TGraph

 Graph default constructor.

 Constructor with only the number of points set
 the arrsys 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"

 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:

                     (y - f1(x))**2
         -----------------------------------
         ey**2 + (0.5*(exl + exh)*f1'(x))**2

 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.

 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.

 This function is dummy in TGraph, but redefined by TGraphErrors

 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.

 Draw this graph with its current attributes.

   Options to draw a graph are described in TGraph::PaintGraph

 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)
             = "B" Use this option when you want to fix one or more parameters
                   and the fitting function is like "gaus","expo","poln","landau".
             = "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, not calculate the chisquare
                    (saves time)
             = "F" If fitting a polN, switch to minuit fitter
             = "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

   When the fit is drawn (by default), the parameter goption may be used
   to specify a list of graphics options. See TGraph::Paint 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 the errors 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 syntaxis: 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 the fitting function GraphFitChisquare is used.
  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);

  How errors are used in the chisquare function (see TFitter GraphFitChisquare)//   Access to the fit results
   ============================================
 In case of a TGraphErrors object, ex, 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 computed as the sum of the quantity below at each point:

                     (y - f(x))**2
         -----------------------------------
         ey**2 + (0.5*(exl + exh)*f'(x))**2

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

  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 .

 Note, that the linear fitter doesn't take into account the errors in x. If errors
 in x are important, go through minuit (use option "F" for polynomial fitting).

   Associated functions
   ====================
  One or more object (typically a TF1*) can be added to the list
  of functions (fFunctions) associated to 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.

 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

 Least squares lpolynomial 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:  number of points to fit
  first:  first point number to fit
  last:   last point to fit O(ndata should be last-first
  ifail:  return parameter indicating the status of the fit (ifail=0, fit is OK)

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

 Draw this graph with its current attributes.

  Paint "stats" box with the fit info

 Control function to draw a graph.

  Draws one dimensional graphs. The aspect of the graph is done
  according to the value of the chopt.

  Input parameters:

  npoints : Number of points in X or in Y.
  x[npoints] or x[2] : X coordinates or (XMIN,XMAX) (WC space).
  y[npoints] or y[2] : Y coordinates or (YMIN,YMAX) (WC space).
  chopt : Option.

  chopt='L' :  A simple polyline between every points is drawn

  chopt='F' :  A fill area is drawn ('CF' draw a smooth fill area)

  chopt='A' :  Axis are drawn around the graph

  chopt='C' :  A smooth Curve is drawn

  chopt='*' :  A Star is plotted at each point

  chopt='P' :  Idem with the current marker

  chopt='B' :  A Bar chart is drawn at each point

  chopt='1' :  ylow=rwymin

  chopt='X+' : The X-axis is drawn on the top side of the plot.

  chopt='Y+' : The Y-axis is drawn on the right side of the plot.

 When a graph is painted with the option "C" or "L" it is possible to draw
 a filled area on one side of the line. This is useful to show exclusion
 zones. This drawing mode is activated when the absolute value of the
 graph line width (set thanks to SetLineWidth) is greater than 99. In that
 case the line width number is interpreted as 100*ff+ll = ffll . The two
 digits number "ll" represent the normal line width whereas "ff" is the
 filled area width. The sign of "ffll" allows to flip the filled area
 from one side of the line to the other. The current fill area attributes
 are used to draw the hatched zone.

 Control function to draw a graphistogram.

   Draws one dimensional graphs. The aspect of the graph is done
 according to the value of the chopt.

 Input parameters:

  npoints : Number of points in X or in Y.
  X(N) or x[1] : X coordinates or (XMIN,XMAX) (WC space).
  Y(N) or y[1] : Y coordinates or (YMIN,YMAX) (WC space).
  chopt : Option.

  chopt='R' :  Graph is drawn horizontaly, parallel to X axis.
               (default is vertically, parallel to Y axis)
               If option R is selected the user must give:
                 2 values for Y (y[0]=YMIN and y[1]=YMAX)
                 N values for X, one for each channel.
               Otherwise the user must give:
                 N values for Y, one for each channel.
                 2 values for X (x[0]=XMIN and x[1]=XMAX)

  chopt='L' :  A simple polyline beetwen every points is drawn

  chopt='H' :  An Histogram with equidistant bins is drawn
               as a polyline.

  chopt='F' :  An histogram with equidistant bins is drawn
               as a fill area. Contour is not drawn unless
               chopt='H' is also selected..

  chopt='N' :  Non equidistant bins (default is equidistant)
               If N is the number of channels array X and Y
               must be dimensionned as follow:
               If option R is not selected (default) then
               the user must give:
                 (N+1) values for X (limits of channels).
                  N values for Y, one for each channel.
               Otherwise the user must give:
                 (N+1) values for Y (limits of channels).
                  N values for X, one for each channel.

  chopt='F1':  Idem as 'F' except that fill area is no more
               reparted arround axis X=0 or Y=0 .

  chopt='F2':  Draw a Fill area polyline connecting the center of bins

  chopt='C' :  A smooth Curve is drawn.

  chopt='*' :  A Star is plotted at the center of each bin.

  chopt='P' :  Idem with the current marker
  chopt='P0':  Idem with the current marker. Empty bins also drawn

  chopt='B' :  A Bar chart with equidistant bins is drawn as fill
               areas (Contours are drawn).

  chopt='9' :  Force graph to be drawn in high resolution mode.
               By default, the graph is drawn in low resolution
               in case the number of points is greater than the number of pixels
               in the current pad.

  chopt='][' : "Cutoff" style. When this option is selected together with
               H option, the first and last vertical lines of the histogram
               are not drawn.

 Draws a polyline with hatches on one side showing an exclusion
 zone. x and y are the the vectors holding the polyline and n the
 number of points in the polyline and w the width of the hatches.
 w can be negative.
 This method is not meant to be used directly. It is called 
 automatically according to the line style convention.

 Convert WC from Log scales.

   Take the LOG10 of gxwork and gywork according to the value of Options
   and put it in gxworkl and gyworkl.

  npoints : Number of points in gxwork and in gywork.

 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)

 Smooth a curve given by N points.

   Underlaying routine for Draw based on the CERN GD3 routine TVIPTE

     Author - Marlow etc.   Modified by - P. Ward     Date -  3.10.1973

   This routine draws a smooth tangentially continuous curve through
 the sequence of data points P(I) I=1,N where P(I)=(X(I),Y(I))
 the curve is approximated by a polygonal arc of short vectors .
 the data points can represent open curves, P(1) != P(N) or closed
 curves P(2) == P(N) . If a tangential discontinuity at P(I) is
 required , then set P(I)=P(I+1) . loops are also allowed .

 Reference Marlow and Powell,Harwell report No.R.7092.1972
 MCCONALOGUE,Computer Journal VOL.13,NO4,NOV1970Pp392 6

 _Input parameters:

  npoints   : Number of data points.
  x         : Abscissa
  y         : Ordinate


 delta is the accuracy required in constructing the curve.
 if it is zero then the routine calculates a value other-
 wise it uses this value. (default is 0.0)

 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.

 Find zero of a continuous function.

  Underlaying routine for PaintGraph
 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

 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.