// @(#)root/hist:$Name: $:$Id: TF1.cxx,v 1.108 2005/09/02 19:18:11 brun Exp $
// Author: Rene Brun 18/08/95
/*************************************************************************
* Copyright (C) 1995-2000, Rene Brun and Fons Rademakers. *
* All rights reserved. *
* *
* For the licensing terms see $ROOTSYS/LICENSE. *
* For the list of contributors see $ROOTSYS/README/CREDITS. *
*************************************************************************/
#include "Riostream.h"
#include "TROOT.h"
#include "TMath.h"
#include "TF1.h"
#include "TH1.h"
#include "TVirtualPad.h"
#include "TStyle.h"
#include "TRandom.h"
#include "Api.h"
#include "TPluginManager.h"
#include "TVirtualUtilPad.h"
#include "TBrowser.h"
#include "TColor.h"
Bool_t TF1::fgAbsValue = kFALSE;
Bool_t TF1::fgRejectPoint = kFALSE;
static TF1 *gHelper = 0;
static Double_t gErrorTF1 = 0;
ClassImp(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)
// ------------------------------------------------------------------
// 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 *TF1::fgCurrent = 0;
//______________________________________________________________________________
TF1::TF1(): TFormula(), TAttLine(), TAttFill(), TAttMarker()
{
//*-*-*-*-*-*-*-*-*-*-*F1 default constructor*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
//*-* ======================
fXmin = 0;
fXmax = 0;
fNpx = 100;
fType = 0;
fNpfits = 0;
fNDF = 0;
fNsave = 0;
fChisquare = 0;
fIntegral = 0;
fFunction = 0;
fParErrors = 0;
fParMin = 0;
fParMax = 0;
fAlpha = 0;
fBeta = 0;
fGamma = 0;
fParent = 0;
fSave = 0;
fHistogram = 0;
fMinimum = -1111;
fMaximum = -1111;
fMethodCall = 0;
SetFillStyle(0);
}
//______________________________________________________________________________
TF1::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
//*-*
//*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
if (xmin < xmax ) {
fXmin = xmin;
fXmax = xmax;
} else {
fXmin = xmax; //when called from TF2,TF3
fXmax = xmin;
}
fNpx = 100;
fType = 0;
fFunction = 0;
if (fNpar) {
fParErrors = new Double_t[fNpar];
fParMin = new Double_t[fNpar];
fParMax = new Double_t[fNpar];
for (int i = 0; i < fNpar; i++) {
fParErrors[i] = 0;
fParMin[i] = 0;
fParMax[i] = 0;
}
} else {
fParErrors = 0;
fParMin = 0;
fParMax = 0;
}
fChisquare = 0;
fIntegral = 0;
fAlpha = 0;
fBeta = 0;
fGamma = 0;
fParent = 0;
fNpfits = 0;
fNDF = 0;
fNsave = 0;
fSave = 0;
fHistogram = 0;
fMinimum = -1111;
fMaximum = -1111;
fMethodCall = 0;
if (fNdim != 1 && xmin < xmax) {
Error("TF1","function: %s/%s has %d parameters instead of 1",name,formula,fNdim);
MakeZombie();
}
if (!gStyle) return;
SetLineColor(gStyle->GetFuncColor());
SetLineWidth(gStyle->GetFuncWidth());
SetLineStyle(gStyle->GetFuncStyle());
SetFillStyle(0);
}
//______________________________________________________________________________
TF1::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.
//*-*
//*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
fXmin = xmin;
fXmax = xmax;
fNpx = 100;
fType = 2;
fFunction = 0;
if (npar > 0 ) fNpar = npar;
if (fNpar) {
fNames = new TString[fNpar];
fParams = new Double_t[fNpar];
fParErrors = new Double_t[fNpar];
fParMin = new Double_t[fNpar];
fParMax = new Double_t[fNpar];
for (int i = 0; i < fNpar; i++) {
fParams[i] = 0;
fParErrors[i] = 0;
fParMin[i] = 0;
fParMax[i] = 0;
}
} else {
fParErrors = 0;
fParMin = 0;
fParMax = 0;
}
fChisquare = 0;
fIntegral = 0;
fAlpha = 0;
fBeta = 0;
fGamma = 0;
fParent = 0;
fNpfits = 0;
fNDF = 0;
fNsave = 0;
fSave = 0;
fHistogram = 0;
fMinimum = -1111;
fMaximum = -1111;
fMethodCall = 0;
fNdim = 1;
TF1 *f1old = (TF1*)gROOT->GetListOfFunctions()->FindObject(name);
if (f1old) delete f1old;
SetName(name);
if (gStyle) {
SetLineColor(gStyle->GetFuncColor());
SetLineWidth(gStyle->GetFuncWidth());
SetLineStyle(gStyle->GetFuncStyle());
}
SetFillStyle(0);
SetTitle(name);
if (name) {
if (*name == '*') return; //case happens via SavePrimitive
fMethodCall = new TMethodCall();
fMethodCall->InitWithPrototype(name,"Double_t*,Double_t*");
fNumber = -1;
gROOT->GetListOfFunctions()->Add(this);
if (! fMethodCall->IsValid() ) {
Error("TF1","No function found with the signature %s(Double_t*,Double_t*)",name);
}
} else {
Error("TF1","requires a proper function name!");
}
}
//______________________________________________________________________________
TF1::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.
//*-*
//*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
fXmin = xmin;
fXmax = xmax;
fNpx = 100;
fType = 2;
fFunction = 0;
if (npar > 0 ) fNpar = npar;
if (fNpar) {
fNames = new TString[fNpar];
fParams = new Double_t[fNpar];
fParErrors = new Double_t[fNpar];
fParMin = new Double_t[fNpar];
fParMax = new Double_t[fNpar];
for (int i = 0; i < fNpar; i++) {
fParams[i] = 0;
fParErrors[i] = 0;
fParMin[i] = 0;
fParMax[i] = 0;
}
} else {
fParErrors = 0;
fParMin = 0;
fParMax = 0;
}
fChisquare = 0;
fIntegral = 0;
fAlpha = 0;
fBeta = 0;
fGamma = 0;
fParent = 0;
fNpfits = 0;
fNDF = 0;
fNsave = 0;
fSave = 0;
fHistogram = 0;
fMinimum = -1111;
fMaximum = -1111;
fMethodCall = 0;
fNdim = 1;
TF1 *f1old = (TF1*)gROOT->GetListOfFunctions()->FindObject(name);
if (f1old) delete f1old;
SetName(name);
if (gStyle) {
SetLineColor(gStyle->GetFuncColor());
SetLineWidth(gStyle->GetFuncWidth());
SetLineStyle(gStyle->GetFuncStyle());
}
SetFillStyle(0);
if (!fcn) return;
char *funcname = G__p2f2funcname(fcn);
SetTitle(funcname);
if (funcname) {
fMethodCall = new TMethodCall();
fMethodCall->InitWithPrototype(funcname,"Double_t*,Double_t*");
fNumber = -1;
gROOT->GetListOfFunctions()->Add(this);
if (! fMethodCall->IsValid() ) {
Error("TF1","No function found with the signature %s(Double_t*,Double_t*)",funcname);
}
} else {
Error("TF1","can not find any function at the address 0x%x. This function requested for %s",fcn,name);
}
}
//______________________________________________________________________________
TF1::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.
//*-*
//*-*
//*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
fXmin = xmin;
fXmax = xmax;
fNpx = 100;
fType = 1;
fMethodCall = 0;
fFunction = fcn;
if (npar > 0 ) fNpar = npar;
if (fNpar) {
fNames = new TString[fNpar];
fParams = new Double_t[fNpar];
fParErrors = new Double_t[fNpar];
fParMin = new Double_t[fNpar];
fParMax = new Double_t[fNpar];
for (int i = 0; i < fNpar; i++) {
fParams[i] = 0;
fParErrors[i] = 0;
fParMin[i] = 0;
fParMax[i] = 0;
}
} else {
fParErrors = 0;
fParMin = 0;
fParMax = 0;
}
fChisquare = 0;
fIntegral = 0;
fAlpha = 0;
fBeta = 0;
fGamma = 0;
fNsave = 0;
fSave = 0;
fParent = 0;
fNpfits = 0;
fNDF = 0;
fHistogram = 0;
fMinimum = -1111;
fMaximum = -1111;
fNdim = 1;
//*-*- Store formula in linked list of formula in ROOT
TF1 *f1old = (TF1*)gROOT->GetListOfFunctions()->FindObject(name);
if (f1old) delete f1old;
SetName(name);
gROOT->GetListOfFunctions()->Add(this);
if (!gStyle) return;
SetLineColor(gStyle->GetFuncColor());
SetLineWidth(gStyle->GetFuncWidth());
SetLineStyle(gStyle->GetFuncStyle());
SetFillStyle(0);
}
//______________________________________________________________________________
TF1::TF1(const char *name,Double_t (*fcn)(const Double_t *, const 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.
//*-*
//*-*
//*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
fXmin = xmin;
fXmax = xmax;
fNpx = 100;
fType = 1;
fMethodCall = 0;
typedef Double_t (*Function_t) (Double_t *, Double_t *);
fFunction = (Function_t)fcn;
if (npar > 0 ) fNpar = npar;
if (fNpar) {
fNames = new TString[fNpar];
fParams = new Double_t[fNpar];
fParErrors = new Double_t[fNpar];
fParMin = new Double_t[fNpar];
fParMax = new Double_t[fNpar];
for (int i = 0; i < fNpar; i++) {
fParams[i] = 0;
fParErrors[i] = 0;
fParMin[i] = 0;
fParMax[i] = 0;
}
} else {
fParErrors = 0;
fParMin = 0;
fParMax = 0;
}
fChisquare = 0;
fIntegral = 0;
fAlpha = 0;
fBeta = 0;
fGamma = 0;
fNsave = 0;
fSave = 0;
fParent = 0;
fNpfits = 0;
fNDF = 0;
fHistogram = 0;
fMinimum = -1111;
fMaximum = -1111;
fNdim = 1;
//*-*- Store formula in linked list of formula in ROOT
TF1 *f1old = (TF1*)gROOT->GetListOfFunctions()->FindObject(name);
if (f1old) delete f1old;
SetName(name);
gROOT->GetListOfFunctions()->Add(this);
if (!gStyle) return;
SetLineColor(gStyle->GetFuncColor());
SetLineWidth(gStyle->GetFuncWidth());
SetLineStyle(gStyle->GetFuncStyle());
SetFillStyle(0);
}
//______________________________________________________________________________
TF1& TF1::operator=(const TF1 &rhs)
{
if (this != &rhs) {
rhs.Copy(*this);
}
return *this;
}
//______________________________________________________________________________
TF1::~TF1()
{
//*-*-*-*-*-*-*-*-*-*-*F1 default destructor*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
//*-* =====================
if (fParMin) delete [] fParMin;
if (fParMax) delete [] fParMax;
if (fParErrors) delete [] fParErrors;
if (fIntegral) delete [] fIntegral;
if (fAlpha) delete [] fAlpha;
if (fBeta) delete [] fBeta;
if (fGamma) delete [] fGamma;
if (fSave) delete [] fSave;
delete fHistogram;
delete fMethodCall;
if (fParent) {
if (fParent->InheritsFrom(TH1::Class())) {
((TH1*)fParent)->GetListOfFunctions()->Remove(this);
return;
}
//parent may be a graph, or ??
//The pad utility manager is required (a plugin)
TVirtualUtilPad *util = (TVirtualUtilPad*)gROOT->GetListOfSpecials()->FindObject("R__TVirtualUtilPad");
if (!util) {
TPluginHandler *h;
if ((h = gROOT->GetPluginManager()->FindHandler("TVirtualUtilPad"))) {
if (h->LoadPlugin() == -1)
return;
h->ExecPlugin(0);
util = (TVirtualUtilPad*)gROOT->GetListOfSpecials()->FindObject("R__TVirtualUtilPad");
}
}
util->RemoveObject(fParent,this);
fParent = 0;
}
}
//______________________________________________________________________________
TF1::TF1(const TF1 &f1) : TFormula(), TAttLine(f1), TAttFill(f1), TAttMarker(f1)
{
fXmin = 0;
fXmax = 0;
fNpx = 100;
fType = 0;
fNpfits = 0;
fNDF = 0;
fNsave = 0;
fChisquare = 0;
fIntegral = 0;
fFunction = 0;
fParErrors = 0;
fParMin = 0;
fParMax = 0;
fAlpha = 0;
fBeta = 0;
fGamma = 0;
fParent = 0;
fSave = 0;
fHistogram = 0;
fMinimum = -1111;
fMaximum = -1111;
fMethodCall = 0;
SetFillStyle(0);
((TF1&)f1).Copy(*this);
}
//______________________________________________________________________________
void TF1::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.
fgAbsValue = flag;
}
//______________________________________________________________________________
void TF1::Browse(TBrowser *b)
{
Draw(b ? b->GetDrawOption() : "");
gPad->Update();
}
//______________________________________________________________________________
void TF1::Copy(TObject &obj) const
{
//*-*-*-*-*-*-*-*-*-*-*Copy this F1 to a new F1*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
//*-* ========================
if (((TF1&)obj).fParMin) delete [] ((TF1&)obj).fParMin;
if (((TF1&)obj).fParMax) delete [] ((TF1&)obj).fParMax;
if (((TF1&)obj).fParErrors) delete [] ((TF1&)obj).fParErrors;
if (((TF1&)obj).fIntegral) delete [] ((TF1&)obj).fIntegral;
if (((TF1&)obj).fAlpha) delete [] ((TF1&)obj).fAlpha;
if (((TF1&)obj).fBeta) delete [] ((TF1&)obj).fBeta;
if (((TF1&)obj).fGamma) delete [] ((TF1&)obj).fGamma;
if (((TF1&)obj).fSave) delete [] ((TF1&)obj).fSave;
delete ((TF1&)obj).fHistogram;
delete ((TF1&)obj).fMethodCall;
TFormula::Copy(obj);
TAttLine::Copy((TF1&)obj);
TAttFill::Copy((TF1&)obj);
TAttMarker::Copy((TF1&)obj);
((TF1&)obj).fXmin = fXmin;
((TF1&)obj).fXmax = fXmax;
((TF1&)obj).fNpx = fNpx;
((TF1&)obj).fType = fType;
((TF1&)obj).fFunction = fFunction;
((TF1&)obj).fChisquare = fChisquare;
((TF1&)obj).fNpfits = fNpfits;
((TF1&)obj).fNDF = fNDF;
((TF1&)obj).fMinimum = fMinimum;
((TF1&)obj).fMaximum = fMaximum;
((TF1&)obj).fParErrors = 0;
((TF1&)obj).fParMin = 0;
((TF1&)obj).fParMax = 0;
((TF1&)obj).fIntegral = 0;
((TF1&)obj).fAlpha = 0;
((TF1&)obj).fBeta = 0;
((TF1&)obj).fGamma = 0;
((TF1&)obj).fParent = fParent;
((TF1&)obj).fNsave = fNsave;
((TF1&)obj).fSave = 0;
((TF1&)obj).fHistogram = 0;
((TF1&)obj).fMethodCall = 0;
if (fNsave) {
((TF1&)obj).fSave = new Double_t[fNsave];
for (Int_t j=0;j<fNsave;j++) ((TF1&)obj).fSave[j] = fSave[j];
}
if (fNpar) {
((TF1&)obj).fParErrors = new Double_t[fNpar];
((TF1&)obj).fParMin = new Double_t[fNpar];
((TF1&)obj).fParMax = new Double_t[fNpar];
Int_t i;
for (i=0;i<fNpar;i++) ((TF1&)obj).fParErrors[i] = fParErrors[i];
for (i=0;i<fNpar;i++) ((TF1&)obj).fParMin[i] = fParMin[i];
for (i=0;i<fNpar;i++) ((TF1&)obj).fParMax[i] = fParMax[i];
}
if (fMethodCall) {
TMethodCall *m = new TMethodCall();
m->InitWithPrototype(fMethodCall->GetMethodName(),fMethodCall->GetProto());
((TF1&)obj).fMethodCall = m;
}
}
//______________________________________________________________________________
Double_t TF1::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
const Double_t kC1 = 1e-15;
if(eps< 1e-10 || eps > 1e-2) {
Warning("Derivative","parameter esp=%g out of allowed range[1e-10,1e-2], reset to 0.001",eps);
eps = 0.001;
}
Double_t xmin, xmax;
GetRange(xmin, xmax);
Double_t h = eps*(xmax-xmin);
Double_t xx[1];
TF1 *func = (TF1*)this;
func->InitArgs(xx, params);
xx[0] = x+h; Double_t f1 = func->EvalPar(xx, params);
xx[0] = x; Double_t fx = func->EvalPar(xx, params);
xx[0] = x-h; Double_t f2 = func->EvalPar(xx, params);
xx[0] = x+h/2; Double_t g1 = func->EvalPar(xx, params);
xx[0] = x-h/2; Double_t g2 = func->EvalPar(xx, params);
//compute the central differences
Double_t h2 = 1/(2.*h);
Double_t d0 = f1 - f2;
Double_t d2 = 2*(g1 - g2);
gErrorTF1 = kC1*h2*fx; //compute the error
Double_t deriv = h2*(4*d2 - d0)/3.;
return deriv;
}
//______________________________________________________________________________
Double_t TF1::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
const Double_t kC1 = 2*1e-15;
if(eps< 1e-6 || eps > 1e-2) {
Warning("Derivative2","parameter esp=%g out of allowed range[1e-6,1e-2], reset to 0.001",eps);
eps = 0.001;
}
Double_t xmin, xmax;
GetRange(xmin, xmax);
Double_t h = eps*(xmax-xmin);
Double_t xx[1];
TF1 *func = (TF1*)this;
func->InitArgs(xx, params);
xx[0] = x+h; Double_t f1 = func->EvalPar(xx, params);
xx[0] = x; Double_t f2 = func->EvalPar(xx, params);
xx[0] = x-h; Double_t f3 = func->EvalPar(xx, params);
xx[0] = x+h/2; Double_t g1 = func->EvalPar(xx, params);
xx[0] = x-h/2; Double_t g3 = func->EvalPar(xx, params);
//compute the central differences
Double_t hh = 1/(h*h);
Double_t d0 = f3 - 2*f2 + f1;
Double_t d2 = 4*g3 - 8*f2 +4*g1;
gErrorTF1 = kC1*hh*f2; //compute the error
Double_t deriv = hh*(4*d2 - d0)/3.;
return deriv;
}
//______________________________________________________________________________
Double_t TF1::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
//const Double_t C1 = (1e-16)*TMath::Sqrt(5./2.)*TMath::Sqrt(16*64 + 1.)/3;
const Double_t kC1 = 1e-15;
if(eps< 1e-4 || eps > 1e-2) {
Warning("Derivative3","parameter esp=%g out of allowed range[1e-4,1e-2], reset to 0.001",eps);
eps = 0.001;
}
Double_t xmin, xmax;
GetRange(xmin, xmax);
Double_t h = eps*(xmax-xmin);
Double_t xx[1];
TF1 *func = (TF1*)this;
func->InitArgs(xx, params);
xx[0] = x+2*h; Double_t f1 = func->EvalPar(xx, params);
xx[0] = x+h; Double_t f2 = func->EvalPar(xx, params);
xx[0] = x-h; Double_t f3 = func->EvalPar(xx, params);
xx[0] = x-2*h; Double_t f4 = func->EvalPar(xx, params);
xx[0] = x; Double_t fx = func->EvalPar(xx, params);
xx[0] = x+h/2; Double_t g2 = func->EvalPar(xx, params);
xx[0] = x-h/2; Double_t g3 = func->EvalPar(xx, params);
//compute the central differences
Double_t hhh = 1/(h*h*h);
Double_t d0 = 0.5*f1 - f2 +f3 - 0.5*f4;
Double_t d2 = 4*f2 - 8*g2 +8*g3 - 4*f3;
gErrorTF1 = kC1*hhh*fx; //compute the error
Double_t deriv = hhh*(4*d2 - d0)/3.;
return deriv;
}
//______________________________________________________________________________
Double_t TF1::DerivativeError()
{
//static function returning the error of the last call to the Derivative functions
return gErrorTF1;
}
//______________________________________________________________________________
Int_t TF1::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.
//*-*
//*-*
//*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
if (!fHistogram) return 9999;
Int_t distance = 9999;
if (px >= 0) {
distance = fHistogram->DistancetoPrimitive(px,py);
if (distance <= 1) return distance;
} else {
px = -px;
}
Double_t xx[1];
Double_t x = gPad->AbsPixeltoX(px);
xx[0] = gPad->PadtoX(x);
if (xx[0] < fXmin || xx[0] > fXmax) return distance;
Double_t fval = Eval(xx[0]);
Double_t y = gPad->YtoPad(fval);
Int_t pybin = gPad->YtoAbsPixel(y);
return TMath::Abs(py - pybin);
}
//______________________________________________________________________________
void TF1::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.
//*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
TString opt = option;
opt.ToLower();
if (gPad && !opt.Contains("same")) gPad->Clear();
AppendPad(option);
}
//______________________________________________________________________________
TF1 *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"
//*-*
//*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
TF1 *newf1 = new TF1();
Copy(*newf1);
newf1->AppendPad(option);
newf1->SetBit(kCanDelete);
return newf1;
}
//______________________________________________________________________________
void TF1::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.
TVirtualPad *pad = gROOT->GetSelectedPad();
TVirtualPad *padsav = gPad;
if (pad) pad->cd();
char cmd[512];
sprintf(cmd,"{TGraph *R__%s_Derivative = new TGraph((TF1*)0x%lx,\"d\");R__%s_Derivative->Draw(\"%s\");}",GetName(),(Long_t)this,GetName(),option);
gROOT->ProcessLine(cmd);
if (padsav) padsav->cd();
}
//______________________________________________________________________________
void TF1::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.
TVirtualPad *pad = gROOT->GetSelectedPad();
TVirtualPad *padsav = gPad;
if (pad) pad->cd();
char cmd[512];
sprintf(cmd,"{TGraph *R__%s_Integral = new TGraph((TF1*)0x%lx,\"i\");R__%s_Integral->Draw(\"%s\");}",GetName(),(Long_t)this,GetName(),option);
gROOT->ProcessLine(cmd);
if (padsav) padsav->cd();
}
//______________________________________________________________________________
void TF1::DrawF1(const char *formula, Double_t xmin, Double_t xmax, Option_t *option)
{
//*-*-*-*-*-*-*-*-*-*Draw formula between xmin and xmax*-*-*-*-*-*-*-*-*-*-*-*
//*-* ==================================
//*-*
if (Compile(formula)) return;
SetRange(xmin, xmax);
Draw(option);
}
//______________________________________________________________________________
void TF1::DrawPanel()
{
//*-*-*-*-*-*-*Display a panel with all function drawing options*-*-*-*-*-*
//*-* =================================================
//*-*
//*-* See class TDrawPanelHist for example
//The pad utility manager is required (a plugin)
TVirtualUtilPad *util = (TVirtualUtilPad*)gROOT->GetListOfSpecials()->FindObject("R__TVirtualUtilPad");
if (!util) {
TPluginHandler *h;
if ((h = gROOT->GetPluginManager()->FindHandler("TVirtualUtilPad"))) {
if (h->LoadPlugin() == -1)
return;
h->ExecPlugin(0);
util = (TVirtualUtilPad*)gROOT->GetListOfSpecials()->FindObject("R__TVirtualUtilPad");
}
}
util->DrawPanel(gPad,this);
}
//______________________________________________________________________________
Double_t TF1::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 xx[4];
xx[0] = x;
xx[1] = y;
xx[2] = z;
xx[3] = t;
((TF1*)this)->InitArgs(xx,fParams);
return ((TF1*)this)->EvalPar(xx,fParams);
}
//______________________________________________________________________________
Double_t TF1::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.
//
fgCurrent = this;
if (fType == 0) return TFormula::EvalPar(x,params);
Double_t result = 0;
if (fType == 1) {
if (fFunction) {
if (params) result = (*fFunction)((Double_t*)x,(Double_t*)params);
else result = (*fFunction)((Double_t*)x,fParams);
}else result = GetSave(x);
}
if (fType == 2) {
if (fMethodCall) fMethodCall->Execute(result);
else result = GetSave(x);
}
return result;
}
//______________________________________________________________________________
void TF1::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
//*-*
//*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
fHistogram->ExecuteEvent(event,px,py);
if (!gPad->GetView()) {
if (event == kMouseMotion) gPad->SetCursor(kHand);
}
}
//______________________________________________________________________________
void TF1::FixParameter(Int_t ipar, Double_t value)
{
// Fix the value of a parameter
// The specified value will be used in a fit operation
if (ipar < 0 || ipar > fNpar-1) return;
SetParameter(ipar,value);
if (value != 0) SetParLimits(ipar,value,value);
else SetParLimits(ipar,1,1);
}
//______________________________________________________________________________
TF1 *TF1::GetCurrent()
{
// static function returning the current function being processed
return fgCurrent;
}
//______________________________________________________________________________
TH1 *TF1::GetHistogram() const
{
// return a pointer to the histogram used to vusualize the function
if (fHistogram) return fHistogram;
// may be function has not yet be painted. force a pad update
//gPad->Modified();
//gPad->Update();
((TF1*)this)->Paint();
return fHistogram;
}
//______________________________________________________________________________
Double_t TF1::GetMaximum(Double_t xmin, Double_t xmax) const
{
// 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
if (xmin >= xmax) {xmin = fXmin; xmax = fXmax;}
Double_t x;
Int_t niter=0;
x = MinimStep(3, xmin, xmax, 0);
Bool_t ok = kTRUE;
x = MinimBrent(3, xmin, xmax, x, 0, ok);
while (!ok){
if (niter>10){
Error("GetMaximum", "maximum search didn't converge");
break;
}
x=MinimStep(3, xmin, xmax,0);
x = MinimBrent(3, xmin, xmax, x, 0, ok);
niter++;
}
return x;
}
//______________________________________________________________________________
Double_t TF1::GetMaximumX(Double_t xmin, Double_t xmax) const
{
// 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
if (xmin >= xmax) {xmin = fXmin; xmax = fXmax;}
Double_t x;
Int_t niter=0;
x = MinimStep(2, xmin, xmax, 0);
Bool_t ok = kTRUE;
x = MinimBrent(2, xmin, xmax, x, 0, ok);
while (!ok){
if (niter>10){
Error("GetMaximumX", "maximum search didn't converge");
break;
}
x=MinimStep(2, xmin, xmax, 0);
x = MinimBrent(2, xmin, xmax, x, 0, ok);
niter++;
}
return x;
}
//______________________________________________________________________________
Double_t TF1::GetMinimum(Double_t xmin, Double_t xmax) const
{
// 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
if (xmin >= xmax) {xmin = fXmin; xmax = fXmax;}
Double_t x;
Int_t niter=0;
x = MinimStep(1, xmin, xmax, 0);
Bool_t ok = kTRUE;
x = MinimBrent(1, xmin, xmax, x, 0, ok);
while (!ok){
if (niter>10){
Error("GetMinimum", "minimum search didn't converge");
break;
}
x=MinimStep(1, xmin, xmax,0);
x = MinimBrent(1, xmin, xmax, x, 0, ok);
niter++;
}
return x;
}
//______________________________________________________________________________
Double_t TF1::GetMinimumX(Double_t xmin, Double_t xmax) const
{
// 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
if (xmin >= xmax) {xmin = fXmin; xmax = fXmax;}
Int_t niter=0;
Double_t x;
x = MinimStep(0, xmin, xmax, 0);
Bool_t ok = kTRUE;
x = MinimBrent(0, xmin, xmax, x, 0, ok);
while (!ok){
if (niter>10){
Error("GetMinimumX", "minimum search didn't converge");
break;
}
x=MinimStep(0, xmin, xmax,0);
x = MinimBrent(0, xmin, xmax, x, 0, ok);
niter++;
}
return x;
}
//______________________________________________________________________________
Double_t TF1::GetX(Double_t fy, Double_t xmin, Double_t xmax) const
{
// 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
if (xmin >= xmax) {xmin = fXmin; xmax = fXmax;}
Int_t niter=0;
Double_t x;
x = MinimStep(4, xmin, xmax, fy);
Bool_t ok = kTRUE;
x = MinimBrent(4, xmin, xmax, x, fy, ok);
while (!ok){
if (niter>10){
Error("GetX", "Search didn't converge");
break;
}
x=MinimStep(4, xmin, xmax, fy);
x = MinimBrent(4, xmin, xmax, x, fy, ok);
niter++;
}
return x;
}
//______________________________________________________________________________
Double_t TF1::MinimStep(Int_t type, Double_t &xmin, Double_t &xmax, Double_t fy) const
{
// 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 x,y, dx;
dx = (xmax-xmin)/(fNpx-1);
Double_t xxmin = xmin;
Double_t yymin;
if (type < 2)
yymin = Eval(xmin);
else if (type < 4)
yymin = -Eval(xmin);
else
yymin = TMath::Abs(Eval(xmin)-fy);
for (Int_t i=1; i<=fNpx-1; i++) {
x = xmin + i*dx;
if (type < 2)
y = Eval(x);
else if (type < 4)
y = -Eval(x);
else
y = TMath::Abs(Eval(x)-fy);
if (y < yymin) {xxmin = x; yymin = y;}
}
xmin = TMath::Max(xmin,xxmin-dx);
xmax = TMath::Min(xmax,xxmin+dx);
return TMath::Min(xxmin, xmax);
}
//______________________________________________________________________________
Double_t TF1::MinimBrent(Int_t type, Double_t &xmin, Double_t &xmax, Double_t xmiddle, Double_t fy, Bool_t &ok) const
{
//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
Double_t eps = 1e-10;
Double_t t = 1e-8;
Int_t itermax = 100;
Double_t c = (3.-TMath::Sqrt(5.))/2.; //comes from golden section
Double_t u, v, w, x, fv, fu, fw, fx, e, p, q, r, t2, d=0, m, tol;
v = w = x = xmiddle;
e=0;
Double_t a=xmin;
Double_t b=xmax;
if (type < 2)
fv = fw = fx = Eval(x);
else if (type < 4)
fv = fw = fx = -Eval(x);
else
fv = fw = fx = TMath::Abs(Eval(x)-fy);
for (Int_t i=0; i<itermax; i++){
m=0.5*(a + b);
tol = eps*(TMath::Abs(x))+t;
t2 = 2*tol;
if (TMath::Abs(x-m) <= (t2-0.5*(b-a))) {
//converged, return x
ok=kTRUE;
if (type==1)
return fx;
else if (type==3)
return -fx;
else
return x;
}
if (TMath::Abs(e)>tol){
//fit parabola
r = (x-w)*(fx-fv);
q = (x-v)*(fx-fw);
p = (x-v)*q - (x-w)*r;
q = 2*(q-r);
if (q>0) p=-p;
else q=-q;
r=e;
e=d;
if (TMath::Abs(p) < TMath::Abs(0.5*q*r) || p < q*(a-x) || p < q*(b-x)) {
//a parabolic interpolation step
d = p/q;
u = x+d;
if (u-a < t2 || b-u < t2)
d=TMath::Sign(tol, m-x);
} else {
e=(x>=m ? a-x : b-x);
d = c*e;
}
} else {
e=(x>=m ? a-x : b-x);
d = c*e;
}
u = (TMath::Abs(d)>=tol ? x+d : x+TMath::Sign(tol, d));
if (type < 2)
fu = Eval(u);
else if (type < 4)
fu = -Eval(u);
else
fu = TMath::Abs(Eval(u)-fy);
//update a, b, v, w and x
if (fu<=fx){
if (u<x) b=x;
else a=x;
v=w; fv=fw; w=x; fw=fx; x=u; fx=fu;
} else {
if (u<x) a=u;
else b=u;
if (fu<=fw || w==x){
v=w; fv=fw; w=u; fw=fu;
}
else if (fu<=fv || v==x || v==w){
v=u; fv=fu;
}
}
}
//didn't converge
ok = kFALSE;
xmin = a;
xmax = b;
return x;
}
//______________________________________________________________________________
Int_t TF1::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.
if (fNDF == 0) return fNpfits-fNpar;
return fNDF;
}
//______________________________________________________________________________
Int_t TF1::GetNumberFreeParameters() const
{
// return the number of free parameters
Int_t nfree = fNpar;
Double_t al,bl;
for (Int_t i=0;i<fNpar;i++) {
((TF1*)this)->GetParLimits(i,al,bl);
if (al*bl != 0 && al >= bl) nfree--;
}
return nfree;
}
//______________________________________________________________________________
char *TF1::GetObjectInfo(Int_t px, Int_t /* py */) const
{
// Redefines TObject::GetObjectInfo.
// Displays the function info (x, function value
// corresponding to cursor position px,py
//
static char info[64];
Double_t x = gPad->PadtoX(gPad->AbsPixeltoX(px));
sprintf(info,"(x=%g, f=%g)",x,((TF1*)this)->Eval(x));
return info;
}
//______________________________________________________________________________
Double_t TF1::GetParError(Int_t ipar) const
{
//return value of parameter number ipar
if (ipar < 0 || ipar > fNpar-1) return 0;
return fParErrors[ipar];
}
//______________________________________________________________________________
void TF1::GetParLimits(Int_t ipar, Double_t &parmin, Double_t &parmax) const
{
//*-*-*-*-*-*Return limits for parameter ipar*-*-*-*
//*-* ================================
parmin = 0;
parmax = 0;
if (ipar < 0 || ipar > fNpar-1) return;
if (fParMin) parmin = fParMin[ipar];
if (fParMax) parmax = fParMax[ipar];
}
//______________________________________________________________________________
Double_t TF1::GetProb() const
{
// return the fit probability
if (fNDF <= 0) return 0;
return TMath::Prob(fChisquare,fNDF);
}
//______________________________________________________________________________
Int_t TF1::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");
const Int_t npx = TMath::Min(250,TMath::Max(50,2*nprobSum));
const Double_t xMin = GetXmin();
const Double_t xMax = GetXmax();
const Double_t dx = (xMax-xMin)/npx;
TArrayD integral(npx+1);
TArrayD alpha(npx);
TArrayD beta(npx);
TArrayD gamma(npx);
integral[0] = 0;
Int_t intNegative = 0;
Int_t i;
for (i = 0; i < npx; i++) {
const Double_t *params = 0;
Double_t integ = Integral(Double_t(xMin+i*dx),Double_t(xMin+i*dx+dx),params);
if (integ < 0) {intNegative++; integ = -integ;}
integral[i+1] = integral[i] + integ;
}
if (intNegative > 0)
Warning("GetQuantiles","function:%s has %d negative values: abs assumed",
GetName(),intNegative);
if (integral[npx] == 0) {
Error("GetQuantiles","Integral of function is zero");
return 0;
}
const Double_t total = integral[npx];
for (i = 1; i <= npx; i++) integral[i] /= total;
//the integral r for each bin is approximated by a parabola
// x = alpha + beta*r +gamma*r**2
// compute the coefficients alpha, beta, gamma for each bin
for (i = 0; i < npx; i++) {
const Double_t x0 = xMin+dx*i;
const Double_t r2 = integral[i+1]-integral[i];
const Double_t r1 = Integral(x0,x0+0.5*dx)/total;
gamma[i] = (2*r2-4*r1)/(dx*dx);
beta[i] = r2/dx-gamma[i]*dx;
alpha[i] = x0;
gamma[i] *= 2;
}
// Be careful because of finite precision in the integral; Use the fact that the integral
// is monotone increasing
for (i = 0; i < nprobSum; i++) {
const Double_t r = probSum[i];
Int_t bin = TMath::Max(TMath::BinarySearch(npx+1,integral.GetArray(),r)-1,(Long64_t)0);
while (bin < npx-1 && integral[bin+1] == r) {
if (integral[bin+2] == r) bin++;
else break;
}
const Double_t rr = r-integral[bin];
if (rr != 0.0) {
Double_t xx;
if (gamma[bin] && beta[bin]*beta[bin]+2*gamma[bin]*rr >= 0.0)
xx = (-beta[bin]+TMath::Sqrt(beta[bin]*beta[bin]+2*gamma[bin]*rr))/gamma[bin];
else
xx = rr/beta[bin];
q[i] = alpha[bin]+xx;
} else {
q[i] = alpha[bin];
if (integral[bin+1] == r) q[i] += dx;
}
}
return nprobSum;
}
//______________________________________________________________________________
Double_t TF1::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.
//*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-**-*-*-*-*-*-*-*
// Check if integral array must be build
if (fIntegral == 0) {
Double_t dx = (fXmax-fXmin)/fNpx;
fIntegral = new Double_t[fNpx+1];
fAlpha = new Double_t[fNpx];
fBeta = new Double_t[fNpx];
fGamma = new Double_t[fNpx];
fIntegral[0] = 0;
Double_t integ;
Int_t intNegative = 0;
Int_t i;
for (i=0;i<fNpx;i++) {
integ = Integral(Double_t(fXmin+i*dx), Double_t(fXmin+i*dx+dx));
if (integ < 0) {intNegative++; integ = -integ;}
fIntegral[i+1] = fIntegral[i] + integ;
}
if (intNegative > 0) {
Warning("GetRandom","function:%s has %d negative values: abs assumed",GetName(),intNegative);
}
if (fIntegral[fNpx] == 0) {
Error("GetRandom","Integral of function is zero");
return 0;
}
Double_t total = fIntegral[fNpx];
for (i=1;i<=fNpx;i++) { // normalize integral to 1
fIntegral[i] /= total;
}
//the integral r for each bin is approximated by a parabola
// x = alpha + beta*r +gamma*r**2
// compute the coefficients alpha, beta, gamma for each bin
Double_t x0,r1,r2,r3;
for (i=0;i<fNpx;i++) {
x0 = fXmin+i*dx;
r2 = fIntegral[i+1] - fIntegral[i];
r1 = Integral(x0,x0+0.5*dx)/total;
r3 = 2*r2 - 4*r1;
if (TMath::Abs(r3) > 1e-8) fGamma[i] = r3/(dx*dx);
else fGamma[i] = 0;
fBeta[i] = r2/dx - fGamma[i]*dx;
fAlpha[i] = x0;
fGamma[i] *= 2;
}
}
// return random number
Double_t r = gRandom->Rndm();
Int_t bin = TMath::BinarySearch(fNpx,fIntegral,r);
Double_t rr = r - fIntegral[bin];
Double_t xx;
if(fGamma[bin] != 0)
xx = (-fBeta[bin] + TMath::Sqrt(fBeta[bin]*fBeta[bin]+2*fGamma[bin]*rr))/fGamma[bin];
else
xx = rr/fBeta[bin];
Double_t x = fAlpha[bin] + xx;
return x;
}
//______________________________________________________________________________
Double_t TF1::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.
//*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-**-*-*-*-*-*-*-*
// Check if integral array must be build
if (fIntegral == 0) {
Double_t dx = (fXmax-fXmin)/fNpx;
fIntegral = new Double_t[fNpx+1];
fAlpha = new Double_t[fNpx];
fBeta = new Double_t[fNpx];
fGamma = new Double_t[fNpx];
fIntegral[0] = 0;
Double_t integ;
Int_t intNegative = 0;
Int_t i;
for (i=0;i<fNpx;i++) {
integ = Integral(Double_t(fXmin+i*dx), Double_t(fXmin+i*dx+dx));
if (integ < 0) {intNegative++; integ = -integ;}
fIntegral[i+1] = fIntegral[i] + integ;
}
if (intNegative > 0) {
Warning("GetRandom","function:%s has %d negative values: abs assumed",GetName(),intNegative);
}
if (fIntegral[fNpx] == 0) {
Error("GetRandom","Integral of function is zero");
return 0;
}
Double_t total = fIntegral[fNpx];
for (i=1;i<=fNpx;i++) { // normalize integral to 1
fIntegral[i] /= total;
}
//the integral r for each bin is approximated by a parabola
// x = alpha + beta*r +gamma*r**2
// compute the coefficients alpha, beta, gamma for each bin
Double_t x0,r1,r2,r3;
for (i=0;i<fNpx;i++) {
x0 = fXmin+i*dx;
r2 = fIntegral[i+1] - fIntegral[i];
r1 = Integral(x0,x0+0.5*dx)/total;
r3 = 2*r2 - 4*r1;
if (TMath::Abs(r3) > 1e-8) fGamma[i] = r3/(dx*dx);
else fGamma[i] = 0;
fBeta[i] = r2/dx - fGamma[i]*dx;
fAlpha[i] = x0;
fGamma[i] *= 2;
}
}
// return random number
Double_t dx = (fXmax-fXmin)/fNpx;
Int_t nbinmin = (Int_t)((xmin-fXmin)/dx);
Int_t nbinmax = (Int_t)((xmax-fXmin)/dx)+2;
if(nbinmax>fNpx) nbinmax=fNpx;
Double_t pmin=fIntegral[nbinmin];
Double_t pmax=fIntegral[nbinmax];
Double_t r,x,xx,rr;
do {
r = gRandom->Uniform(pmin,pmax);
Int_t bin = TMath::BinarySearch(fNpx,fIntegral,r);
rr = r - fIntegral[bin];
if(fGamma[bin] != 0)
xx = (-fBeta[bin] + TMath::Sqrt(fBeta[bin]*fBeta[bin]+2*fGamma[bin]*rr))/fGamma[bin];
else
xx = rr/fBeta[bin];
x = fAlpha[bin] + xx;
} while(x<xmin || x>xmax);
return x;
}
//______________________________________________________________________________
void TF1::GetRange(Double_t &xmin, Double_t &xmax) const
{
//*-*-*-*-*-*-*-*-*-*-*Return range of a 1-D function*-*-*-*-*-*-*-*-*-*-*-*
//*-* ==============================
xmin = fXmin;
xmax = fXmax;
}
//______________________________________________________________________________
void TF1::GetRange(Double_t &xmin, Double_t &ymin, Double_t &xmax, Double_t &ymax) const
{
//*-*-*-*-*-*-*-*-*-*-*Return range of a 2-D function*-*-*-*-*-*-*-*-*-*-*-*-*
//*-* ==============================
xmin = fXmin;
xmax = fXmax;
ymin = 0;
ymax = 0;
}
//______________________________________________________________________________
void TF1::GetRange(Double_t &xmin, Double_t &ymin, Double_t &zmin, Double_t &xmax, Double_t &ymax, Double_t &zmax) const
{
//*-*-*-*-*-*-*-*-*-*-*Return range of function*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
//*-* ========================
xmin = fXmin;
xmax = fXmax;
ymin = 0;
ymax = 0;
zmin = 0;
zmax = 0;
}
//______________________________________________________________________________
Double_t TF1::GetSave(const Double_t *xx)
{
// Get value corresponding to X in array of fSave values
if (fNsave <= 0) return 0;
if (fSave == 0) return 0;
Int_t np = fNsave - 3;
Double_t xmin = Double_t(fSave[np+1]);
Double_t xmax = Double_t(fSave[np+2]);
Double_t x = Double_t(xx[0]);
Double_t dx = (xmax-xmin)/np;
if (x < xmin || x > xmax) return 0;
if (dx <= 0) return 0;
Int_t bin = Int_t((x-xmin)/dx);
Double_t xlow = xmin + bin*dx;
Double_t xup = xlow + dx;
Double_t ylow = fSave[bin];
Double_t yup = fSave[bin+1];
Double_t y = ((xup*ylow-xlow*yup) + x*(yup-ylow))/dx;
return y;
}
//______________________________________________________________________________
TAxis *TF1::GetXaxis() const
{
// Get x axis of the function.
//if (!gPad) return 0;
TH1 *h = GetHistogram();
if (!h) return 0;
return h->GetXaxis();
}
//______________________________________________________________________________
TAxis *TF1::GetYaxis() const
{
// Get y axis of the function.
//if (!gPad) return 0;
TH1 *h = GetHistogram();
if (!h) return 0;
return h->GetYaxis();
}
//______________________________________________________________________________
TAxis *TF1::GetZaxis() const
{
// Get z axis of the function. (In case this object is a TF2 or TF3)
//if (!gPad) return 0;
TH1 *h = GetHistogram();
if (!h) return 0;
return h->GetZaxis();
}
//______________________________________________________________________________
void TF1::InitArgs(const Double_t *x, const Double_t *params)
{
//*-*-*-*-*-*-*-*-*-*-*Initialize parameters addresses*-*-*-*-*-*-*-*-*-*-*-*
//*-* ===============================
if (fMethodCall) {
Long_t args[2];
args[0] = (Long_t)x;
if (params) args[1] = (Long_t)params;
else args[1] = (Long_t)fParams;
fMethodCall->SetParamPtrs(args);
}
}
//______________________________________________________________________________
void TF1::InitStandardFunctions()
{
// Create the basic function objects
TF1 *f1;
if (!gROOT->GetListOfFunctions()->FindObject("gaus")) {
f1 = new TF1("gaus","gaus",-1,1); f1->SetParameters(1,0,1);
f1 = new TF1("gausn","gausn",-1,1); f1->SetParameters(1,0,1);
f1 = new TF1("landau","landau",-1,1); f1->SetParameters(1,0,1);
f1 = new TF1("landaun","landaun",-1,1); f1->SetParameters(1,0,1);
f1 = new TF1("expo","expo",-1,1); f1->SetParameters(1,1);
for (Int_t i=0;i<10;i++) {
f1 = new TF1(Form("pol%d",i),Form("pol%d",i),-1,1);
f1->SetParameters(1,1,1,1,1,1,1,1,1,1);
}
}
}
//______________________________________________________________________________
Double_t TF1::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.
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.
*/
//
//
// 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.
//
//=========================================================================
Double_t ctr[15], wth[15], wthl[15], z[15];
const Double_t xl2 = 0.358568582800318073;
const Double_t xl4 = 0.948683298050513796;
const Double_t xl5 = 0.688247201611685289;
const Double_t w2 = 980./6561;
const Double_t w4 = 200./19683;
const Double_t wp2 = 245./486;
const Double_t wp4 = 25./729;
Double_t wn1[14] = { -0.193872885230909911, -0.555606360818980835,
-0.876695625666819078, -1.15714067977442459, -1.39694152314179743,
-1.59609815576893754, -1.75461057765584494, -1.87247878880251983,
-1.94970278920896201, -1.98628257887517146, -1.98221815780114818,
-1.93750952598689219, -1.85215668343240347, -1.72615963013768225};
Double_t wn3[14] = { 0.0518213686937966768, 0.0314992633236803330,
0.0111771579535639891,-0.00914494741655235473,-0.0294670527866686986,
-0.0497891581567850424,-0.0701112635269013768, -0.0904333688970177241,
-0.110755474267134071, -0.131077579637250419, -0.151399685007366752,
-0.171721790377483099, -0.192043895747599447, -0.212366001117715794};
Double_t wn5[14] = { 0.871183254585174982e-01, 0.435591627292587508e-01,
0.217795813646293754e-01, 0.108897906823146873e-01, 0.544489534115734364e-02,
0.272244767057867193e-02, 0.136122383528933596e-02, 0.680611917644667955e-03,
0.340305958822333977e-03, 0.170152979411166995e-03, 0.850764897055834977e-04,
0.425382448527917472e-04, 0.212691224263958736e-04, 0.106345612131979372e-04};
Double_t wpn1[14] = { -1.33196159122085045, -2.29218106995884763,
-3.11522633744855959, -3.80109739368998611, -4.34979423868312742,
-4.76131687242798352, -5.03566529492455417, -5.17283950617283939,
-5.17283950617283939, -5.03566529492455417, -4.76131687242798352,
-4.34979423868312742, -3.80109739368998611, -3.11522633744855959};
Double_t wpn3[14] = { 0.0445816186556927292, -0.0240054869684499309,
-0.0925925925925925875, -0.161179698216735251, -0.229766803840877915,
-0.298353909465020564, -0.366941015089163228, -0.435528120713305891,
-0.504115226337448555, -0.572702331961591218, -0.641289437585733882,
-0.709876543209876532, -0.778463648834019195, -0.847050754458161859};
Double_t result = 0;
Double_t abserr = 0;
ifail = 3;
nfnevl = 0;
relerr = 0;
if (n < 2 || n > 15) return 0;
Double_t twondm = TMath::Power(2,n);
Int_t ifncls = 0;
Bool_t ldv = kFALSE;
Int_t irgnst = 2*n+3;
Int_t irlcls = Int_t(twondm) +2*n*(n+1)+1;
Int_t isbrgn = irgnst;
Int_t isbrgs = irgnst;
if (minpts < 1) minpts = irlcls;
if (maxpts < minpts) maxpts = 10*minpts;
// The original agorithm expected a working space array WK of length IWK
// with IWK Length ( >= (2N + 3) * (1 + MAXPTS/(2**N + 2N(N + 1) + 1))/2).
// Here, this array is allocated dynamically
Int_t iwk = irgnst*(1 +maxpts/irlcls)/2;
Double_t *wk = new Double_t[iwk+10];
Int_t j;
for (j=0;j<n;j++) {
ctr[j] = (b[j] + a[j])*0.5;
wth[j] = (b[j] - a[j])*0.5;
}
Double_t rgnvol, sum1, sum2, sum3, sum4, sum5, difmax, f2, f3, dif, aresult;
Double_t rgncmp=0, rgnval, rgnerr;
Int_t j1, k, l, m, idvaxn=0, idvax0=0, isbtmp, isbtpp;
InitArgs(z,fParams);
L20:
rgnvol = twondm;
for (j=0;j<n;j++) {
rgnvol *= wth[j];
z[j] = ctr[j];
}
sum1 = EvalPar(z,fParams); //evaluate function
difmax = 0;
sum2 = 0;
sum3 = 0;
for (j=0;j<n;j++) {
z[j] = ctr[j] - xl2*wth[j];
if (fgAbsValue) f2 = TMath::Abs(EvalPar(z,fParams));
else f2 = EvalPar(z,fParams);
z[j] = ctr[j] + xl2*wth[j];
if (fgAbsValue) f2 += TMath::Abs(EvalPar(z,fParams));
else f2 += EvalPar(z,fParams);
wthl[j] = xl4*wth[j];
z[j] = ctr[j] - wthl[j];
if (fgAbsValue) f3 = TMath::Abs(EvalPar(z,fParams));
else f3 = EvalPar(z,fParams);
z[j] = ctr[j] + wthl[j];
if (fgAbsValue) f3 += TMath::Abs(EvalPar(z,fParams));
else f3 += EvalPar(z,fParams);
sum2 += f2;
sum3 += f3;
dif = TMath::Abs(7*f2-f3-12*sum1);
if (dif >= difmax) {
difmax=dif;
idvaxn=j+1;
}
z[j] = ctr[j];
}
sum4 = 0;
for (j=1;j<n;j++) {
j1 = j-1;
for (k=j;k<n;k++) {
for (l=0;l<2;l++) {
wthl[j1] = -wthl[j1];
z[j1] = ctr[j1] + wthl[j1];
for (m=0;m<2;m++) {
wthl[k] = -wthl[k];
z[k] = ctr[k] + wthl[k];
if (fgAbsValue) sum4 += TMath::Abs(EvalPar(z,fParams));
else sum4 += EvalPar(z,fParams);
}
}
z[k] = ctr[k];
}
z[j1] = ctr[j1];
}
sum5 = 0;
for (j=0;j<n;j++) {
wthl[j] = -xl5*wth[j];
z[j] = ctr[j] + wthl[j];
}
L90:
if (fgAbsValue) sum5 += TMath::Abs(EvalPar(z,fParams));
else sum5 += EvalPar(z,fParams);
for (j=0;j<n;j++) {
wthl[j] = -wthl[j];
z[j] = ctr[j] + wthl[j];
if (wthl[j] > 0) goto L90;
}
rgncmp = rgnvol*(wpn1[n-2]*sum1+wp2*sum2+wpn3[n-2]*sum3+wp4*sum4);
rgnval = wn1[n-2]*sum1+w2*sum2+wn3[n-2]*sum3+w4*sum4+wn5[n-2]*sum5;
rgnval *= rgnvol;
rgnerr = TMath::Abs(rgnval-rgncmp);
result += rgnval;
abserr += rgnerr;
ifncls += irlcls;
aresult = TMath::Abs(result);
//if (result > 0 && aresult< 1e-100) {
// delete [] wk;
// ifail = 0; //function is probably symmetric ==> integral is null: not an error
// return result;
//}
if (ldv) {
L110:
isbtmp = 2*isbrgn;
if (isbtmp > isbrgs) goto L160;
if (isbtmp < isbrgs) {
isbtpp = isbtmp + irgnst;
if (wk[isbtmp-1] < wk[isbtpp-1]) isbtmp = isbtpp;
}
if (rgnerr >= wk[isbtmp-1]) goto L160;
for (k=0;k<irgnst;k++) {
wk[isbrgn-k-1] = wk[isbtmp-k-1];
}
isbrgn = isbtmp;
goto L110;
}
L140:
isbtmp = (isbrgn/(2*irgnst))*irgnst;
if (isbtmp >= irgnst && rgnerr > wk[isbtmp-1]) {
for (k=0;k<irgnst;k++) {
wk[isbrgn-k-1] = wk[isbtmp-k-1];
}
isbrgn = isbtmp;
goto L140;
}
L160:
wk[isbrgn-1] = rgnerr;
wk[isbrgn-2] = rgnval;
wk[isbrgn-3] = Double_t(idvaxn);
for (j=0;j<n;j++) {
isbtmp = isbrgn-2*j-4;
wk[isbtmp] = ctr[j];
wk[isbtmp-1] = wth[j];
}
if (ldv) {
ldv = kFALSE;
ctr[idvax0-1] += 2*wth[idvax0-1];
isbrgs += irgnst;
isbrgn = isbrgs;
goto L20;
}
relerr = abserr/aresult;
if (relerr < 1e-1 && aresult < 1e-20) ifail = 0;
if (relerr < 1e-3 && aresult < 1e-10) ifail = 0;
if (relerr < 1e-5 && aresult < 1e-5) ifail = 0;
if (isbrgs+irgnst > iwk) ifail = 2;
if (ifncls+2*irlcls > maxpts) ifail = 1;
if (relerr < eps && ifncls >= minpts) ifail = 0;
if (ifail == 3) {
ldv = kTRUE;
isbrgn = irgnst;
abserr -= wk[isbrgn-1];
result -= wk[isbrgn-2];
idvax0 = Int_t(wk[isbrgn-3]);
for (j=0;j<n;j++) {
isbtmp = isbrgn-2*j-4;
ctr[j] = wk[isbtmp];
wth[j] = wk[isbtmp-1];
}
wth[idvax0-1] = 0.5*wth[idvax0-1];
ctr[idvax0-1] -= wth[idvax0-1];
goto L20;
}
delete [] wk;
nfnevl = ifncls; //number of function evaluations performed.
return result; //an approximate value of the integral
}
//______________________________________________________________________________
Bool_t TF1::IsInside(const Double_t *x) const
{
// Return kTRUE is the point is inside the function range
if (x[0] < fXmin || x[0] > fXmax) return kFALSE;
return kTRUE;
}
//______________________________________________________________________________
void TF1::Paint(Option_t *option)
{
//*-*-*-*-*-*-*-*-*-*-*Paint this function with its current attributes*-*-*-*-*
//*-* ===============================================
Int_t i;
Double_t xv[1];
fgCurrent = this;
TString opt = option;
opt.ToLower();
Double_t xmin=fXmin, xmax=fXmax, pmin=fXmin, pmax=fXmax;
if (gPad) {
pmin = gPad->PadtoX(gPad->GetUxmin());
pmax = gPad->PadtoX(gPad->GetUxmax());
}
if (opt.Contains("same")) {
if (xmax < pmin) return; // Otto: completely outside
if (xmin > pmax) return;
if (xmin < pmin) xmin = pmin;
if (xmax > pmax) xmax = pmax;
}
// Create a temporary histogram and fill each channel with the function value
// Preserve axis titles
TString xtitle = "";
TString ytitle = "";
if (fHistogram) {
xtitle = fHistogram->GetXaxis()->GetTitle();
ytitle = fHistogram->GetYaxis()->GetTitle();
if (!gPad->GetLogx() && fHistogram->TestBit(TH1::kLogX)) { delete fHistogram; fHistogram = 0;}
if ( gPad->GetLogx() && !fHistogram->TestBit(TH1::kLogX)) { delete fHistogram; fHistogram = 0;}
}
if (fHistogram) {
fHistogram->GetXaxis()->SetLimits(xmin,xmax);
} else {
// if logx, we must bin in logx and not in x !!!
// otherwise if several decades, one gets crazy results
if (xmin > 0 && gPad && gPad->GetLogx()) {
Axis_t *xbins = new Axis_t[fNpx+1];
Double_t xlogmin = TMath::Log10(xmin);
Double_t xlogmax = TMath::Log10(xmax);
Double_t dlogx = (xlogmax-xlogmin)/((Double_t)fNpx);
for (i=0;i<=fNpx;i++) {
xbins[i] = gPad->PadtoX(xlogmin+ i*dlogx);
}
fHistogram = new TH1D("Func",GetTitle(),fNpx,xbins);
fHistogram->SetBit(TH1::kLogX);
delete [] xbins;
} else {
fHistogram = new TH1D("Func",GetTitle(),fNpx,xmin,xmax);
}
if (!fHistogram) return;
if (fMinimum != -1111) fHistogram->SetMinimum(fMinimum);
if (fMaximum != -1111) fHistogram->SetMaximum(fMaximum);
fHistogram->SetDirectory(0);
}
//restore axis titles
fHistogram->GetXaxis()->SetTitle(xtitle.Data());
fHistogram->GetYaxis()->SetTitle(ytitle.Data());
InitArgs(xv,fParams);
for (i=1;i<=fNpx;i++) {
xv[0] = fHistogram->GetBinCenter(i);
fHistogram->SetBinContent(i,EvalPar(xv,fParams));
}
//*-*- Copy Function attributes to histogram attributes
Double_t minimum = fHistogram->GetMinimumStored();
Double_t maximum = fHistogram->GetMaximumStored();
if (minimum <= 0 && gPad && gPad->GetLogy()) minimum = -1111; //this can happen when switching from lin to log scale
if (minimum == -1111) { //this can happen after unzooming
if (fHistogram->TestBit(TH1::kIsZoomed)) {
minimum = fHistogram->GetYaxis()->GetXmin();
} else {
minimum = fMinimum;
if (minimum == -1111) {
Double_t hmin = fHistogram->GetMinimum();
if (hmin > 0) {
Double_t hmax = fHistogram->GetMaximum();
hmin -= 0.05*(hmax-hmin);
if (hmin < 0) hmin = 0;
if (hmin <= 0 && gPad && gPad->GetLogy()) hmin = 0.001*hmax;
minimum = hmin;
}
}
}
fHistogram->SetMinimum(minimum);
}
if (maximum == -1111) {
if (fHistogram->TestBit(TH1::kIsZoomed)) {
maximum = fHistogram->GetYaxis()->GetXmax();
} else {
maximum = fMaximum;
}
fHistogram->SetMaximum(maximum);
}
fHistogram->SetBit(TH1::kNoStats);
fHistogram->SetLineColor(GetLineColor());
fHistogram->SetLineStyle(GetLineStyle());
fHistogram->SetLineWidth(GetLineWidth());
fHistogram->SetFillColor(GetFillColor());
fHistogram->SetFillStyle(GetFillStyle());
fHistogram->SetMarkerColor(GetMarkerColor());
fHistogram->SetMarkerStyle(GetMarkerStyle());
fHistogram->SetMarkerSize(GetMarkerSize());
//*-*- Draw the histogram
if (!gPad) return;
if (opt.Length() == 0) fHistogram->Paint("lf");
else if (opt == "same") fHistogram->Paint("lfsame");
else fHistogram->Paint(option);
}
//______________________________________________________________________________
void TF1::Print(Option_t *option) const
{
//*-*-*-*-*-*-*-*-*-*-*Dump this function with its attributes*-*-*-*-*-*-*-*-*-*
//*-* ==================================
TFormula::Print(option);
if (fHistogram) fHistogram->Print(option);
}
//______________________________________________________________________________
void TF1::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.
if (ipar < 0 || ipar > fNpar-1) return;
SetParLimits(ipar,0,0);
}
//______________________________________________________________________________
void TF1::Save(Double_t xmin, Double_t xmax, Double_t, Double_t, Double_t, Double_t)
{
// Save values of function in array fSave
if (fSave != 0) {delete [] fSave; fSave = 0;}
fNsave = fNpx+3;
if (fNsave <= 3) {fNsave=0; return;}
fSave = new Double_t[fNsave];
Int_t i;
Double_t dx = (xmax-xmin)/fNpx;
if (dx <= 0) {
dx = (fXmax-fXmin)/fNpx;
fNsave--;
xmin = fXmin +0.5*dx;
xmax = fXmax -0.5*dx;
}
Double_t xv[1];
InitArgs(xv,fParams);
for (i=0;i<=fNpx;i++) {
xv[0] = xmin + dx*i;
fSave[i] = EvalPar(xv,fParams);
}
fSave[fNpx+1] = xmin;
fSave[fNpx+2] = xmax;
}
//______________________________________________________________________________
void TF1::SavePrimitive(ofstream &out, Option_t *option)
{
// Save primitive as a C++ statement(s) on output stream out
Int_t i;
char quote = '"';
out<<" "<<endl;
if (!fMethodCall) {
out<<" TF1 *"<<GetName()<<" = new TF1("<<quote<<GetName()<<quote<<","<<quote<<GetExpFormula()<<quote<<","<<fXmin<<","<<fXmax<<");"<<endl;
if (fNpx != 100) {
out<<" "<<GetName()<<"->SetNpx("<<fNpx<<");"<<endl;
}
} else {
out<<" TF1 *"<<GetName()<<" = new TF1("<<quote<<"*"<<GetName()<<quote<<","<<fXmin<<","<<fXmax<<","<<GetNpar()<<");"<<endl;
out<<" //The original function : "<<GetTitle()<<" had originally been created by:" <<endl;
out<<" //TF1 *"<<GetName()<<" = new TF1("<<quote<<GetName()<<quote<<","<<GetTitle()<<","<<fXmin<<","<<fXmax<<","<<GetNpar()<<");"<<endl;
out<<" "<<GetName()<<"->SetRange("<<fXmin<<","<<fXmax<<");"<<endl;
out<<" "<<GetName()<<"->SetName("<<quote<<GetName()<<quote<<");"<<endl;
out<<" "<<GetName()<<"->SetTitle("<<quote<<GetTitle()<<quote<<");"<<endl;
if (fNpx != 100) {
out<<" "<<GetName()<<"->SetNpx("<<fNpx<<");"<<endl;
}
Double_t dx = (fXmax-fXmin)/fNpx;
Double_t xv[1];
InitArgs(xv,fParams);
for (i=0;i<=fNpx;i++) {
xv[0] = fXmin + dx*i;
Double_t save = EvalPar(xv,fParams);
out<<" "<<GetName()<<"->SetSavedPoint("<<i<<","<<save<<");"<<endl;
}
out<<" "<<GetName()<<"->SetSavedPoint("<<fNpx+1<<","<<fXmin<<");"<<endl;
out<<" "<<GetName()<<"->SetSavedPoint("<<fNpx+2<<","<<fXmax<<");"<<endl;
}
if (TestBit(kNotDraw)) {
out<<" "<<GetName()<<"->SetBit(TF1::kNotDraw);"<<endl;
}
if (GetFillColor() != 0) {
if (GetFillColor() > 228) {
TColor::SaveColor(out, GetFillColor());
out<<" "<<GetName()<<"->SetFillColor(ci);" << endl;
} else
out<<" "<<GetName()<<"->SetFillColor("<<GetFillColor()<<");"<<endl;
}
if (GetFillStyle() != 1001) {
out<<" "<<GetName()<<"->SetFillStyle("<<GetFillStyle()<<");"<<endl;
}
if (GetMarkerColor() != 1) {
if (GetMarkerColor() > 228) {
TColor::SaveColor(out, GetMarkerColor());
out<<" "<<GetName()<<"->SetMarkerColor(ci);" << endl;
} else
out<<" "<<GetName()<<"->SetMarkerColor("<<GetMarkerColor()<<");"<<endl;
}
if (GetMarkerStyle() != 1) {
out<<" "<<GetName()<<"->SetMarkerStyle("<<GetMarkerStyle()<<");"<<endl;
}
if (GetMarkerSize() != 1) {
out<<" "<<GetName()<<"->SetMarkerSize("<<GetMarkerSize()<<");"<<endl;
}
if (GetLineColor() != 1) {
if (GetLineColor() > 228) {
TColor::SaveColor(out, GetLineColor());
out<<" "<<GetName()<<"->SetLineColor(ci);" << endl;
} else
out<<" "<<GetName()<<"->SetLineColor("<<GetLineColor()<<");"<<endl;
}
if (GetLineWidth() != 4) {
out<<" "<<GetName()<<"->SetLineWidth("<<GetLineWidth()<<");"<<endl;
}
if (GetLineStyle() != 1) {
out<<" "<<GetName()<<"->SetLineStyle("<<GetLineStyle()<<");"<<endl;
}
if (GetChisquare() != 0) {
out<<" "<<GetName()<<"->SetChisquare("<<GetChisquare()<<");"<<endl;
}
GetXaxis()->SaveAttributes(out,GetName(),"->GetXaxis()");
GetYaxis()->SaveAttributes(out,GetName(),"->GetYaxis()");
Double_t parmin, parmax;
for (i=0;i<fNpar;i++) {
out<<" "<<GetName()<<"->SetParameter("<<i<<","<<GetParameter(i)<<");"<<endl;
out<<" "<<GetName()<<"->SetParError("<<i<<","<<GetParError(i)<<");"<<endl;
GetParLimits(i,parmin,parmax);
out<<" "<<GetName()<<"->SetParLimits("<<i<<","<<parmin<<","<<parmax<<");"<<endl;
}
if (!strstr(option,"nodraw")) {
out<<" "<<GetName()<<"->Draw("
<<quote<<option<<quote<<");"<<endl;
}
}
//______________________________________________________________________________
void TF1::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.
fgCurrent = f1;
}
//______________________________________________________________________________
void TF1::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
fMaximum = maximum;
if (fHistogram) fHistogram->SetMaximum(maximum);
if (gPad) gPad->Modified();
}
//______________________________________________________________________________
void TF1::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
fMinimum = minimum;
if (fHistogram) fHistogram->SetMinimum(minimum);
if (gPad) gPad->Modified();
}
//______________________________________________________________________________
void TF1::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
fNDF = ndf;
}
//______________________________________________________________________________
void TF1::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
if (npx < 4) {
Warning("SetNpx","Number of points must be >4 && < 100000, fNpx set to 4");
fNpx = 4;
} else if(npx > 100000) {
Warning("SetNpx","Number of points must be >4 && < 100000, fNpx set to 100000");
fNpx = 100000;
} else {
fNpx = npx;
}
Update();
}
//______________________________________________________________________________
void TF1::SetParError(Int_t ipar, Double_t error)
{
// set error for parameter number ipar
if (ipar < 0 || ipar > fNpar-1) return;
fParErrors[ipar] = error;
}
//______________________________________________________________________________
void TF1::SetParErrors(const Double_t *errors)
{
// set errors for all active parameters
// when calling this function, the array errors must have at least fNpar values
if (!errors) return;
for (Int_t i=0;i<fNpar;i++) fParErrors[i] = errors[i];
}
//______________________________________________________________________________
void TF1::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
if (ipar < 0 || ipar > fNpar-1) return;
Int_t i;
if (!fParMin) {fParMin = new Double_t[fNpar]; for (i=0;i<fNpar;i++) fParMin[i]=0;}
if (!fParMax) {fParMax = new Double_t[fNpar]; for (i=0;i<fNpar;i++) fParMax[i]=0;}
fParMin[ipar] = parmin;
fParMax[ipar] = parmax;
}
//______________________________________________________________________________
void TF1::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.
fXmin = xmin;
fXmax = xmax;
Update();
}
//______________________________________________________________________________
void TF1::SetSavedPoint(Int_t point, Double_t value)
{
// Restore value of function saved at point
if (!fSave) {
fNsave = fNpx+3;
fSave = new Double_t[fNsave];
}
if (point < 0 || point >= fNsave) return;
fSave[point] = value;
}
//_______________________________________________________________________
void TF1::Streamer(TBuffer &b)
{
//*-*-*-*-*-*-*-*-*Stream a class object*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
//*-* =========================================
if (b.IsReading()) {
UInt_t R__s, R__c;
Version_t v = b.ReadVersion(&R__s, &R__c);
if (v > 4) {
TF1::Class()->ReadBuffer(b, this, v, R__s, R__c);
if (v == 5 && fNsave > 0) {
//correct badly saved fSave in 3.00/06
Int_t np = fNsave - 3;
fSave[np] = fSave[np-1];
fSave[np+1] = fXmin;
fSave[np+2] = fXmax;
}
return;
}
//====process old versions before automatic schema evolution
TFormula::Streamer(b);
TAttLine::Streamer(b);
TAttFill::Streamer(b);
TAttMarker::Streamer(b);
if (v < 4) {
Float_t xmin,xmax;
b >> xmin; fXmin = xmin;
b >> xmax; fXmax = xmax;
} else {
b >> fXmin;
b >> fXmax;
}
b >> fNpx;
b >> fType;
b >> fChisquare;
b.ReadArray(fParErrors);
if (v > 1) {
b.ReadArray(fParMin);
b.ReadArray(fParMax);
} else {
fParMin = new Double_t[fNpar+1];
fParMax = new Double_t[fNpar+1];
}
b >> fNpfits;
if (v == 1) {
b >> fHistogram;
delete fHistogram; fHistogram = 0;
}
if (v > 1) {
if (v < 4) {
Float_t minimum,maximum;
b >> minimum; fMinimum =minimum;
b >> maximum; fMaximum =maximum;
} else {
b >> fMinimum;
b >> fMaximum;
}
}
if (v > 2) {
b >> fNsave;
if (fNsave > 0) {
fSave = new Double_t[fNsave+10];
b.ReadArray(fSave);
//correct fSave limits to match new version
fSave[fNsave] = fSave[fNsave-1];
fSave[fNsave+1] = fSave[fNsave+2];
fSave[fNsave+2] = fSave[fNsave+3];
fNsave += 3;
} else fSave = 0;
}
b.CheckByteCount(R__s, R__c, TF1::IsA());
//====end of old versions
} else {
Int_t saved = 0;
if (fType > 0 && fNsave <= 0) { saved = 1; Save(fXmin,fXmax,0,0,0,0);}
TF1::Class()->WriteBuffer(b,this);
if (saved) {delete [] fSave; fSave = 0; fNsave = 0;}
}
}
//_______________________________________________________________________
void TF1::Update()
{
// called by functions such as SetRange, SetNpx, SetParameters
// to force the deletion of the associated histogram or Integral
delete fHistogram;
fHistogram = 0;
if (fIntegral) {
delete [] fIntegral; fIntegral = 0;
delete [] fAlpha; fAlpha = 0;
delete [] fBeta; fBeta = 0;
delete [] fGamma; fGamma = 0;
}
}
//_______________________________________________________________________
void TF1::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.
fgRejectPoint = reject;
}
//_______________________________________________________________________
Bool_t TF1::RejectedPoint()
{
// see TF1::RejectPoint above
return fgRejectPoint;
}
//______________________________________________________________________________
Double_t TF1_ExpValHelperx(Double_t *x, Double_t *par) {
return x[0]*gHelper->EvalPar(x,par);
}
//______________________________________________________________________________
Double_t TF1_ExpValHelper(Double_t *x, Double_t *par) {
Int_t npar = gHelper->GetNpar();
Double_t xbar = par[npar];
Double_t n = par[npar+1];
return TMath::Power(x[0]-xbar,n)*gHelper->EvalPar(x,par);
}
//______________________________________________________________________________
Double_t TF1::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>
fgAbsValue = kTRUE;
Double_t norm = Integral(a,b,params,epsilon);
if (norm == 0) {
fgAbsValue = kFALSE;
Error("Moment", "Integral zero over range");
return 0;
}
gHelper = this;
//TF1 fnc("TF1_ExpValHelper",Form("%s*pow(x,%f)",GetName(),n));
TF1 fnc("TF1_ExpValHelper",TF1_ExpValHelper,fXmin,fXmax,fNpar+2);
for (Int_t i=0;i<fNpar;i++) {
if(params) fnc.SetParameter(i,params[i]);
else fnc.SetParameter(i,fParams[i]);
}
fnc.SetParameter(fNpar,0);
fnc.SetParameter(fNpar+1,n);
Double_t res = fnc.Integral(a,b,params,epsilon)/norm;
fgAbsValue = kFALSE;
return res;
}
//______________________________________________________________________________
Double_t TF1::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>
fgAbsValue = kTRUE;
Double_t norm = Integral(a,b,params,epsilon);
if (norm == 0) {
fgAbsValue = kFALSE;
Error("CentralMoment", "Integral zero over range");
return 0;
}
gHelper = this;
//TF1 fncx("TF1_ExpValHelperx",Form("%s*x",GetName()));
TF1 fncx("TF1_ExpValHelperx",TF1_ExpValHelperx,fXmin,fXmax,fNpar);
Int_t i;
for (i=0;i<fNpar;i++) {
if(params) fncx.SetParameter(i,params[i]);
else fncx.SetParameter(i,fParams[i]);
}
Double_t xbar = fncx.Integral(a,b,params,epsilon)/norm;
//TF1 fnc("TF1_ExpValHelper",Form("%s*pow(x-%f,%f)",GetName(),xbar,n));
TF1 fnc("TF1_ExpValHelper",TF1_ExpValHelper,fXmin,fXmax,fNpar+2);
for (i=0;i<fNpar;i++) {
if(params) fnc.SetParameter(i,params[i]);
else fnc.SetParameter(i,fParams[i]);
}
fnc.SetParameter(fNpar,0);
fnc.SetParameter(fNpar+1,n);
fnc.SetParameter(fNpar,xbar);
fnc.SetParameter(fNpar+1,n);
Double_t res = fnc.Integral(a,b,params,epsilon)/norm;
fgAbsValue = kFALSE;
return res;
}
//--------------------------------------------------------------------
// some useful static utility functions to compute sampling points for IntegralFast
//--------------------------------------------------------------------
//______________________________________________________________________________
#ifdef INTHEFUTURE
void TF1::CalcGaussLegendreSamplingPoints(TGraph *g, Double_t eps)
{
//type safe interface (static method)
// The number of sampling points are taken from the TGraph
if (!g) return;
CalcGaussLegendreSamplingPoints(g->GetN(), g->GetX(), g->GetY(), eps);
}
//______________________________________________________________________________
TGraph *TF1::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
if (num<=0)
return 0;
TGraph *g = new TGraph(num);
CalcGaussLegendreSamplingPoints(g->GetN(), g->GetX(), g->GetY(), eps);
return g;
}
#endif
//______________________________________________________________________________
void TF1::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
//
if (num<=0 || eps<=0)
return;
// The roots of symmetric is the interval, so we only have to find half of them
const UInt_t m = (num+1)/2;
Double_t z, pp, p1,p2, p3;
// Loop over the disired roots
for (UInt_t i=0; i<m; i++) {
z = TMath::Cos(TMath::Pi()*(i+0.75)/(num+0.5));
// Starting with the above approximation to the i-th root, we enter
// the main loop of refinement by Newton's method
do {
p1=1.0;
p2=0.0;
// Loop up the recurrence relation to get the Legendre
// polynomial evaluated at z
for (int j=0; j<num; j++)
{
p3 = p2;
p2 = p1;
p1 = ((2.0*j+1.0)*z*p2-j*p3)/(j+1.0);
}
// p1 is now the desired Legendre polynomial. We next compute pp, its
// derivative, by a standard relation involving also p2, the polynomial
// of one lower order
pp = num*(z*p1-p2)/(z*z-1.0);
// Newton's method
z -= p1/pp;
} while (TMath::Abs(p1/pp) > eps);
// Put root and its symmetric counterpart
x[i] = -z;
x[num-i-1] = z;
// Compute the weight and put its symmetric counterpart
w[i] = 2.0/((1.0-z*z)*pp*pp);
w[num-i-1] = w[i];
}
}