class TGraph: public TNamed, public TAttLine, public TAttFill, public TAttMarker

 Graph default constructor.

 Constructor with only the number of points set
 the arrays x and y will be set later

 Graph normal constructor with ints.

 Graph normal constructor with floats.

 Graph normal constructor with doubles.

 Copy constructor for this graph

 Equal operator for this graph

 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.

 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.

 Graph constructor importing its parameters from the TH1 object passed as argument

 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.

 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 instanciation is about 2 times slower.

 Graph default destructor.

 Allocate arrays.

 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

 Browse

 Return the chisquare of this graph with respect to f1.
 The chisquare is computed as the sum of the quantity below at each point:


 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

 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.

 Return kTRUE if fX[left] > fX[right]. Can be used by Sort.

 Return kTRUE if fY[left] > fY[right]. Can be used by Sort.

 Return kTRUE if point number "left"'s distance to origin is bigger than
 that of point number "right". Can be used by Sort.

 Compute the x/y range of the points in this graph

 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

 Copy points from fX and fY to arrays[0] and arrays[1]
 or to fX and fY if arrays == 0 and ibegin != iend.

 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

The options to draw a graph are described in TGraphPainter class.

 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.

 Draw this graph with new attributes.

 Draw this graph with new attributes.

 Draw this graph with new attributes.

 Display a panel with all graph drawing options.

 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.

 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.

 If array sizes <= newsize, expand storage to 2*newsize.

 If graph capacity is less than newsize points then make array sizes
 equal to least multiple of step to contain newsize points.
 Returns kTRUE if size was altered

 if size > fMaxSize allocate new arrays of 2*size points
  and copy oend first points.
 Return pointer to new arrays.

 Set zero values for point arrays in the range [begin, end)
 Should be redefined in descendant classes

 Search object named name in the list of functions

 Search object obj in the list of functions

 Fit this graph with function with name fname.

  interface to TGraph::Fit(TF1 *f1...

      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.
      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)"

 Fit this graph with function f1.

   f1 is an already predefined function created by TF1.
   Predefined functions such as gaus, expo and poln are automatically
   created by ROOT.

   The list of fit options is given in parameter option.
      option = "W" Set all weights to 1; ignore error bars
             = "U" Use a User specified fitting algorithm (via SetFCN)
             = "Q" Quiet mode (minimum printing)
             = "V" Verbose mode (default is between Q and V)
             = "E"  Perform better Errors estimation using Minos technique
             = "B"  User defined parameter settings are used for predefined functions
                    like "gaus", "expo", "poln", "landau".
                    Use this option when you want to fix one or more parameters for these functions.
             = "M"  More. Improve fit results.
                    It uses the IMPROVE command of TMinuit (see TMinuit::mnimpr)
                    This algorithm attempts to improve the found local minimum by
                    searching for a better one.
             = "R" Use the Range specified in the function range
             = "N" Do not store the graphics function, do not draw
             = "0" Do not plot the result of the fit. By default the fitted function
                   is drawn unless the option "N" above is specified.
             = "+" Add this new fitted function to the list of fitted functions
                   (by default, any previous function is deleted)
             = "C" In case of linear fitting, do not calculate the chisquare
                    (saves time)
             = "F" If fitting a polN, use the minuit fitter
             = "EX0" When fitting a TGraphErrors or TGraphAsymErrors do not consider errors in the coordinate
             = "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
             = "S"  The result of the fit is returned in the TFitResultPtr
                     (see below Access to the Fit Result)

   When the fit is drawn (by default), the parameter goption may be used
   to specify a list of graphics options. See TGraphPainter for a complete
   list of these options.

   In order to use the Range option, one must first create a function
   with the expression to be fitted. For example, if your graph
   has a defined range between -4 and 4 and you want to fit a gaussian
   only in the interval 1 to 3, you can do:
        TF1 *f1 = new TF1("f1","gaus",1,3);
        graph->Fit("f1","R");


 Who is calling this function:

   Note that this function is called when calling TGraphErrors::Fit
   or TGraphAsymmErrors::Fit ot TGraphBentErrors::Fit
   See the discussion below on error calulation.

 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 "[0]*x + [1]*sin(x)", create a
    TF1 *f1=new TF1("f1", "x++sin(x)", xmin, xmax);
   For such a TF1 you don't have to set the initial conditions.
   Going via the linear fitter for functions, linear in parameters, gives a
   considerable advantage in speed.

 Setting initial conditions:


   Parameters must be initialized before invoking the Fit function.
   The setting of the parameter initial values is automatic for the
   predefined functions : poln, expo, gaus, landau. One can however disable
   this automatic computation by specifying the option "B".
   You can specify boundary limits for some or all parameters via
        f1->SetParLimits(p_number, parmin, parmax);
   If parmin>=parmax, the parameter is fixed
   Note that you are not forced to fix the limits for all parameters.
   For example, if you fit a function with 6 parameters, you can do:
     func->SetParameters(0,3.1,1.e-6,0.1,-8,100);
     func->SetParLimits(4,-10,-4);
     func->SetParLimits(5, 1,1);
   With this setup, parameters 0->3 can vary freely.
   Parameter 4 has boundaries [-10,-4] with initial value -8.
   Parameter 5 is fixed to 100.

 Fit range:


   The fit range can be specified in two ways:
     - specify rxmax > rxmin (default is rxmin=rxmax=0)
     - specify the option "R". In this case, the function will be taken
       instead of the full graph range.

 Changing the fitting function:


   By default a chi2 fitting function is used for fitting a TGraph.
   The function is implemented in FitUtil::EvaluateChi2.
   In case of TGraphErrors an effective chi2 is used (see below TGraphErrors fit)
   To specify a User defined fitting function, specify option "U" and
   call the following functions:
     TVirtualFitter::Fitter(mygraph)->SetFCN(MyFittingFunction)
   where MyFittingFunction is of type:
   extern void MyFittingFunction(Int_t &npar, Double_t *gin, Double_t &f,
                                 Double_t *u, Int_t flag);


 TGraphErrors fit:


   In case of a TGraphErrors object, when x errors are present, the error along x,
   is projected along the y-direction by calculating the function at the points x-exlow and
   x+exhigh. The chisquare is then computed as the sum of the quantity below at each point:




   where x and y are the point coordinates, and f'(x) is the derivative of the
   function f(x).

   In case the function lies below (above) the data point, ey is ey_low (ey_high).

   thanks to Andy Haas (haas@yahoo.com) for adding the case with TGraphAsymmErrors
             University of Washington

   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.
   The improvement, compared to the first method (f(x+ exhigh) - f(x-exlow))/2
   is of (error of x)**2 order. This approach is called "effective variance method".
   This improvement has been made in version 4.00/08 by Anna Kreshuk.
   The implementation is provided in the function FitUtil::EvaluateChi2Effective

 NOTE:
   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 willnot be considere.
        If errors in x are important, go through minuit (use option "F" for polynomial 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:
      errorp *= sqrt(chisquare/(ndf-1))

   Access to the fit result

  The function returns a TFitResultPtr which can hold a  pointer to a TFitResult object.
  By default the TFitResultPtr contains only the status of the fit which is return by an
  automatic conversion of the TFitResultPtr to an integer. One can write in this case
  directly:
  Int_t fitStatus =  h->Fit(myFunc)

  If the option "S" is instead used, TFitResultPtr contains the TFitResult and behaves
  as a smart pointer to it. For example one can do:
  TFitResultPtr r = h->Fit(myFunc,"S");
  TMatrixDSym cov = r->GetCovarianceMatrix();  //  to access the covariance matrix
  Double_t chi2   = r->Chi2(); // to retrieve the fit chi2
  Double_t par0   = r->Value(0); // retrieve the value for the parameter 0
  Double_t err0   = r->ParError(0); // retrieve the error for the parameter 0
  r->Print("V");     // print full information of fit including covariance matrix
  r->Write();        // store the result in a file

  The fit parameters, error and chi2 (but not covariance matrix) can be retrieved also
  from the fitted function.
  If the histogram is made persistent, the list of
  associated functions is also persistent. Given a pointer (see above)
  to an associated function myfunc, one can retrieve the function/fit
  parameters with calls such as:
    Double_t chi2 = myfunc->GetChisquare();
    Double_t par0 = myfunc->GetParameter(0); //value of 1st parameter
    Double_t err0 = myfunc->GetParError(0);  //error on first parameter


  Access to the fit status

  The status of the fit can be obtained converting the TFitResultPtr to an integer
  indipendently if the fit option "S" is used or not:
  TFitResultPtr r = h->Fit(myFunc,opt);
  Int_t fitStatus = r;

  The fitStatus is 0 if the fit is OK (i.e. no error occurred).
  The value of the fit status code is negative in case of an error not connected with the
  minimization procedure, for example when a wrong function is used.
  Otherwise the return value is the one returned from the minimization procedure.
  When TMinuit (default case) or Minuit2 are used as minimizer the status returned is :
  fitStatus =  migradResult + 10*minosResult + 100*hesseResult + 1000*improveResult.
  TMinuit will return 0 (for migrad, minos, hesse or improve) in case of success and 4 in
  case of error (see the documentation of TMinuit::mnexcm). So for example, for an error
  only in Minos but not in Migrad a fitStatus of 40 will be returned.
  Minuit2 will return also 0 in case of success and different values in migrad, minos or
  hesse depending on the error.   See in this case the documentation of
  Minuit2Minimizer::Minimize for the migradResult, Minuit2Minimizer::GetMinosError for the
  minosResult and Minuit2Minimizer::Hesse for the hesseResult.
  If other minimizers are used see their specific documentation for the status code
  returned. For example in the case of Fumili, for the status returned see TFumili::Minimize.

 Associated functions:


   One or more object (typically a TF1*) can be added to the list
   of functions (fFunctions) associated with each graph.
   When TGraph::Fit is invoked, the fitted function is added to this list.
   Given a graph gr, one can retrieve an associated function
   with:  TF1 *myfunc = gr->GetFunction("myfunc");

   If the graph is made persistent, the list of associated functions is also
   persistent. Given a pointer (see above) to an associated function myfunc,
   one can retrieve the function/fit parameters with calls such as:
     Double_t chi2 = myfunc->GetChisquare();
     Double_t par0 = myfunc->GetParameter(0); //value of 1st parameter
     Double_t err0 = myfunc->GetParError(0);  //error on first parameter

 Fit Statistics


   You can change the statistics box to display the fit parameters with
   the TStyle::SetOptFit(mode) method. This mode has four digits.
   mode = pcev  (default = 0111)
     v = 1;  print name/values of parameters
     e = 1;  print errors (if e=1, v must be 1)
     c = 1;  print Chisquare/Number of degress of freedom
     p = 1;  print Probability

   For example: gStyle->SetOptFit(1011);
   prints the fit probability, parameter names/values, and errors.
   You can change the position of the statistics box with these lines
   (where g is a pointer to the TGraph):

   Root > TPaveStats *st = (TPaveStats*)g->GetListOfFunctions()->FindObject("stats")
   Root > st->SetX1NDC(newx1); //new x start position
   Root > st->SetX2NDC(newx2); //new x end position

 Display a GUI panel with all graph fit options.

   See class TFitEditor for example

 Return graph correlation factor

 Return covariance of vectors x,y

 Return mean value of X (axis=1)  or Y (axis=2)

 Return RMS of X (axis=1)  or Y (axis=2)

 This function is called by GraphFitChisquare.
 It always returns a negative value. Real implementation in TGraphErrors

 This function is called by GraphFitChisquare.
 It always returns a negative value. Real implementation in TGraphErrors

 This function is called by GraphFitChisquare.
 It always returns a negative value. Real implementation in TGraphErrors
 and TGraphAsymmErrors

 This function is called by GraphFitChisquare.
 It always returns a negative value. Real implementation in TGraphErrors
 and TGraphAsymmErrors

 This function is called by GraphFitChisquare.
 It always returns a negative value. Real implementation in TGraphErrors
 and TGraphAsymmErrors

 This function is called by GraphFitChisquare.
 It always returns a negative value. Real implementation in TGraphErrors
 and TGraphAsymmErrors

 Return pointer to function with name.

 Functions such as TGraph::Fit store the fitted function in the list of
 functions of this graph.

 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

 Get x and y values for point number i.
 The function returns -1 in case of an invalid request or the point number otherwise

 Get x axis of the graph.

 Get y axis of the graph.

 Compute Initial values of parameters for a gaussian.

 Compute Initial values of parameters for an exponential.

 Compute Initial values of parameters for a polynom.

 Insert a new point at the mouse position

 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

 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.

 Least squares polynomial fitting without weights.

  m     number of parameters
  a     array of parameters
  first 1st point number to fit (default =0)
  last  last point number to fit (default=fNpoints-1)

   based on CERNLIB routine LSQ: Translated to C++ by Rene Brun

 Least square linear fit without weights.

  Fit a straight line (a0 + a1*x) to the data in this graph.
  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.
  a0:     constant
  a1:     slope
  ifail:  return parameter indicating the status of the fit (ifail=0, fit is OK)
  xmin, xmax: fitting range

  extracted from CERNLIB LLSQ: Translated to C++ by Rene Brun

 Draw this graph with its current attributes.

 Draw the (x,y) as a graph.

 Draw the (x,y) as a histogram.

 Draw the stats

 Print graph values.

 Recursively remove object from the list of functions

 Delete point close to the mouse position

 Delete point number ipoint

 Save primitive as a C++ statement(s) on output stream out

 Set number of points in the graph
 Existing coordinates are preserved
 New coordinates above fNpoints are preset to 0.

 Return kTRUE if kNotEditable bit is not set, kFALSE otherwise.

 if editable=kFALSE, the graph cannot be modified with the mouse
  by default a TGraph is editable

 Set the maximum of the graph.

 Set the minimum of the graph.

 Set x and y values for point number i.

 Set graph title.

 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)

 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:
   // 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);

 Stream an object of class TGraph.

 Swap points.

 Swap values.

 Set current style settings in this graph
 This function is called when either TCanvas::UseCurrentStyle
 or TROOT::ForceStyle have been invoked.

 Adds all graphs from the collection to this graph.
 Returns the total number of poins in the result or -1 in case of an error.

  protected function to perform the merge operation of a graph

 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