TF1

 a TF1 object is a 1-Dim function defined between a lower and upper limit.
 The function may be a simple function (see TFormula) or a precompiled
 user function.
 The function may have associated parameters.
 TF1 graphics function is via the TH1/TGraph drawing functions.

  The following types of functions can be created:
    A- Expression using variable x and no parameters
    B- Expression using variable x with parameters
    C- A general C function with parameters

         +++++++++++++++++++++++++++++++++++
 ===>    + Example of a function of type A +
         +++++++++++++++++++++++++++++++++++

  Case A1 (inline expression using standard C++ functions/operators)
  ------------------------------------------------------------------
   TF1 *fa1 = new TF1("fa1","sin(x)/x",0,10);
   fa1->Draw();

  Case A2 (inline expression using TMath functions without parameters)
  --------------------------------------------------------------------
   TF1 *fa2 = new TF1("fa2","TMath::DiLog(x)",0,10);
   fa2->Draw();

  Case A3 (inline expression using a CINT function by name
  --------------------------------------------------------
   Double_t myFunc(x) {
      return x+sin(x);
   }
   TF1 *fa3 = new TF1("fa4","myFunc(x)",-3,5);
   fa3->Draw();


         +++++++++++++++++++++++++++++++++++
 ===>    + Example of a function of type B+
         +++++++++++++++++++++++++++++++++++

  Case B1 (inline expression using standard C++ functions/operators)
  ------------------------------------------------------------------
  Example B1a
  -----------
   TF1 *fa = new TF1("fa","[0]*x*sin([1]*x)",-3,3);
    This creates a function of variable x with 2 parameters.
    The parameters must be initialized via:
      fa->SetParameter(0,value_first_parameter);
      fa->SetParameter(1,value_second_parameter);
    Parameters may be given a name:
      fa->SetParName(0,"Constant");

  Example B1b
  -----------
   TF1 *fb = new TF1("fb","gaus(0)*expo(3)",0,10);
     gaus(0) is a substitute for [0]*exp(-0.5*((x-[1])/[2])**2)
        and (0) means start numbering parameters at 0
     expo(3) is a substitute for exp([3]+[4]*x)

  Case B2 (inline expression using TMath functions with parameters)
  --------------------------------------------------------------------
   TF1 *fb2 = new TF1("fa3","TMath::Landau(x,[0],[1],0)",-5,10);
   fb2->SetParameters(0.2,1.3);
   fb2->Draw();


         +++++++++++++++++++++++++++++++++++
 ===>    + Example of a function of type C+
         +++++++++++++++++++++++++++++++++++

   Consider the macro myfunc.C below
-------------macro myfunc.C-----------------------------
Double_t myfunction(Double_t *x, Double_t *par)
{
   Float_t xx =x[0];
   Double_t f = TMath::Abs(par[0]*sin(par[1]*xx)/xx);
   return f;
}
void myfunc()
{
   TF1 *f1 = new TF1("myfunc",myfunction,0,10,2);
   f1->SetParameters(2,1);
   f1->SetParNames("constant","coefficient");
   f1->Draw();
}
void myfit()
{
   TH1F *h1=new TH1F("h1","test",100,0,10);
   h1->FillRandom("myfunc",20000);
   TF1 *f1=gROOT->GetFunction("myfunc");
   f1->SetParameters(800,1);
   h1.Fit("myfunc");
}
--------end of macro myfunc.C---------------------------------

 In an interactive session you can do:
   Root > .L myfunc.C
   Root > myfunc();
   Root > myfit();


  TF1 objects can reference other TF1 objects (thanks John Odonnell)
  of type A or B defined above.This excludes CINT interpreted functions
  and compiled functions.
  However, there is a restriction. A function cannot reference a basic
  function if the basic function is a polynomial polN.
  Example:
{
  TF1 *fcos = new TF1 ("fcos", "[0]*cos(x)", 0., 10.);
  fcos->SetParNames( "cos");
  fcos->SetParameter( 0, 1.1);

  TF1 *fsin = new TF1 ("fsin", "[0]*sin(x)", 0., 10.);
  fsin->SetParNames( "sin");
  fsin->SetParameter( 0, 2.1);

  TF1 *fsincos = new TF1 ("fsc", "fcos+fsin");

  TF1 *fs2 = new TF1 ("fs2", "fsc+fsc");
}

     WHY TF1 CANNOT ACCEPT A CLASS MEMBER FUNCTION ?
     ===============================================
 This is a frequently asked question.
 C++ is a strongly typed language. There is no way for TF1 (without
 recompiling this class) to know about all possible user defined data types.
 This also apply to the case of a static class function.

------------------------------------------------------------------------

TF1()

*-*-*-*-*-*-*-*-*-*-*F1 default constructor*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
*-*                  ======================

TF1(const char *name,const char *formula, Double_t xmin, Double_t xmax)

 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.

TF1(const char *name, Double_t xmin, Double_t xmax, Int_t npar)

*-*-*-*-*-*-*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.
*-*
*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*

TF1(const char *name,void *fcn, Double_t xmin, Double_t xmax, Int_t npar)

*-*-*-*-*-*-*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.
*-*
*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*

TF1(const char *name,Double_t (*fcn)(Double_t *, Double_t *), Double_t xmin, Double_t xmax, Int_t npar)

*-*-*-*-*-*-*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.
*-*
*-*
*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*

TF1& operator=(const TF1 &rhs)

 Operator =

~TF1()

*-*-*-*-*-*-*-*-*-*-*F1 default destructor*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
*-*                  =====================

void AbsValue(Bool_t flag)

 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.

void Browse(TBrowser *b)

 Browse.

void Copy(TObject &obj)

*-*-*-*-*-*-*-*-*-*-*Copy this F1 to a new F1*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
*-*                  ========================

Double_t Derivative(Double_t x, Double_t *params, Double_t eps)

 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
 D(h) = (f(x+h) - f(x-h))/2h
 the final estimate D = (4*D(h/2) - D(h))/3
  "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:
    err = k*Sqrt(f(x)*f(x) + x*x*deriv*deriv)*Sqrt(Sum(ai)*(ai)),
 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

Double_t Derivative2(Double_t x, Double_t *params, Double_t eps)

 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
    D(h) = (f(x+h) - 2*f(x) + f(x-h))/(h*h)
 the final estimate D = (4*D(h/2) - D(h))/3
  "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:
    err = k*Sqrt(f(x)*f(x) + x*x*deriv*deriv)*Sqrt(Sum(ai)*(ai)),
 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

Double_t Derivative3(Double_t x, Double_t *params, Double_t eps)

 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
    D(h) = (f(x+2h) - 2*f(x+h) + 2*f(x-h) - f(x-2h))/(2*h*h*h)
 the final estimate D = (4*D(h/2) - D(h))/3
  "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:
    err = k*Sqrt(f(x)*f(x) + x*x*deriv*deriv)*Sqrt(Sum(ai)*(ai)),
 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

Double_t DerivativeError()

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

Int_t DistancetoPrimitive(Int_t px, Int_t py)

*-*-*-*-*-*-*-*-*-*-*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.
*-*
*-*
*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*

void Draw(Option_t *option)

*-*-*-*-*-*-*-*-*-*-*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.
*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*

TF1 * DrawCopy(Option_t *option)

*-*-*-*-*-*-*-*Draw a copy of this function with its current attributes*-*-*
*-*            ========================================================
*-*
*-*  This function MUST be used instead of Draw when you want to draw
*-*  the same function with different parameters settings in the same canvas.
*-*
*-* Possible option values are:
*-*   "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"
*-*
*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*

void DrawDerivative(Option_t *option)

 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.

void DrawIntegral(Option_t *option)

 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.

void DrawF1(const char *formula, Double_t xmin, Double_t xmax, Option_t *option)

*-*-*-*-*-*-*-*-*-*Draw formula between xmin and xmax*-*-*-*-*-*-*-*-*-*-*-*
*-*                ==================================
*-*

Double_t Eval(Double_t x, Double_t y, Double_t z, Double_t t)

*-*-*-*-*-*-*-*-*-*-*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
*-*
*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*

Double_t EvalPar(const Double_t *x, const Double_t *params)

*-*-*-*-*-*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.

void ExecuteEvent(Int_t event, Int_t px, Int_t py)

*-*-*-*-*-*-*-*-*-*-*Execute action corresponding to one event*-*-*-*
*-*                  =========================================
*-*  This member function is called when a F1 is clicked with the locator
*-*
*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*

void FixParameter(Int_t ipar, Double_t value)

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

TF1 * GetCurrent()

 static function returning the current function being processed

TH1 * GetHistogram()

 return a pointer to the histogram used to vusualize the function

Double_t GetMaximum(Double_t xmin, Double_t xmax)

 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

Double_t GetMaximumX(Double_t xmin, Double_t xmax)

 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

Double_t GetMinimum(Double_t xmin, Double_t xmax)

 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

Double_t GetMinimumX(Double_t xmin, Double_t xmax)

 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

Double_t GetX(Double_t fy, Double_t xmin, Double_t xmax)

 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

Double_t MinimStep(Int_t type, Double_t &xmin, Double_t &xmax, Double_t fy)

   Grid search implementation, used to bracket the minimum and later
   use Brent's method with the bracketed interval
   The step of the search is set to (xmax-xmin)/fNpx
   type: 0-returns MinimumX
         1-returns Minimum
         2-returns MaximumX
         3-returns Maximum
         4-returns X corresponding to fy

Double_t MinimBrent(Int_t type, Double_t &xmin, Double_t &xmax, Double_t xmiddle, Double_t fy, Bool_t &ok)

Finds a minimum of a function, if the function is unimodal  between xmin and xmax
This method uses a combination of golden section search and parabolic interpolation
Details about convergence and properties of this algorithm can be
found in the book by R.P.Brent "Algorithms for Minimization Without Derivatives"
or in the "Numerical Recipes", chapter 10.2

type: 0-returns MinimumX
      1-returns Minimum
      2-returns MaximumX
      3-returns Maximum
      4-returns X corresponding to fy
if ok=true the method has converged

Int_t GetNDF()

 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.

Int_t GetNumberFreeParameters()

 return the number of free parameters

char * GetObjectInfo(Int_t px, Int_t /* py */)

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

Double_t GetParError(Int_t ipar)

return value of parameter number ipar

void GetParLimits(Int_t ipar, Double_t &parmin, Double_t &parmax)

*-*-*-*-*-*Return limits for parameter ipar*-*-*-*
*-*        ================================

Double_t GetProb()

 return the fit probability

Int_t GetQuantiles(Int_t nprobSum, Double_t *q, const Double_t *probSum)

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

        F(x_q) = Integral_{xmin}^(x_q) f dx = q with 0 <= q <= 1.

     For instance the median x_0.5 of a distribution is defined as that value
     of the random variable for which the distribution function equals 0.5:

        F(x_0.5) = Probability(x < x_0.5) = 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");

Double_t GetRandom()

 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.
*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-**-*-*-*-*-*-*-*

Double_t GetRandom(Double_t xmin, Double_t xmax)

 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.
*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-**-*-*-*-*-*-*-*

void GetRange(Double_t &xmin, Double_t &xmax)

*-*-*-*-*-*-*-*-*-*-*Return range of a 1-D function*-*-*-*-*-*-*-*-*-*-*-*
*-*                  ==============================

void GetRange(Double_t &xmin, Double_t &ymin, Double_t &xmax, Double_t &ymax)

*-*-*-*-*-*-*-*-*-*-*Return range of a 2-D function*-*-*-*-*-*-*-*-*-*-*-*-*
*-*                  ==============================

void GetRange(Double_t &xmin, Double_t &ymin, Double_t &zmin, Double_t &xmax, Double_t &ymax, Double_t &zmax)

*-*-*-*-*-*-*-*-*-*-*Return range of function*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
*-*                  ========================

Double_t GetSave(const Double_t *xx)

 Get value corresponding to X in array of fSave values

TAxis * GetXaxis()

 Get x axis of the function.

TAxis * GetYaxis()

 Get y axis of the function.

TAxis * GetZaxis()

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

void GradientPar(const Double_t *x, Double_t *grad, Double_t eps)

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

void InitArgs(const Double_t *x, const Double_t *params)

*-*-*-*-*-*-*-*-*-*-*Initialize parameters addresses*-*-*-*-*-*-*-*-*-*-*-*
*-*                  ===============================

void InitStandardFunctions()

     Create the basic function objects

Double_t Integral(Double_t a, Double_t b, const Double_t *params, Double_t epsilon)

*-*-*-*-*-*-*-*-*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 and to be the 8-point and 16-point Gaussian quadrature approximations to

and define

Then,

where, starting with and finishing with , the subdivision points are given by

with equal to the first member of the sequence for which . 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 relative error of I in the case |I|>1, and a bound on the absolute error in the case |I|<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.

Note1:

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.

---

Note2:

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

---

*/

---------------------------------------------------------------

Double_t Integral(Double_t, Double_t, Double_t, Double_t, Double_t)

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

Double_t Integral(Double_t, Double_t, Double_t, Double_t, Double_t, Double_t, Double_t)

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

Double_t IntegralFast(const TGraph *g, Double_t a, Double_t b, Double_t *params)

 Gauss-Legendre integral, see CalcGaussLegendreSamplingPoints

Double_t IntegralMultiple(Int_t n, const Double_t *a, const Double_t *b, Double_t eps, Double_t &relerr)

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

Double_t IntegralMultiple(Int_t n, const Double_t *a, const Double_t *b, Int_t minpts, Int_t maxpts, Double_t eps, Double_t &relerr,Int_t &nfnevl, Int_t &ifail)

  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 parameter
 ================
 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.

=========================================================================

Bool_t IsInside(const Double_t *x)

 Return kTRUE is the point is inside the function range

void Paint(Option_t *option)

*-*-*-*-*-*-*-*-*-*-*Paint this function with its current attributes*-*-*-*-*
*-*                  ===============================================

void Print(Option_t *option)

*-*-*-*-*-*-*-*-*-*-*Dump this function with its attributes*-*-*-*-*-*-*-*-*-*
*-*                  ==================================

void ReleaseParameter(Int_t ipar)

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

void Save(Double_t xmin, Double_t xmax, Double_t, Double_t, Double_t, Double_t)

 Save values of function in array fSave

void SavePrimitive(ostream &out, Option_t *option /*= ""*/)

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

void SetCurrent(TF1 *f1)

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

void SetMaximum(Double_t maximum)

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

void SetMinimum(Double_t minimum)

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

void SetNDF(Int_t ndf)

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

void SetNpx(Int_t npx)

 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

void SetParError(Int_t ipar, Double_t error)

 set error for parameter number ipar

void SetParErrors(const Double_t *errors)

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

void SetParLimits(Int_t ipar, Double_t parmin, Double_t parmax)

*-*-*-*-*-*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

void SetRange(Double_t xmin, Double_t xmax)

*-*-*-*-*-*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.

void SetSavedPoint(Int_t point, Double_t value)

 Restore value of function saved at point

void SetTitle(const char *title)

 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.

void Streamer(TBuffer &b)

*-*-*-*-*-*-*-*-*Stream a class object*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
*-*              =========================================

void Update()

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

void RejectPoint(Bool_t reject)

 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.

Bool_t RejectedPoint()

 see TF1::RejectPoint above

Double_t Moment(Double_t n, Double_t a, Double_t b, const Double_t *params, Double_t epsilon)

 Return nth moment of function between a and b

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

Double_t CentralMoment(Double_t n, Double_t a, Double_t b, const Double_t *params, Double_t epsilon)

 Return nth central moment of function between a and b

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

void CalcGaussLegendreSamplingPoints(TGraph *g, Double_t eps)

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

void SetRange(Double_t xmin, Double_t, Double_t xmax, Double_t)

void SetRange(Double_t xmin, Double_t, Double_t, Double_t xmax, Double_t, Double_t)

Double_t GetChisquare()

Int_t GetNpx()

TMethodCall * GetMethodCall()

Int_t GetNumberFitPoints()

TObject * GetParent()

Double_t * GetParErrors()

Double_t GetXmin()

Double_t GetXmax()

void SetChisquare(Double_t chi2)

void SetFunction(Double_t (*fcn)(Double_t *, Double_t *))

void SetNumberFitPoints(Int_t npfits)

void SetParent(TObject *p=0)

Double_t Mean(Double_t a, Double_t b, const Double_t *params=0, Double_t epsilon=0.000001)

Double_t Variance(Double_t a, Double_t b, const Double_t *params=0, Double_t epsilon=0.000001)