class TF1: public TFormula, public TAttLine, public TAttFill, public TAttMarker

 F1 default constructor.

 F1 constructor using a formula definition

  See TFormula constructor for explanation of the formula syntax.

  See tutorials: fillrandom, first, fit1, formula1, multifit
  for real examples.

  Creates a function of type A or B between xmin and xmax

  if formula has the form "fffffff;xxxx;yyyy", it is assumed that
  the formula string is "fffffff" and "xxxx" and "yyyy" are the
  titles for the X and Y axis respectively.

 F1 constructor using name of an interpreted function.

  Creates a function of type C between xmin and xmax.
  name is the name of an interpreted CINT cunction.
  The function is defined with npar parameters
  fcn must be a function of type:
     Double_t fcn(Double_t *x, Double_t *params)

  This constructor is called for functions of type C by CINT.

 WARNING! A function created with this constructor cannot be Cloned.

 F1 constructor using pointer to an interpreted function.

  See TFormula constructor for explanation of the formula syntax.

  Creates a function of type C between xmin and xmax.
  The function is defined with npar parameters
  fcn must be a function of type:
     Double_t fcn(Double_t *x, Double_t *params)

  see tutorial; myfit for an example of use
  also test/stress.cxx (see function stress1)


  This constructor is called for functions of type C by CINT.

  WARNING! A function created with this constructor cannot be Cloned.

 F1 constructor using a pointer to a real function.

   npar is the number of free parameters used by the function

   This constructor creates a function of type C when invoked
   with the normal C++ compiler.

   see test program test/stress.cxx (function stress1) for an example.
   note the interface with an intermediate pointer.

 WARNING! A function created with this constructor cannot be Cloned.

 F1 constructor using a pointer to real function.

   npar is the number of free parameters used by the function

   This constructor creates a function of type C when invoked
   with the normal C++ compiler.

   see test program test/stress.cxx (function stress1) for an example.
   note the interface with an intermediate pointer.

 WARNING! A function created with this constructor cannot be Cloned.

 F1 constructor using the Functor class.

   xmin and xmax define the plotting range of the function
   npar is the number of free parameters used by the function

   This constructor can be used only in compiled code

 WARNING! A function created with this constructor cannot be Cloned.

 Internal Function to Create a TF1  using a Functor.

          Used by the template constructors

 F1 constructor from an interpreted class defining the operator() or Eval().
 This constructor emulate the syntax of the template constructor using a C++ callable object (functor)
 which can be used only in C++ compiled mode.
 The class name is required to get the type of class given the void pointer ptr.
 For the method name is used the operator() (double *, double * ).
 Use the other constructor taking the method name for different method names.

  xmin and xmax specify the function plotting range
  npar are the number of function parameters

  see tutorial  math.exampleFunctor.C for an example of using this constructor

  This constructor is used only when using CINT.
  In compiled mode the template constructor is used and in that case className is not needed

 Internal function used to create from TF1 from an interpreter CINT class
 with the specified type (className) and member function name (methodName).

 Operator =

 TF1 default destructor.

 Static function: set the fgAbsValue flag.
 By default TF1::Integral uses the original function value to compute the integral
 However, TF1::Moment, CentralMoment require to compute the integral
 using the absolute value of the function.

 Browse.

 Copy this F1 to a new F1.

 Returns the first derivative of the function at point x,
 computed by Richardson's extrapolation method (use 2 derivative estimates
 to compute a third, more accurate estimation)
 first, derivatives with steps h and h/2 are computed by central difference formulas


 the final estimate 
  "Numerical Methods for Scientists and Engineers", H.M.Antia, 2nd edition"

 if the argument params is null, the current function parameters are used,
 otherwise the parameters in params are used.

 the argument eps may be specified to control the step size (precision).
 the step size is taken as eps*(xmax-xmin).
 the default value (0.001) should be good enough for the vast majority
 of functions. Give a smaller value if your function has many changes
 of the second derivative in the function range.

 Getting the error via TF1::DerivativeError:
   (total error = roundoff error + interpolation error)
 the estimate of the roundoff error is taken as follows:


 where k is the double precision, ai are coefficients used in
 central difference formulas
 interpolation error is decreased by making the step size h smaller.

 Author: Anna Kreshuk

 Returns the second derivative of the function at point x,
 computed by Richardson's extrapolation method (use 2 derivative estimates
 to compute a third, more accurate estimation)
 first, derivatives with steps h and h/2 are computed by central difference formulas


 the final estimate 
  "Numerical Methods for Scientists and Engineers", H.M.Antia, 2nd edition"

 if the argument params is null, the current function parameters are used,
 otherwise the parameters in params are used.

 the argument eps may be specified to control the step size (precision).
 the step size is taken as eps*(xmax-xmin).
 the default value (0.001) should be good enough for the vast majority
 of functions. Give a smaller value if your function has many changes
 of the second derivative in the function range.

 Getting the error via TF1::DerivativeError:
   (total error = roundoff error + interpolation error)
 the estimate of the roundoff error is taken as follows:


 where k is the double precision, ai are coefficients used in
 central difference formulas
 interpolation error is decreased by making the step size h smaller.

 Author: Anna Kreshuk

 Returns the third derivative of the function at point x,
 computed by Richardson's extrapolation method (use 2 derivative estimates
 to compute a third, more accurate estimation)
 first, derivatives with steps h and h/2 are computed by central difference formulas


 the final estimate 
  "Numerical Methods for Scientists and Engineers", H.M.Antia, 2nd edition"

 if the argument params is null, the current function parameters are used,
 otherwise the parameters in params are used.

 the argument eps may be specified to control the step size (precision).
 the step size is taken as eps*(xmax-xmin).
 the default value (0.001) should be good enough for the vast majority
 of functions. Give a smaller value if your function has many changes
 of the second derivative in the function range.

 Getting the error via TF1::DerivativeError:
   (total error = roundoff error + interpolation error)
 the estimate of the roundoff error is taken as follows:


 where k is the double precision, ai are coefficients used in
 central difference formulas
 interpolation error is decreased by making the step size h smaller.

 Author: Anna Kreshuk

 Static function returning the error of the last call to the Derivative
 functions

 Compute distance from point px,py to a function.

  Compute the closest distance of approach from point px,py to this
  function. The distance is computed in pixels units.

  Note that px is called with a negative value when the TF1 is in
  TGraph or TH1 list of functions. In this case there is no point
  looking at the histogram axis.

 Draw this function with its current attributes.

 Possible option values are:
   "SAME"  superimpose on top of existing picture
   "L"     connect all computed points with a straight line
   "C"     connect all computed points with a smooth curve.
   "FC"    draw a fill area below a smooth curve

 Note that the default value is "L". Therefore to draw on top
 of an existing picture, specify option "LSAME"

 NB. You must use DrawCopy if you want to draw several times the same
     function in the current canvas.

 Draw a copy of this function with its current attributes.

  This function MUST be used instead of Draw when you want to draw
  the same function with different parameters settings in the same canvas.

 Possible option values are:
   "SAME"  superimpose on top of existing picture
   "L"     connect all computed points with a straight line
   "C"     connect all computed points with a smooth curve.
   "FC"    draw a fill area below a smooth curve

 Note that the default value is "L". Therefore to draw on top
 of an existing picture, specify option "LSAME"

 Draw derivative of this function

 An intermediate TGraph object is built and drawn with option.

 The resulting graph will be drawn into the current pad.
 If this function is used via the context menu, it recommended
 to create a new canvas/pad before invoking this function.

 Draw integral of this function

 An intermediate TGraph object is built and drawn with option.

 The resulting graph will be drawn into the current pad.
 If this function is used via the context menu, it recommended
 to create a new canvas/pad before invoking this function.

 Draw formula between xmin and xmax.

 Evaluate this formula.

   Computes the value of this function (general case for a 3-d function)
   at point x,y,z.
   For a 1-d function give y=0 and z=0
   The current value of variables x,y,z is passed through x, y and z.
   The parameters used will be the ones in the array params if params is given
    otherwise parameters will be taken from the stored data members fParams

 Evaluate function with given coordinates and parameters.

 Compute the value of this function at point defined by array x
 and current values of parameters in array params.
 If argument params is omitted or equal 0, the internal values
 of parameters (array fParams) will be used instead.
 For a 1-D function only x[0] must be given.
 In case of a multi-dimemsional function, the arrays x must be
 filled with the corresponding number of dimensions.

 WARNING. In case of an interpreted function (fType=2), it is the
 user's responsability to initialize the parameters via InitArgs
 before calling this function.
 InitArgs should be called at least once to specify the addresses
 of the arguments x and params.
 InitArgs should be called everytime these addresses change.

 Execute action corresponding to one event.

  This member function is called when a F1 is clicked with the locator

 Fix the value of a parameter
 The specified value will be used in a fit operation

 Static function returning the current function being processed

 Return a pointer to the histogram used to vusualize the function

 Return the maximum value of the function
 Method:
  First, the grid search is used to bracket the maximum
  with the step size = (xmax-xmin)/fNpx.
  This way, the step size can be controlled via the SetNpx() function.
  If the function is unimodal or if its extrema are far apart, setting
  the fNpx to a small value speeds the algorithm up many times.
  Then, Brent's method is applied on the bracketed interval

 Return the X value corresponding to the maximum value of the function
 Method:
  First, the grid search is used to bracket the maximum
  with the step size = (xmax-xmin)/fNpx.
  This way, the step size can be controlled via the SetNpx() function.
  If the function is unimodal or if its extrema are far apart, setting
  the fNpx to a small value speeds the algorithm up many times.
  Then, Brent's method is applied on the bracketed interval

 Returns the minimum value of the function on the (xmin, xmax) interval
 Method:
  First, the grid search is used to bracket the maximum
  with the step size = (xmax-xmin)/fNpx. This way, the step size
  can be controlled via the SetNpx() function. If the function is
  unimodal or if its extrema are far apart, setting the fNpx to
  a small value speeds the algorithm up many times.
  Then, Brent's method is applied on the bracketed interval

 Returns the X value corresponding to the minimum value of the function
 on the (xmin, xmax) interval
 Method:
  First, the grid search is used to bracket the maximum
  with the step size = (xmax-xmin)/fNpx. This way, the step size
  can be controlled via the SetNpx() function. If the function is
  unimodal or if its extrema are far apart, setting the fNpx to
  a small value speeds the algorithm up many times.
  Then, Brent's method is applied on the bracketed interval

 Returns the X value corresponding to the function value fy for (xmin<x<xmax).
 Method:
  First, the grid search is used to bracket the maximum
  with the step size = (xmax-xmin)/fNpx. This way, the step size
  can be controlled via the SetNpx() function. If the function is
  unimodal or if its extrema are far apart, setting the fNpx to
  a small value speeds the algorithm up many times.
  Then, Brent's method is applied on the bracketed interval

 Return the number of degrees of freedom in the fit
 the fNDF parameter has been previously computed during a fit.
 The number of degrees of freedom corresponds to the number of points
 used in the fit minus the number of free parameters.

 Return the number of free parameters

 Redefines TObject::GetObjectInfo.
 Displays the function info (x, function value)
 corresponding to cursor position px,py

 Return value of parameter number ipar

 Return limits for parameter ipar.

 Return the fit probability

  Compute Quantiles for density distribution of this function
     Quantile x_q of a probability distribution Function F is defined as


     For instance the median  of a distribution is defined as that value
     of the random variable for which the distribution function equals 0.5:


  code from Eddy Offermann, Renaissance

 input parameters
   - this TF1 function
   - nprobSum maximum size of array q and size of array probSum
   - probSum array of positions where quantiles will be computed.
     It is assumed to contain at least nprobSum values.
  output
   - return value nq (<=nprobSum) with the number of quantiles computed
   - array q filled with nq quantiles

  Getting quantiles from two histograms and storing results in a TGraph,
   a so-called QQ-plot

     TGraph *gr = new TGraph(nprob);
     f1->GetQuantiles(nprob,gr->GetX());
     f2->GetQuantiles(nprob,gr->GetY());
     gr->Draw("alp");

 Return a random number following this function shape

   The distribution contained in the function fname (TF1) is integrated
   over the channel contents.
   It is normalized to 1.
   For each bin the integral is approximated by a parabola.
   The parabola coefficients are stored as non persistent data members
   Getting one random number implies:
     - Generating a random number between 0 and 1 (say r1)
     - Look in which bin in the normalized integral r1 corresponds to
     - Evaluate the parabolic curve in the selected bin to find
       the corresponding X value.
   The parabolic approximation is very good as soon as the number
   of bins is greater than 50.

 Return a random number following this function shape in [xmin,xmax]

   The distribution contained in the function fname (TF1) is integrated
   over the channel contents.
   It is normalized to 1.
   For each bin the integral is approximated by a parabola.
   The parabola coefficients are stored as non persistent data members
   Getting one random number implies:
     - Generating a random number between 0 and 1 (say r1)
     - Look in which bin in the normalized integral r1 corresponds to
     - Evaluate the parabolic curve in the selected bin to find
       the corresponding X value.
   The parabolic approximation is very good as soon as the number
   of bins is greater than 50.

  IMPORTANT NOTE
  The integral of the function is computed at fNpx points. If the function
  has sharp peaks, you should increase the number of points (SetNpx)
  such that the peak is correctly tabulated at several points.

 Return range of a 1-D function.

 Return range of a 2-D function.

 Return range of function.

 Get value corresponding to X in array of fSave values

 Get x axis of the function.

 Get y axis of the function.

 Get z axis of the function. (In case this object is a TF2 or TF3)

 Compute the gradient (derivative) wrt a parameter ipar
 Parameters:
 ipar - index of parameter for which the derivative is computed
 x - point, where the derivative is computed
 eps - if the errors of parameters have been computed, the step used in
 numerical differentiation is eps*parameter_error.
 if the errors have not been computed, step=eps is used
 default value of eps = 0.01
 Method is the same as in Derivative() function

 If a paramter is fixed, the gradient on this parameter = 0

 Compute the gradient wrt parameters
 Parameters:
 x - point, were the gradient is computed
 grad - used to return the computed gradient, assumed to be of at least fNpar size
 eps - if the errors of parameters have been computed, the step used in
 numerical differentiation is eps*parameter_error.
 if the errors have not been computed, step=eps is used
 default value of eps = 0.01
 Method is the same as in Derivative() function

 If a paramter is fixed, the gradient on this parameter = 0

 Initialize parameters addresses.

 Create the basic function objects

 Return Integral of function between a and b.

   based on original CERNLIB routine DGAUSS by Sigfried Kolbig
   converted to C++ by Rene Brun

 This function computes, to an attempted specified accuracy, the value
 of the integral.


 Usage:
   In any arithmetic expression, this function has the approximate value
   of the integral I.
   - A, B: End-points of integration interval. Note that B may be less
           than A.
   - params: Array of function parameters. If 0, use current parameters.
   - epsilon: Accuracy parameter (see Accuracy).

Method:
   For any interval [a,b] we define g8(a,b) and g16(a,b) to be the 8-point
   and 16-point Gaussian quadrature approximations to


   and define


   Then,


   where, starting with x0 = A and finishing with xk = B,
   the subdivision points xi(i=1,2,...) are given by


    is equal to the first member of the
   sequence 1,1/2,1/4,... for which r(xi-1, xi) < EPS.
   If, at any stage in the process of subdivision, the ratio


   is so small that 1+0.005q is indistinguishable from 1 to
   machine accuracy, an error exit occurs with the function value
   set equal to zero.

 Accuracy:
   Unless there is severe cancellation of positive and negative values of
   f(x) over the interval [A,B], the argument EPS may be considered as
   specifying a bound on the <I>relative</I> error of I in the case
   |I|&gt;1, and a bound on the absolute error in the case |I|&lt;1. More
   precisely, if k is the number of sub-intervals contributing to the
   approximation (see Method), and if


   then the relation


   will nearly always be true, provided the routine terminates without
   printing an error message. For functions f having no singularities in
   the closed interval [A,B] the accuracy will usually be much higher than
   this.

 Error handling:
   The requested accuracy cannot be obtained (see Method).
   The function value is set equal to zero.

 Note 1:
   Values of the function f(x) at the interval end-points A and B are not
   required. The subprogram may therefore be used when these values are
   undefined.

 Note 2:
   Instead of TF1::Integral, you may want to use the combination of
   TF1::CalcGaussLegendreSamplingPoints and TF1::IntegralFast.
   See an example with the following script:

   void gint() {
      TF1 *g = new TF1("g","gaus",-5,5);
      g->SetParameters(1,0,1);
      //default gaus integration method uses 6 points
      //not suitable to integrate on a large domain
      double r1 = g->Integral(0,5);
      double r2 = g->Integral(0,1000);

      //try with user directives computing more points
      Int_t np = 1000;
      double *x=new double[np];
      double *w=new double[np];
      g->CalcGaussLegendreSamplingPoints(np,x,w,1e-15);
      double r3 = g->IntegralFast(np,x,w,0,5);
      double r4 = g->IntegralFast(np,x,w,0,1000);
      double r5 = g->IntegralFast(np,x,w,0,10000);
      double r6 = g->IntegralFast(np,x,w,0,100000);
      printf("g->Integral(0,5)               = %g\n",r1);
      printf("g->Integral(0,1000)            = %g\n",r2);
      printf("g->IntegralFast(n,x,w,0,5)     = %g\n",r3);
      printf("g->IntegralFast(n,x,w,0,1000)  = %g\n",r4);
      printf("g->IntegralFast(n,x,w,0,10000) = %g\n",r5);
      printf("g->IntegralFast(n,x,w,0,100000)= %g\n",r6);
      delete [] x;
      delete [] w;
   }

   This example produces the following results:

      g->Integral(0,5)               = 1.25331
      g->Integral(0,1000)            = 1.25319
      g->IntegralFast(n,x,w,0,5)     = 1.25331
      g->IntegralFast(n,x,w,0,1000)  = 1.25331
      g->IntegralFast(n,x,w,0,10000) = 1.25331
      g->IntegralFast(n,x,w,0,100000)= 1.253

 Return Integral of a 2d function in range [ax,bx],[ay,by]

 Return Integral of a 3d function in range [ax,bx],[ay,by],[az,bz]

 Return Error on Integral of a parameteric function between a and b due to the parameters uncertainties
 It is assumed the parameters are estimated from a fit and the covariance matrix resulting from the fit is used in
 estimating this error.
 IMPORTANT NOTE: The calculation is valid assuming the parameters are resulting from the latest fit. If in the meantime
 a fit is done using another function, the routine will signal  an error and return zero.

 Gauss-Legendre integral, see CalcGaussLegendreSamplingPoints

  See more general prototype below.
  This interface kept for back compatibility

  Adaptive Quadrature for Multiple Integrals over N-Dimensional
  Rectangular Regions




 Author(s): A.C. Genz, A.A. Malik
 converted/adapted by R.Brun to C++ from Fortran CERNLIB routine RADMUL (D120)
 The new code features many changes compared to the Fortran version.
 Note that this function is currently called only by TF2::Integral (n=2)
 and TF3::Integral (n=3).

 This function computes, to an attempted specified accuracy, the value of
 the integral over an n-dimensional rectangular region.

 Input parameters:

    n     : Number of dimensions [2,15]
    a,b   : One-dimensional arrays of length >= N . On entry A[i],  and  B[i],
            contain the lower and upper limits of integration, respectively.
    minpts: Minimum number of function evaluations requested. Must not exceed maxpts.
            if minpts < 1 minpts is set to 2^n +2*n*(n+1) +1
    maxpts: Maximum number of function evaluations to be allowed.
            maxpts >= 2^n +2*n*(n+1) +1
            if maxpts<minpts, maxpts is set to 10*minpts
    eps   : Specified relative accuracy.

 Output parameters:

    relerr : Contains, on exit, an estimation of the relative accuracy of the result.
    nfnevl : number of function evaluations performed.
    ifail  :
        0 Normal exit.  . At least minpts and at most maxpts calls to the function were performed.
        1 maxpts is too small for the specified accuracy eps.
          The result and relerr contain the values obtainable for the
          specified value of maxpts.
        3 n<2 or n>15

 Method:

    An integration rule of degree seven is used together with a certain
    strategy of subdivision.
    For a more detailed description of the method see References.

 Notes:

   1.Multi-dimensional integration is time-consuming. For each rectangular
     subregion, the routine requires function evaluations.
     Careful programming of the integrand might result in substantial saving
     of time.
   2.Numerical integration usually works best for smooth functions.
     Some analysis or suitable transformations of the integral prior to
     numerical work may contribute to numerical efficiency.

 References:

   1.A.C. Genz and A.A. Malik, Remarks on algorithm 006:
     An adaptive algorithm for numerical integration over
     an N-dimensional rectangular region, J. Comput. Appl. Math. 6 (1980) 295-302.
   2.A. van Doren and L. de Ridder, An adaptive algorithm for numerical
     integration over an n-dimensional cube, J.Comput. Appl. Math. 2 (1976) 207-217.

 Return kTRUE is the point is inside the function range

 Paint this function with its current attributes.

 Dump this function with its attributes.

 Release parameter number ipar If used in a fit, the parameter
 can vary freely. The parameter limits are reset to 0,0.

 Save values of function in array fSave

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

 Static function setting the current function.
 the current function may be accessed in static C-like functions
 when fitting or painting a function.

 Set the maximum value along Y for this function
 In case the function is already drawn, set also the maximum in the
 helper histogram

 Set the minimum value along Y for this function
 In case the function is already drawn, set also the minimum in the
 helper histogram

 Set the number of degrees of freedom
 ndf should be the number of points used in a fit - the number of free parameters

 Set the number of points used to draw the function

 The default number of points along x is 100 for 1-d functions and 30 for 2-d/3-d functions
 You can increase this value to get a better resolution when drawing
 pictures with sharp peaks or to get a better result when using TF1::GetRandom
 the minimum number of points is 4, the maximum is 100000 for 1-d and 10000 for 2-d/3-d functions

 Set error for parameter number ipar

 Set errors for all active parameters
 when calling this function, the array errors must have at least fNpar values

 Set limits for parameter ipar.

 The specified limits will be used in a fit operation
 when the option "B" is specified (Bounds).
 To fix a parameter, use TF1::FixParameter

 Initialize the upper and lower bounds to draw the function.

 The function range is also used in an histogram fit operation
 when the option "R" is specified.

 Restore value of function saved at point

 Set function title
  if title has the form "fffffff;xxxx;yyyy", it is assumed that
  the function title is "fffffff" and "xxxx" and "yyyy" are the
  titles for the X and Y axis respectively.

 Stream a class object.

 Called by functions such as SetRange, SetNpx, SetParameters
 to force the deletion of the associated histogram or Integral

 Static function to set the global flag to reject points
 the fgRejectPoint global flag is tested by all fit functions
 if TRUE the point is not included in the fit.
 This flag can be set by a user in a fitting function.
 The fgRejectPoint flag is reset by the TH1 and TGraph fitting functions.

 See TF1::RejectPoint above

 Return nth moment of function between a and b

 See TF1::Integral() for parameter definitions
   Author: Gene Van Buren <gene@bnl.gov>

 Return nth central moment of function between a and b

 See TF1::Integral() for parameter definitions
   Author: Gene Van Buren <gene@bnl.gov>

 Type safe interface (static method)
 The number of sampling points are taken from the TGraph

 for using TF1 as a callable object (functor)