the x coordinate of the current vertex and multiply it by the y coordinate of the next
/// vertex. Then it subtracts from it the result of the y coordinate of the current vertex multiplied
/// by the x coordinate of the next vertex. Then divide the result by 2 to get the surface/area.
///
/// ### Sources
///  - http://forums.wolfram.com/mathgroup/archive/1998/Mar/msg00462.html
///  - http://stackoverflow.com/questions/451426/how-do-i-calculate-the-surface-area-of-a-2d-polygon
 
Double_t TGraph::Integral(Int_t first, Int_t last) const
{
   if (first < 0) first = 0;
   if (last < 0) last = fNpoints - 1;
   if (last >= fNpoints) last = fNpoints - 1;
   if (first >= last) return 0;
   Int_t np = last - first + 1;
   Double_t sum = 0.0;
   //for(Int_t i=first;i<=last;i++) {
   //   Int_t j = first + (i-first+1)%np;
   //   sum += TMath::Abs(fX[i]*fY[j]);
   //   sum -= TMath::Abs(fY[i]*fX[j]);
   //}
   for (Int_t i = first; i <= last; i++) {
      Int_t j = first + (i - first + 1) % np;
      sum += (fY[i] + fY[j]) * (fX[j] - fX[i]);
   }
   return 0.5 * TMath::Abs(sum);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return 1 if the point (x,y) is inside the polygon defined by
/// the graph vertices 0 otherwise.
///
/// Algorithm:
///
/// The loop is executed with the end-point coordinates of a line segment
/// (X1,Y1)-(X2,Y2) and the Y-coordinate of a horizontal line.
/// The counter inter is incremented if the line (X1,Y1)-(X2,Y2) intersects
/// the horizontal line. In this case XINT is set to the X-coordinate of the
/// intersection point. If inter is an odd number, then the point x,y is within
/// the polygon.
 
Int_t TGraph::IsInside(Double_t x, Double_t y) const
{
   return (Int_t)TMath::IsInside(x, y, fNpoints, fX, fY);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Least squares polynomial fitting without weights.
///
/// \param [in] m     number of parameters
/// \param [in] a     array of parameters
/// \param [in] xmin  1st point number to fit (default =0)
/// \param [in] xmax  last point number to fit (default=fNpoints-1)
///
/// based on CERNLIB routine LSQ: Translated to C++ by Rene Brun
 
void TGraph::LeastSquareFit(Int_t m, Double_t *a, Double_t xmin, Double_t xmax)
{
   const Double_t zero = 0.;
   const Double_t one = 1.;
   const Int_t idim = 20;
 
   Double_t  b[400]        /* was [20][20] */;
   Int_t i, k, l, ifail;
   Double_t power;
   Double_t da[20], xk, yk;
   Int_t n = fNpoints;
   if (xmax <= xmin) {
      xmin = fX[0];
      xmax = fX[fNpoints-1];
   }
 
   if (m <= 2) {
      LeastSquareLinearFit(n, a[0], a[1], ifail, xmin, xmax);
      return;
   }
   if (m > idim || m > n) return;
   da[0] = zero;
   for (l = 2; l <= m; ++l) {
      b[l-1]           = zero;
      b[m + l*20 - 21] = zero;
      da[l-1]          = zero;
   }
   Int_t np = 0;
   for (k = 0; k < fNpoints; ++k) {
      xk     = fX[k];
      if (xk < xmin || xk > xmax) continue;
      np++;
      yk     = fY[k];
      power  = one;
      da[0] += yk;
      for (l = 2; l <= m; ++l) {
         power   *= xk;
         b[l-1]  += power;
         da[l-1] += power * yk;
      }
      for (l = 2; l <= m; ++l) {
         power            *= xk;
         b[m + l*20 - 21] += power;
      }
   }
   b[0]  = Double_t(np);
   for (i = 3; i <= m; ++i) {
      for (k = i; k <= m; ++k) {
         b[k - 1 + (i-1)*20 - 21] = b[k + (i-2)*20 - 21];
      }
   }
   H1LeastSquareSeqnd(m, b, idim, ifail, 1, da);
 
   if (ifail < 0) {
      a[0] = fY[0];
      for (i = 1; i < m; ++i) a[i] = 0;
      return;
   }
   for (i = 0; i < m; ++i) a[i] = da[i];
}
 
////////////////////////////////////////////////////////////////////////////////
/// Least square linear fit without weights.
///
///  Fit a straight line (a0 + a1*x) to the data in this graph.
///
/// \param [in] ndata        if ndata<0, fits the logarithm of the graph (used in InitExpo() to set
///                          the initial parameter values for a fit with exponential function.
/// \param [in] a0           constant
/// \param [in] a1           slope
/// \param [in] ifail        return parameter indicating the status of the fit (ifail=0, fit is OK)
/// \param [in] xmin, xmax   fitting range
///
///  extracted from CERNLIB LLSQ: Translated to C++ by Rene Brun
 
void TGraph::LeastSquareLinearFit(Int_t ndata, Double_t &a0, Double_t &a1, Int_t &ifail, Double_t xmin, Double_t xmax)
{
   Double_t xbar, ybar, x2bar;
   Int_t i;
   Double_t xybar;
   Double_t fn, xk, yk;
   Double_t det;
   if (xmax <= xmin) {
      xmin = fX[0];
      xmax = fX[fNpoints-1];
   }
 
   ifail = -2;
   xbar  = ybar = x2bar = xybar = 0;
   Int_t np = 0;
   for (i = 0; i < fNpoints; ++i) {
      xk = fX[i];
      if (xk < xmin || xk > xmax) continue;
      np++;
      yk = fY[i];
      if (ndata < 0) {
         if (yk <= 0) yk = 1e-9;
         yk = TMath::Log(yk);
      }
      xbar  += xk;
      ybar  += yk;
      x2bar += xk * xk;
      xybar += xk * yk;
   }
   fn    = Double_t(np);
   det   = fn * x2bar - xbar * xbar;
   ifail = -1;
   if (det <= 0) {
      if (fn > 0) a0 = ybar / fn;
      else        a0 = 0;
      a1 = 0;
      return;
   }
   ifail = 0;
   a0 = (x2bar * ybar - xbar * xybar) / det;
   a1 = (fn * xybar - xbar * ybar) / det;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Draw this graph with its current attributes.
 
void TGraph::Paint(Option_t *option)
{
   TVirtualGraphPainter *painter = TVirtualGraphPainter::GetPainter();
   if (painter) painter->PaintHelper(this, option);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Draw the (x,y) as a graph.
 
void TGraph::PaintGraph(Int_t npoints, const Double_t *x, const Double_t *y, Option_t *chopt)
{
   TVirtualGraphPainter *painter = TVirtualGraphPainter::GetPainter();
   if (painter) painter->PaintGraph(this, npoints, x, y, chopt);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Draw the (x,y) as a histogram.
 
void TGraph::PaintGrapHist(Int_t npoints, const Double_t *x, const Double_t *y, Option_t *chopt)
{
   TVirtualGraphPainter *painter = TVirtualGraphPainter::GetPainter();
   if (painter) painter->PaintGrapHist(this, npoints, x, y, chopt);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Draw the stats
 
void TGraph::PaintStats(TF1 *fit)
{
   TVirtualGraphPainter *painter = TVirtualGraphPainter::GetPainter();
   if (painter) painter->PaintStats(this, fit);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Print graph values.
 
void TGraph::Print(Option_t *) const
{
   for (Int_t i = 0; i < fNpoints; i++) {
      printf("x[%d]=%g, y[%d]=%g\n", i, fX[i], i, fY[i]);
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Recursively remove object from the list of functions
 
void TGraph::RecursiveRemove(TObject *obj)
{
   if (fFunctions) {
      if (!fFunctions->TestBit(kInvalidObject))
         fFunctions->RecursiveRemove(obj);
   }
   if (fHistogram == obj)
      fHistogram = nullptr;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Delete point close to the mouse position
/// Returns index of removed point (or -1 if nothing was changed)
 
Int_t TGraph::RemovePoint()
{
   if (!gPad) {
      Error("RemovePoint", "Cannot be used without gPad, requires last mouse position");
      return -1;
   }
 
   Int_t px = gPad->GetEventX();
   Int_t py = gPad->GetEventY();
 
   //localize point to be deleted
   Int_t ipoint = -2;
   // start with a small window (in case the mouse is very close to one point)
   for (Int_t i = 0; i < fNpoints; i++) {
      Int_t dpx = px - gPad->XtoAbsPixel(gPad->XtoPad(fX[i]));
      Int_t dpy = py - gPad->YtoAbsPixel(gPad->YtoPad(fY[i]));
      if (dpx * dpx + dpy * dpy < 100) {
         ipoint = i;
         break;
      }
   }
   return RemovePoint(ipoint);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Delete point number ipoint
/// Returns index of removed point (or -1 if nothing was changed)
 
Int_t TGraph::RemovePoint(Int_t ipoint)
{
   if ((ipoint < 0) || (ipoint >= fNpoints))
      return -1;
 
   Double_t **ps = ShrinkAndCopy(fNpoints - 1, ipoint);
   CopyAndRelease(ps, ipoint + 1, fNpoints--, ipoint);
   if (gPad) gPad->Modified();
   return ipoint;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Save the graph as .csv, .tsv or .txt. In case of any other extension, fall
/// back to TObject::SaveAs
///
/// The result can be immediately imported into Excel, gnuplot, Python or whatever,
/// without the needing to install pyroot, etc.
///
/// \param filename the name of the file where to store the graph
/// \param option some tuning options
///
/// The file extension defines the delimiter used:
///  - `.csv` : comma
///  - `.tsv` : tab
///  - `.txt` : space
///
/// If option = "title" a title line is generated with the axis titles.
 
void TGraph::SaveAs(const char *filename, Option_t *option) const
{
   char del = '\0';
   TString ext = "";
   TString fname = filename;
   TString opt = option;
 
   if (filename) {
      if      (fname.EndsWith(".csv")) {del = ',';  ext = "csv";}
      else if (fname.EndsWith(".tsv")) {del = '\t'; ext = "tsv";}
      else if (fname.EndsWith(".txt")) {del = ' ';  ext = "txt";}
   }
   if (del) {
      std::ofstream out;
      out.open(filename, std::ios::out);
      if (!out.good ()) {
         Error("SaveAs", "cannot open file: %s", filename);
         return;
      }
      if (InheritsFrom(TGraphErrors::Class()) ) {
         if(opt.Contains("title"))
         out << "# " << GetXaxis()->GetTitle() << "\tex\t" << GetYaxis()->GetTitle() << "\tey" << std::endl;
         double *ex = this->GetEX();
         double *ey = this->GetEY();
         for(int i=0 ; i<fNpoints ; i++)
         out << fX[i] << del << (ex?ex[i]:0) << del << fY[i] << del << (ey?ey[i]:0) << std::endl;
      } else if (InheritsFrom(TGraphAsymmErrors::Class()) || InheritsFrom(TGraphBentErrors::Class())) {
         if(opt.Contains("title"))
         out << "# " << GetXaxis()->GetTitle() << "\texl\t" << "\texh\t" << GetYaxis()->GetTitle() << "\teyl" << "\teyh" << std::endl;
         double *exl = this->GetEXlow();
         double *exh = this->GetEXhigh();
         double *eyl = this->GetEYlow();
         double *eyh = this->GetEYhigh();
         for(int i=0 ; i<fNpoints ; i++)
         out << fX[i] << del << (exl?exl[i]:0) << del << (exh?exh[i]:0) << del << fY[i] << del << (eyl?eyl[i]:0) << del << (eyh?eyh[i]:0) << std::endl;
      } else {
         if(opt.Contains("title"))
         out << "# " << GetXaxis()->GetTitle() << "\t" << GetYaxis()->GetTitle() << std::endl;
         for (int i=0 ; i<fNpoints ; i++)
         out << fX[i] << del << fY[i] << std::endl;
      }
      out.close();
      Info("SaveAs", "%s file: %s has been generated", ext.Data(), filename);
   } else {
      TObject::SaveAs(filename, option);
   }
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Save primitive as a C++ statement(s) on output stream out
 
void TGraph::SavePrimitive(std::ostream &out, Option_t *option /*= ""*/)
{
   out << "   " << std::endl;
   static Int_t frameNumber = 0;
   frameNumber++;
 
   TString fXName, fYName;
 
   if (fNpoints >= 1) {
      fXName = SaveArray(out, "fx", frameNumber, fX);
      fYName = SaveArray(out, "fy", frameNumber, fY);
   }
 
   if (gROOT->ClassSaved(TGraph::Class()))
      out << "   ";
   else
      out << "   TGraph *";
 
   if (fNpoints >= 1)
      out << "graph = new TGraph(" << fNpoints << "," << fXName << "," << fYName << ");" << std::endl;
   else
      out << "graph = new TGraph();" << std::endl;
 
   SaveHistogramAndFunctions(out, "graph", frameNumber, option);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Save array as C++ code
/// Returns name of created array
 
TString TGraph::SaveArray(std::ostream &out, const char *suffix, Int_t frameNumber, Double_t *arr)
{
   const char *name = GetName();
   if (!name || !*name)
      name = "Graph";
   TString arrname = TString::Format("%s_%s%d", name, suffix, frameNumber);
 
   out << "   Double_t " << arrname << "[" << fNpoints << "] = { ";
   for (Int_t i = 0; i < fNpoints-1; i++) {
      out << arr[i] << ",";
      if (i && (i % 16 == 0))
         out << std::endl << "   ";
      else
         out << " ";
   }
   out << arr[fNpoints-1] << " };" << std::endl;
 
   return arrname;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Save histogram and list of functions of TGraph as C++ statement
/// Used in all TGraph-derived classes
 
void TGraph::SaveHistogramAndFunctions(std::ostream &out, const char *varname, Int_t &frameNumber, Option_t *option)
{
   char quote = '"';
 
   out << "   "<<varname<<"->SetName(" << quote << GetName() << quote << ");" << std::endl;
   out << "   "<<varname<<"->SetTitle(" << quote << GetTitle() << quote << ");" << std::endl;
 
   SaveFillAttributes(out, varname, 0, 1001);
   SaveLineAttributes(out, varname, 1, 1, 1);
   SaveMarkerAttributes(out, varname, 1, 1, 1);
 
   if (fHistogram) {
      TString hname = fHistogram->GetName();
      fHistogram->SetName(TString::Format("Graph_%s%d", hname.Data(), frameNumber).Data());
      fHistogram->SavePrimitive(out, "nodraw");
      out << "   "<<varname<<"->SetHistogram(" << fHistogram->GetName() << ");" << std::endl;
      out << "   " << std::endl;
      fHistogram->SetName(hname.Data());
   }
 
   // save list of functions
   TIter next(fFunctions);
   while (auto obj = next()) {
      obj->SavePrimitive(out, TString::Format("nodraw #%d\n", ++frameNumber).Data());
      if (obj->InheritsFrom("TPaveStats")) {
         out << "   "<<varname<<"->GetListOfFunctions()->Add(ptstats);" << std::endl;
         out << "   ptstats->SetParent("<<varname<<"->GetListOfFunctions());" << std::endl;
      } else {
         auto objname = TString::Format("%s%d",obj->GetName(), frameNumber);
         if (obj->InheritsFrom("TF1")) {
            out << "   " << objname << "->SetParent("<<varname<<");\n";
         }
         out << "   "<<varname<<"->GetListOfFunctions()->Add(" << objname << ");" << std::endl;
      }
   }
 
   const char *soption = option ? option : "";
   const char *l = strstr(soption, "multigraph");
   if (l) {
      out << "   multigraph->Add("<<varname<<"," << quote << l + 10 << quote << ");" << std::endl;
      return;
   }
   l = strstr(soption, "th2poly");
   if (l) {
      out << "   " << l + 7 << "->AddBin("<<varname<<");" << std::endl;
      return;
   }
   out << "   "<<varname<<"->Draw(" << quote << soption << quote << ");" << std::endl;
 
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Multiply the values of a TGraph by a constant c1.
///
/// If option contains "x" the x values are scaled
/// If option contains "y" the y values are scaled
/// If option contains "xy" both x and y values are scaled
 
void TGraph::Scale(Double_t c1, Option_t *option)
{
   TString opt = option; opt.ToLower();
   if (opt.Contains("x")) {
      for (Int_t i=0; i<GetN(); i++)
         GetX()[i] *= c1;
   }
   if (opt.Contains("y")) {
      for (Int_t i=0; i<GetN(); i++)
         GetY()[i] *= c1;
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set number of points in the graph
/// Existing coordinates are preserved
/// New coordinates above fNpoints are preset to 0.
 
void TGraph::Set(Int_t n)
{
   if (n < 0) n = 0;
   if (n == fNpoints) return;
   Double_t **ps = Allocate(n);
   CopyAndRelease(ps, 0, TMath::Min(fNpoints, n), 0);
   if (n > fNpoints) {
      FillZero(fNpoints, n, kFALSE);
   }
   fNpoints = n;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return kTRUE if kNotEditable bit is not set, kFALSE otherwise.
 
Bool_t TGraph::GetEditable() const
{
   return TestBit(kNotEditable) ? kFALSE : kTRUE;
}
 
////////////////////////////////////////////////////////////////////////////////
/// if editable=kFALSE, the graph cannot be modified with the mouse
///  by default a TGraph is editable
 
void TGraph::SetEditable(Bool_t editable)
{
   if (editable) ResetBit(kNotEditable);
   else          SetBit(kNotEditable);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set highlight (enable/disble) mode for the graph
/// by default highlight mode is disable
 
void TGraph::SetHighlight(Bool_t set)
{
   if (IsHighlight() == set) return;
 
   TVirtualGraphPainter *painter = TVirtualGraphPainter::GetPainter();
   if (!painter) return;
   SetBit(kIsHighlight, set);
   painter->SetHighlight(this);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set the maximum of the graph.
 
void TGraph::SetMaximum(Double_t maximum)
{
   fMaximum = maximum;
   GetHistogram()->SetMaximum(maximum);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set the minimum of the graph.
 
void TGraph::SetMinimum(Double_t minimum)
{
   fMinimum = minimum;
   GetHistogram()->SetMinimum(minimum);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set x and y values for point number i.
 
void TGraph::SetPoint(Int_t i, Double_t x, Double_t y)
{
   if (i < 0) return;
   if (fHistogram) SetBit(kResetHisto);
 
   if (i >= fMaxSize) {
      Double_t **ps = ExpandAndCopy(i + 1, fNpoints);
      CopyAndRelease(ps, 0, 0, 0);
   }
   if (i >= fNpoints) {
      // points above i can be not initialized
      // set zero up to i-th point to avoid redefinition
      // of this method in descendant classes
      FillZero(fNpoints, i + 1);
      fNpoints = i + 1;
   }
   fX[i] = x;
   fY[i] = y;
   if (gPad) gPad->Modified();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set x value for point i.
 
void TGraph::SetPointX(Int_t i, Double_t x)
{
    SetPoint(i, x, GetPointY(i));
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set y value for point i.
 
void TGraph::SetPointY(Int_t i, Double_t y)
{
    SetPoint(i, GetPointX(i), y);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set graph name.
void TGraph::SetName(const char *name)
{
   fName = name;
   if (fHistogram) fHistogram->SetName(name);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Change (i.e. set) the title
///
/// if title is in the form `stringt;stringx;stringy;stringz`
/// the graph title is set to `stringt`, the x axis title to `stringx`,
/// the y axis title to `stringy`, and the z axis title to `stringz`.
///
/// To insert the character `;` in one of the titles, one should use `#;`
/// or `#semicolon`.
 
void TGraph::SetTitle(const char* title)
{
   fTitle = title;
   fTitle.ReplaceAll("#;",2,"#semicolon",10);
   Int_t p = fTitle.Index(";");
 
   if (p>0) {
      if (!fHistogram) GetHistogram();
      fHistogram->SetTitle(title);
      Int_t n = fTitle.Length()-p;
      if (p>0) fTitle.Remove(p,n);
      fTitle.ReplaceAll("#semicolon",10,"#;",2);
   } else {
      if (fHistogram) fHistogram->SetTitle(title);
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set graph name and title
 
void TGraph::SetNameTitle(const char *name, const char *title)
{
   SetName(name);
   SetTitle(title);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set statistics option on/off.
///
/// By default, the statistics box is drawn.
/// The paint options can be selected via gStyle->SetOptStat.
/// This function sets/resets the kNoStats bit in the graph object.
/// It has priority over the Style option.
 
void TGraph::SetStats(Bool_t stats)
{
   ResetBit(kNoStats);
   if (!stats) {
      SetBit(kNoStats);
      //remove the "stats" object from the list of functions
      if (fFunctions) {
         TObject *obj = fFunctions->FindObject("stats");
         if (obj) {
            fFunctions->Remove(obj);
            delete obj;
         }
      }
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// if size*2 <= fMaxSize allocate new arrays of size points,
/// copy points [0,oend).
/// Return newarray (passed or new instance if it was zero
/// and allocations are needed)
 
Double_t **TGraph::ShrinkAndCopy(Int_t size, Int_t oend)
{
   if (size * 2 > fMaxSize || !fMaxSize)
      return nullptr;
 
   Double_t **newarrays = Allocate(size);
   CopyPoints(newarrays, 0, oend, 0);
   return newarrays;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Sorts the points of this TGraph using in-place quicksort (see e.g. older glibc).
/// To compare two points the function parameter greaterfunc is used (see TGraph::CompareX for an
/// example of such a method, which is also the default comparison function for Sort). After
/// the sort, greaterfunc(this, i, j) will return kTRUE for all i>j if ascending == kTRUE, and
/// kFALSE otherwise.
///
/// The last two parameters are used for the recursive quick sort, stating the range to be sorted
///
/// Examples:
/// ~~~ {.cpp}
///   // sort points along x axis
///   graph->Sort();
///   // sort points along their distance to origin
///   graph->Sort(&TGraph::CompareRadius);
///
///   Bool_t CompareErrors(const TGraph* gr, Int_t i, Int_t j) {
///     const TGraphErrors* ge=(const TGraphErrors*)gr;
///     return (ge->GetEY()[i]>ge->GetEY()[j]); }
///   // sort using the above comparison function, largest errors first
///   graph->Sort(&CompareErrors, kFALSE);
/// ~~~
 
void TGraph::Sort(Bool_t (*greaterfunc)(const TGraph*, Int_t, Int_t) /*=TGraph::CompareX()*/,
                  Bool_t ascending /*=kTRUE*/, Int_t low /* =0 */, Int_t high /* =-1111 */)
{
 
   // set the bit in case of an ascending =sort in X
   if (greaterfunc == TGraph::CompareX && ascending  && low == 0 && high == -1111)
      SetBit(TGraph::kIsSortedX);
 
   if (high == -1111) high = GetN() - 1;
   //  Termination condition
   if (high <= low) return;
 
   int left, right;
   left = low; // low is the pivot element
   right = high;
   while (left < right) {
      // move left while item < pivot
      while (left <= high && greaterfunc(this, left, low) != ascending)
         left++;
      // move right while item > pivot
      while (right > low && greaterfunc(this, right, low) == ascending)
         right--;
      if (left < right && left < high && right > low)
         SwapPoints(left, right);
   }
   // right is final position for the pivot
   if (right > low)
      SwapPoints(low, right);
   Sort(greaterfunc, ascending, low, right - 1);
   Sort(greaterfunc, ascending, right + 1, high);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Stream an object of class TGraph.
 
void TGraph::Streamer(TBuffer &b)
{
   if (b.IsReading()) {
      UInt_t R__s, R__c;
      Version_t R__v = b.ReadVersion(&R__s, &R__c);
      if (R__v > 2) {
         b.ReadClassBuffer(TGraph::Class(), this, R__v, R__s, R__c);
         if (fHistogram) fHistogram->SetDirectory(nullptr);
         TIter next(fFunctions);
         TObject *obj;
         while ((obj = next())) {
            if (obj->InheritsFrom(TF1::Class())) {
               TF1 *f1 = (TF1*)obj;
               f1->SetParent(this);
            }
         }
         fMaxSize = fNpoints;
         return;
      }
      //====process old versions before automatic schema evolution
      TNamed::Streamer(b);
      TAttLine::Streamer(b);
      TAttFill::Streamer(b);
      TAttMarker::Streamer(b);
      b >> fNpoints;
      fMaxSize = fNpoints;
      fX = new Double_t[fNpoints];
      fY = new Double_t[fNpoints];
      if (R__v < 2) {
         Float_t *x = new Float_t[fNpoints];
         Float_t *y = new Float_t[fNpoints];
         b.ReadFastArray(x, fNpoints);
         b.ReadFastArray(y, fNpoints);
         for (Int_t i = 0; i < fNpoints; i++) {
            fX[i] = x[i];
            fY[i] = y[i];
         }
         delete [] y;
         delete [] x;
      } else {
         b.ReadFastArray(fX, fNpoints);
         b.ReadFastArray(fY, fNpoints);
      }
      b >> fFunctions;
      b >> fHistogram;
      if (fHistogram) fHistogram->SetDirectory(nullptr);
      if (R__v < 2) {
         Float_t mi, ma;
         b >> mi;
         b >> ma;
         fMinimum = mi;
         fMaximum = ma;
      } else {
         b >> fMinimum;
         b >> fMaximum;
      }
      b.CheckByteCount(R__s, R__c, TGraph::IsA());
      //====end of old versions
 
   } else {
      b.WriteClassBuffer(TGraph::Class(), this);
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Swap points.
 
void TGraph::SwapPoints(Int_t pos1, Int_t pos2)
{
   SwapValues(fX, pos1, pos2);
   SwapValues(fY, pos1, pos2);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Swap values.
 
void TGraph::SwapValues(Double_t* arr, Int_t pos1, Int_t pos2)
{
   Double_t tmp = arr[pos1];
   arr[pos1] = arr[pos2];
   arr[pos2] = tmp;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set current style settings in this graph
/// This function is called when either TCanvas::UseCurrentStyle
/// or TROOT::ForceStyle have been invoked.
 
void TGraph::UseCurrentStyle()
{
   if (gStyle->IsReading()) {
      SetFillColor(gStyle->GetHistFillColor());
      SetFillStyle(gStyle->GetHistFillStyle());
      SetLineColor(gStyle->GetHistLineColor());
      SetLineStyle(gStyle->GetHistLineStyle());
      SetLineWidth(gStyle->GetHistLineWidth());
      SetMarkerColor(gStyle->GetMarkerColor());
      SetMarkerStyle(gStyle->GetMarkerStyle());
      SetMarkerSize(gStyle->GetMarkerSize());
   } else {
      gStyle->SetHistFillColor(GetFillColor());
      gStyle->SetHistFillStyle(GetFillStyle());
      gStyle->SetHistLineColor(GetLineColor());
      gStyle->SetHistLineStyle(GetLineStyle());
      gStyle->SetHistLineWidth(GetLineWidth());
      gStyle->SetMarkerColor(GetMarkerColor());
      gStyle->SetMarkerStyle(GetMarkerStyle());
      gStyle->SetMarkerSize(GetMarkerSize());
   }
   if (fHistogram) fHistogram->UseCurrentStyle();
 
   TIter next(GetListOfFunctions());
   TObject *obj;
 
   while ((obj = next())) {
      obj->UseCurrentStyle();
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Adds all graphs from the collection to this graph.
/// Returns the total number of points in the result or -1 in case of an error.
 
Int_t TGraph::Merge(TCollection* li)
{
   TIter next(li);
   while (TObject* o = next()) {
      TGraph *g = dynamic_cast<TGraph*>(o);
      if (!g) {
         Error("Merge",
               "Cannot merge - an object which doesn't inherit from TGraph found in the list");
         return -1;
      }
      DoMerge(g);
   }
   return GetN();
}
 
////////////////////////////////////////////////////////////////////////////////
///  protected function to perform the merge operation of a graph
 
Bool_t TGraph::DoMerge(const TGraph* g)
{
   Double_t x = 0, y = 0;
   for (Int_t i = 0 ; i < g->GetN(); i++) {
      g->GetPoint(i, x, y);
      SetPoint(GetN(), x, y);
   }
   return kTRUE;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Move all graph points on specified values dx,dy
/// If log argument specified, calculation done in logarithmic scale like:
///  new_value = exp( log(old_value) + delta );
 
void TGraph::MovePoints(Double_t dx, Double_t dy, Bool_t logx, Bool_t logy)
{
   Double_t x = 0, y = 0;
   for (Int_t i = 0 ; i < GetN(); i++) {
      GetPoint(i, x, y);
      if (!logx) {
         x += dx;
      } else if (x > 0) {
         x = TMath::Exp(TMath::Log(x) + dx);
      }
      if (!logy) {
         y += dy;
      } else if (y > 0) {
         y = TMath::Exp(TMath::Log(y) + dy);
      }
      SetPoint(i, x, y);
   }
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Find zero of a continuous function.
/// This function finds a real zero of the continuous real
/// function Y(X) in a given interval (A,B). See accompanying
/// notes for details of the argument list and calling sequence
 
void TGraph::Zero(Int_t &k, Double_t AZ, Double_t BZ, Double_t E2, Double_t &X, Double_t &Y
                  , Int_t maxiterations)
{
   static Double_t a, b, ya, ytest, y1, x1, h;
   static Int_t j1, it, j3, j2;
   Double_t yb, x2;
   yb = 0;
 
   //       Calculate Y(X) at X=AZ.
   if (k <= 0) {
      a  = AZ;
      b  = BZ;
      X  = a;
      j1 = 1;
      it = 1;
      k  = j1;
      return;
   }
 
   //       Test whether Y(X) is sufficiently small.
 
   if (TMath::Abs(Y) <= E2) {
      k = 2;
      return;
   }
 
   //       Calculate Y(X) at X=BZ.
 
   if (j1 == 1) {
      ya = Y;
      X  = b;
      j1 = 2;
      return;
   }
   //       Test whether the signs of Y(AZ) and Y(BZ) are different.
   //       if not, begin the binary subdivision.
 
   if (j1 != 2) goto L100;
   if (ya * Y < 0) goto L120;
   x1 = a;
   y1 = ya;
   j1 = 3;
   h  = b - a;
   j2 = 1;
   x2 = a + 0.5 * h;
   j3 = 1;
   it++;      //*-*-   Check whether (maxiterations) function values have been calculated.
   if (it >= maxiterations) k = j1;
   else                     X = x2;
   return;
 
   //      Test whether a bracket has been found .
   //      If not,continue the search
 
L100:
   if (j1 > 3) goto L170;
   if (ya*Y >= 0) {
      if (j3 >= j2) {
         h  = 0.5 * h;
         j2 = 2 * j2;
         a  = x1;
         ya = y1;
         x2 = a + 0.5 * h;
         j3 = 1;
      } else {
         a  = X;
         ya = Y;
         x2 = X + h;
         j3++;
      }
      it++;
      if (it >= maxiterations) k = j1;
      else                     X = x2;
      return;
   }
 
   //       The first bracket has been found.calculate the next X by the
   //       secant method based on the bracket.
 
L120:
   b  = X;
   yb = Y;
   j1 = 4;
L130:
   if (TMath::Abs(ya) > TMath::Abs(yb)) {
      x1 = a;
      y1 = ya;
      X  = b;
      Y  = yb;
   } else                                 {
      x1 = b;
      y1 = yb;
      X  = a;
      Y  = ya;
   }
 
   //       Use the secant method based on the function values y1 and Y.
   //       check that x2 is inside the interval (a,b).
 
L150:
   x2    = X - Y * (X - x1) / (Y - y1);
   x1    = X;
   y1    = Y;
   ytest = 0.5 * TMath::Min(TMath::Abs(ya), TMath::Abs(yb));
   if ((x2 - a)*(x2 - b) < 0) {
      it++;
      if (it >= maxiterations) k = j1;
      else                     X = x2;
      return;
   }
 
   //       Calculate the next value of X by bisection . Check whether
   //       the maximum accuracy has been achieved.
 
L160:
   x2    = 0.5 * (a + b);
   ytest = 0;
   if ((x2 - a)*(x2 - b) >= 0) {
      k = 2;
      return;
   }
   it++;
   if (it >= maxiterations) k = j1;
   else                     X = x2;
   return;
 
 
   //       Revise the bracket (a,b).
 
L170:
   if (j1 != 4) return;
   if (ya * Y < 0) {
      b  = X;
      yb = Y;
   } else          {
      a  = X;
      ya = Y;
   }
 
   //       Use ytest to decide the method for the next value of X.
 
   if (ytest <= 0) goto L130;
   if (TMath::Abs(Y) - ytest <= 0) goto L150;
   goto L160;
}