library: libHist #include "TF1.h"

# TF1

class description - source file - inheritance tree (.pdf)

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

| Inheritance Chart:

`TObject`

<-

`TNamed`

<-

`TFormula`

`TAttLine`

`TAttFill`

`TAttMarker`

<-

`TF1`

<-

`TF12` |

`TF2`

<-

`TF3` |

        public:

                            TF1()
                            TF1(const char* name, const char* formula, Double_t xmin = 0, Double_t xmax = 1)
                            TF1(const char* name, Double_t xmin, Double_t xmax, Int_t npar)
                            TF1(const char* name, void* fcn, Double_t xmin, Double_t xmax, Int_t npar)
                            TF1(const char* name, Double_t (*)(Double_t*, Double_t*) fcn, Double_t xmin = 0, Double_t xmax = 1, Int_t npar = 0)
                            TF1(const TF1& f1)
                    virtual ~TF1()
                static void AbsValue(Bool_t reject = kTRUE)
               virtual void Browse(TBrowser* b)
                static void CalcGaussLegendreSamplingPoints(Int_t num, Double_t* x, Double_t* w, Double_t eps = 3.0e-11)
           virtual Double_t CentralMoment(Double_t n, Double_t a, Double_t b, const Double_t* params = 0, Double_t epsilon = 0.000001)
             static TClass* Class()
               virtual void Copy(TObject& f1) const
           virtual Double_t Derivative(Double_t x, Double_t* params = 0, Double_t epsilon = 0.001) const
           virtual Double_t Derivative2(Double_t x, Double_t* params = 0, Double_t epsilon = 0.001) const
           virtual Double_t Derivative3(Double_t x, Double_t* params = 0, Double_t epsilon = 0.001) const
            static Double_t DerivativeError()
              virtual Int_t DistancetoPrimitive(Int_t px, Int_t py)
               virtual void Draw(Option_t* option = "")
               virtual TF1* DrawCopy(Option_t* option = "") const
               virtual void DrawDerivative(Option_t* option = "al")
               virtual void DrawF1(const char* formula, Double_t xmin, Double_t xmax, Option_t* option = "")
               virtual void DrawIntegral(Option_t* option = "al")
               virtual void DrawPanel()
           virtual Double_t Eval(Double_t x, Double_t y = 0, Double_t z = 0, Double_t t = 0) const
           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
                static TF1* GetCurrent()
                       TH1* GetHistogram() const
           virtual Double_t GetMaximum(Double_t xmin = 0, Double_t xmax = 0) const
           virtual Double_t GetMaximumX(Double_t xmin = 0, Double_t xmax = 0) const
               TMethodCall* GetMethodCall() const
           virtual Double_t GetMinimum(Double_t xmin = 0, Double_t xmax = 0) const
           virtual Double_t GetMinimumX(Double_t xmin = 0, Double_t xmax = 0) const
              virtual Int_t GetNDF() const
              virtual Int_t GetNpx() const
              virtual Int_t GetNumberFitPoints() const
              virtual Int_t GetNumberFreeParameters() const
              virtual char* GetObjectInfo(Int_t px, Int_t py) const
                   TObject* GetParent() const
           virtual Double_t GetParError(Int_t ipar) const
          virtual Double_t* GetParErrors() const
               virtual void GetParLimits(Int_t ipar, Double_t& parmin, Double_t& parmax) const
           virtual Double_t GetProb() const
              virtual Int_t GetQuantiles(Int_t nprobSum, Double_t* q, const Double_t* probSum)
           virtual Double_t GetRandom()
           virtual Double_t GetRandom(Double_t xmin, Double_t xmax)
               virtual void GetRange(Double_t& xmin, Double_t& xmax) const
               virtual void GetRange(Double_t& xmin, Double_t& ymin, Double_t& xmax, Double_t& ymax) const
               virtual void GetRange(Double_t& xmin, Double_t& ymin, Double_t& zmin, Double_t& xmax, Double_t& ymax, Double_t& zmax) const
           virtual Double_t GetSave(const Double_t* x)
           virtual Double_t GetX(Double_t y, Double_t xmin = 0, Double_t xmax = 0) const
                     TAxis* GetXaxis() const
           virtual Double_t GetXmax() const
           virtual Double_t GetXmin() const
                     TAxis* GetYaxis() const
                     TAxis* GetZaxis() 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 = 1e-12)
           virtual Double_t Integral(Double_t ax, Double_t bx, Double_t ay, Double_t by, Double_t epsilon = 1e-12)
           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 = 1e-12)
           virtual Double_t IntegralFast(Int_t num, Double_t* x, Double_t* w, Double_t a, Double_t b, Double_t* params = 0)
           virtual Double_t IntegralMultiple(Int_t n, const Double_t* a, const Double_t* b, Int_t minpts, Int_t maxpts, Double_t epsilon, Double_t& relerr, Int_t& nfnevl, Int_t& ifail)
           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 Bool_t IsInside(const Double_t* x) const
           virtual Double_t Mean(Double_t a, Double_t b, const Double_t* params = 0, Double_t epsilon = 0.000001)
           virtual Double_t Moment(Double_t n, Double_t a, Double_t b, const Double_t* params = 0, Double_t epsilon = 0.000001)
                       TF1& operator=(const TF1& rhs)
               virtual void Paint(Option_t* option = "")
               virtual void Print(Option_t* option = "") const
              static Bool_t RejectedPoint()
                static void RejectPoint(Bool_t reject = kTRUE)
               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)
                static void SetCurrent(TF1* f1)
               virtual void SetFunction(Double_t (*)(Double_t*, Double_t*) fcn)
               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 SetParErrors(const Double_t* errors)
               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 SetSavedPoint(Int_t point, Double_t value)
               virtual void ShowMembers(TMemberInspector& insp, char* parent)
               virtual void Streamer(TBuffer& b)
                       void StreamerNVirtual(TBuffer& b)
               virtual void Update()
           virtual Double_t Variance(Double_t a, Double_t b, const Double_t* params = 0, Double_t epsilon = 0.000001)

### Data Members

        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
                               static Bool_t fgAbsValue     use absolute value of function when computing integral
                               static Bool_t fgRejectPoint  True if point must be rejected in a fit
                                 static TF1* fgCurrent      pointer to current function being processed

        public:

          static const enum TF1:: kNotDraw

## 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 +
             +++++++++++++++++++++++++++++++++++

      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)
      ------------------------------------------------------------------
       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");

      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(): TFormula(), TAttLine(), TAttFill(), TAttMarker()

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

TF1(const char *name,const char *formula, Double_t xmin, Double_t xmax) :TFormula(name,formula), TAttLine(), TAttFill(), TAttMarker()

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

TF1(const char *name, Double_t xmin, Double_t xmax, Int_t npar) :TFormula(), TAttLine(), TAttFill(), TAttMarker()

    *-*-*-*-*-*-*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) :TFormula(), TAttLine(), TAttFill(), TAttMarker()

    *-*-*-*-*-*-*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) :TFormula(), TAttLine(), TAttFill(), TAttMarker()

    *-*-*-*-*-*-*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()

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

TF1(const TF1 &f1) : TFormula(), TAttLine(f1), TAttFill(f1), TAttMarker(f1)

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)

void Copy(TObject &obj) const

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

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

     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) const

     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) - 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) const

     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+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) const

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

void DrawPanel()

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

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

    *-*-*-*-*-*-*-*-*-*-*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() const

     return a pointer to the histogram used to vusualize the function

Double_t GetMaximum(Double_t xmin, Double_t xmax) const

     return the maximum value of the function
     Method:
       the function is computed at fNpx points between xmin and xmax
       xxmax is the X value corresponding to the maximum function value
       An iterative procedure computes the maximum around xxmax
       until dx is less than 1e-9 *(fXmax-fXmin)

Double_t GetMaximumX(Double_t xmin, Double_t xmax) const

     return the X value corresponding to the maximum value of the function
     Method:
       the function is computed at fNpx points between xmin and xmax
       xxmax is the X value corresponding to the maximum function value
       An iterative procedure computes the maximum around xxmax
       until dx is less than 1e-9 *(fXmax-fXmin)

Double_t GetMinimum(Double_t xmin, Double_t xmax) const

     return the minimum value of the function
     Method:
       the function is computed at fNpx points between xmin and xmax
       xxmax is the X value corresponding to the minimum function value
       An iterative procedure computes the minimum around xxmax
       until dx is less than 1e-9 *(fXmax-fXmin)

Double_t GetMinimumX(Double_t xmin, Double_t xmax) const

     return the X value corresponding to the minimum value of the function
     Method:
       the function is computed at fNpx points between xmin and xmax
       xxmax is the X value corresponding to the minimum function value
       An iterative procedure computes the minimum around xxmax
       until dx is less than 1e-9 *(fXmax-fXmin)

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

     return the X value corresponding to the function value fy for (xmin<x<xmax).
     Method:
       the function z = abs(f-fy) is computed at fNpx points between xmin and xmax
       xxmax is the X value corresponding to the minimum function value
       An iterative procedure computes the minimum around xxmax
       until dx is less than 1e-9 *(fXmax-fXmin).
      In case the problem has several solutions in X, the higher X solution is returned
      If the problem has no solution, the function returns the value of X
      where f(X) is closer to fy.

Int_t GetNDF() const

     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() const

     return the number of free parameters

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

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

Double_t GetParError(Int_t ipar) const

    return value of parameter number ipar

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

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

Double_t GetProb() const

     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) const

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

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

    *-*-*-*-*-*-*-*-*-*-*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) const

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

Double_t GetSave(const Double_t *xx)

     Get value corresponding to X in array of fSave values

TAxis* GetXaxis() const

     Get x axis of the function.

TAxis* GetYaxis() const

     Get y axis of the function.

TAxis* GetZaxis() const

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

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.

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.

*/

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

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 CalcIntegralSamplingPoints

Double_t IntegralFast(Int_t num, Double_t *x, Double_t *w, Double_t a, Double_t b, Double_t *params)

     Gauss-Legendre integral, see CalcIntegralSamplingPoints

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) const

     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) const

    *-*-*-*-*-*-*-*-*-*-*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(ofstream &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 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

TGraph* CalcGaussLegendreSamplingPoints(Int_t num, Double_t eps)

    type safe interface (static method)
     A TGraph is created with new with num points and the pointer to the
     graph is returned by the function. It is the responsibility of the
     user to delete the object.
     if num is invalid (<=0) NULL is returned

void CalcGaussLegendreSamplingPoints(Int_t num, Double_t *x, Double_t *w, Double_t eps)

     Type: unsafe but fast interface filling the arrays x and w (static method)

     Given the number of sampling points this routine fills the arrays x and w
     of length num, containing the abscissa and weight of the Gauss-Legendre
     n-point quadrature formula.

     Gauss-Legendre: W(x)=1  -1<x<1
                     (j+1)P_{j+1} = (2j+1)xP_j-jP_{j-1}

     num is the number of sampling points (>0)
     x and w are arrays of size num
     eps is the relative precision

     If num<=0 or eps<=0 no action is done.

     Reference: Numerical Recipes in C, Second Edition

### Inline Functions

                    TF1& operator=(const TF1& rhs)
                Double_t GetChisquare() const
                   Int_t GetNpx() const
            TMethodCall* GetMethodCall() const
                   Int_t GetNumberFitPoints() const
                TObject* GetParent() const
               Double_t* GetParErrors() const
                Double_t GetXmin() const
                Double_t GetXmax() const
                    void SetChisquare(Double_t chi2)
                    void SetFunction(Double_t (*)(Double_t*, Double_t*) fcn)
                    void SetNumberFitPoints(Int_t npfits)
                    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)
                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)
                 TClass* Class()
                 TClass* IsA() const
                    void ShowMembers(TMemberInspector& insp, char* parent)
                    void StreamerNVirtual(TBuffer& b)

Author: Rene Brun 18/08/95
Last update: root/hist:$Name: $:$Id: TF1.cxx,v 1.103 2005/04/30 12:51:00 brun Exp $
Copyright (C) 1995-2000, Rene Brun and Fons Rademakers. *