TF1

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

    private:


    public:

                   TF1 TF1()
                   TF1 TF1(const char* name, const char* formula, Double_t xmin = 0, Double_t xmax = 1)
                   TF1 TF1(const char* name, Double_t xmin, Double_t xmax, Int_t npar)
                   TF1 TF1(const char* name, void* fcn, Double_t xmin, Double_t xmax, Int_t npar)
                   TF1 TF1(const char* name, Double_t (*)(Double_t*, Double_t*) fcn, Double_t xmin = 0, Double_t xmax = 1, Int_t npar = 0)
                   TF1 TF1(const TF1& f1)
          virtual void ~TF1()
          virtual void Browse(TBrowser* b)
        static TClass* Class()
          virtual void Copy(TObject& f1)
      virtual Double_t Derivative(Double_t x, Double_t* params = 0, Double_t epsilon = 0)
         virtual Int_t DistancetoPrimitive(Int_t px, Int_t py)
          virtual void Draw(Option_t* option)
          virtual TF1* DrawCopy(Option_t* option)
          virtual void DrawF1(const char* formula, Double_t xmin, Double_t xmax, Option_t* option)
          virtual void DrawPanel()
      virtual Double_t Eval(Double_t x, Double_t y = 0, Double_t z = 0)
      virtual Double_t EvalPar(const Double_t* x, const Double_t* params = 0)
          virtual void ExecuteEvent(Int_t event, Int_t px, Int_t py)
          virtual void FixParameter(Int_t ipar, Double_t value)
              Double_t GetChisquare() const
                  TH1* GetHistogram() const
          TMethodCall* GetMethodCall() const
                 Int_t GetNDF() const
                 Int_t GetNpx() const
                 Int_t GetNumberFitPoints() const
         virtual char* GetObjectInfo(Int_t px, Int_t py) const
              TObject* GetParent() const
              Double_t GetParError(Int_t ipar) const
             Double_t* GetParErrors() const
          virtual void GetParLimits(Int_t ipar, Double_t& parmin, Double_t& parmax)
      virtual Double_t GetProb() const
      virtual Double_t GetRandom()
          virtual void GetRange(Double_t& xmin, Double_t& xmax)
          virtual void GetRange(Double_t& xmin, Double_t& ymin, Double_t& xmax, Double_t& ymax)
          virtual void GetRange(Double_t& xmin, Double_t& ymin, Double_t& zmin, Double_t& xmax, Double_t& ymax, Double_t& zmax)
      virtual Double_t GetSave(const Double_t* x)
              Double_t GetXmax() const
              Double_t GetXmin() const
          virtual void InitArgs(const Double_t* x, const Double_t* params)
           static void InitStandardFunctions()
      virtual Double_t Integral(Double_t a, Double_t b, const Double_t* params = 0, Double_t epsilon = 0.000001)
      virtual Double_t Integral(Double_t ax, Double_t bx, Double_t ay, Double_t by, Double_t epsilon = 0.000001)
      virtual Double_t Integral(Double_t ax, Double_t bx, Double_t ay, Double_t by, Double_t az, Double_t bz, Double_t epsilon = 0.000001)
      virtual Double_t IntegralMultiple(Int_t n, const Double_t* a, const Double_t* b, Double_t epsilon, Double_t& relerr)
       virtual TClass* IsA() const
          virtual void Paint(Option_t* option)
          virtual void Print(Option_t* option) const
          virtual void ReleaseParameter(Int_t ipar)
          virtual void Save(Double_t xmin, Double_t xmax, Double_t ymin, Double_t ymax, Double_t zmin, Double_t zmax)
          virtual void SavePrimitive(ofstream& out, Option_t* option)
          virtual void SetChisquare(Double_t chi2)
          virtual void SetMaximum(Double_t maximum = -1111)
          virtual void SetMinimum(Double_t minimum = -1111)
          virtual void SetNDF(Int_t ndf)
          virtual void SetNpx(Int_t npx = 100)
          virtual void SetNumberFitPoints(Int_t npfits)
          virtual void SetParent(TObject* p = 0)
          virtual void SetParError(Int_t ipar, Double_t error)
          virtual void SetParLimits(Int_t ipar, Double_t parmin, Double_t parmax)
          virtual void SetRange(Double_t xmin, Double_t xmax)
          virtual void SetRange(Double_t xmin, Double_t ymin, Double_t xmax, Double_t ymax)
          virtual void SetRange(Double_t xmin, Double_t ymin, Double_t zmin, Double_t xmax, Double_t ymax, Double_t zmax)
          virtual void ShowMembers(TMemberInspector& insp, char* parent)
          virtual void Streamer(TBuffer& b)
                  void StreamerNVirtual(TBuffer& b)
          virtual void Update()

Data Members

    private:


    protected:

                                Double_t fXmin        Lower bounds for the range
                                Double_t fXmax        Upper bounds for the range
                                   Int_t fNpx         Number of points used for the graphical representation
                                   Int_t fType        (=0 for standard functions, 1 if pointer to function)
                                   Int_t fNpfits      Number of points used in the fit
                                   Int_t fNDF         Number of degrees of freedom in the fit
                                   Int_t fNsave       Number of points used to fill array fSave
                                Double_t fChisquare   Function fit chisquare
                               Double_t* fIntegral    ![fNpx] Integral of function binned on fNpx bins
                               Double_t* fParErrors   [fNpar] Array of errors of the fNpar parameters
                               Double_t* fParMin      [fNpar] Array of lower limits of the fNpar parameters
                               Double_t* fParMax      [fNpar] Array of upper limits of the fNpar parameters
                               Double_t* fSave        [fNsave] Array of fNsave function values
                               Double_t* fAlpha       !Array alpha. for each bin in x the deconvolution r of fIntegral
                               Double_t* fBeta        !Array beta.  is approximated by x = alpha +beta*r *gamma*r**2
                               Double_t* fGamma       !Array gamma.
                                TObject* fParent      !Parent object hooking this function (if one)
                                    TH1* fHistogram   !Pointer to histogram used for visualisation
                                Double_t fMaximum     Maximum value for plotting
                                Double_t fMinimum     Minimum value for plotting
                            TMethodCall* fMethodCall  !Pointer to MethodCall in case of interpreted function
      Double_t (*)(Double_t*, Double_t*) fFunction    !Pointer to function

    public:

      static const enum TObject:: kNotDraw  

See also

TF2

Class Description

 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

   TF1 *f1 = new TF1("f1","sin(x)/x",0,10);
   f1->Draw();

/*

*/


      Example of a function of type B
   TF1 *f1 = new TF1("f1","[0]*x*sin([1]*x)",-3,3);
    This creates a function of variable x with 2 parameters.
    The parameters must be initialized via:
      f1->SetParameter(0,value_first_parameter);
      f1->SetParameter(1,value_second_parameter);
    Parameters may be given a name:
      f1->SetParName(0,"Constant");

     Example of 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)
  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");
}

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

*-*-*-*-*-*-*F1 constructor using name 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.
*-*
*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*

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

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

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

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

*-*-*-*-*-*-*-*-*Return derivative of function at point x*-*-*-*-*-*-*-*

    The derivative is computed by computing the value of the function
   at points x-epsilon*range and x+epsilon*range (range=fXmax-fXmin).
   if params is NULL, use the current values of parameters

*-*-*-*-*-*-*-*-*-*-*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.
*-*
*-*  Algorithm:
*-*
*-*
*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*

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

*-*-*-*-*-*-*Display a panel with all function drawing options*-*-*-*-*-*
*-*          =================================================
*-*
*-*   See class TDrawPanelHist for example

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

 return a pointer to the histogram used to vusualize the function

 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.

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

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

*-*-*-*-*-*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 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

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

Notes:

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.


*/

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

 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]

  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.

 N Number of dimensions.
 A,B One-dimensional arrays of length >= N . On entry A[i],  and  B[i],
     contain the lower and upper limits of integration, respectively.
 EPS    Specified relative accuracy.
 RELERR Contains, on exit, an estimation of the relative accuray of RESULT.

 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.

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

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

 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*-*-*-*-*-*
*-*            ==================================================

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

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

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

Inline Functions

            Double_t GetChisquare() const
               Int_t GetNpx() const
        TMethodCall* GetMethodCall() const
               Int_t GetNumberFitPoints() const
            TObject* GetParent() const
            Double_t GetParError(Int_t ipar) const
           Double_t* GetParErrors() const
            Double_t GetProb() const
            Double_t GetXmin() const
            Double_t GetXmax() const
                void SetChisquare(Double_t chi2)
                void SetMaximum(Double_t maximum = -1111)
                void SetMinimum(Double_t minimum = -1111)
                void SetNumberFitPoints(Int_t npfits)
                void SetParError(Int_t ipar, Double_t error)
                void SetParent(TObject* p = 0)
                void SetRange(Double_t xmin, Double_t ymin, Double_t xmax, Double_t ymax)
                void SetRange(Double_t xmin, Double_t ymin, Double_t zmin, Double_t xmax, Double_t ymax, Double_t zmax)
             TClass* Class()
             TClass* IsA() const
                void ShowMembers(TMemberInspector& insp, char* parent)
                void StreamerNVirtual(TBuffer& b)