[u/mrichter/AliRoot.git] / RALICE / AliMath.cxx

/**************************************************************************
 * Copyright(c) 1998-1999, ALICE Experiment at CERN, All rights reserved. *
 *                                                                        *
 * Author: The ALICE Off-line Project.                                    *
 * Contributors are mentioned in the code where appropriate.              *
 *                                                                        *
 * Permission to use, copy, modify and distribute this software and its   *
 * documentation strictly for non-commercial purposes is hereby granted   *
 * without fee, provided that the above copyright notice appears in all   *
 * copies and that both the copyright notice and this permission notice   *
 * appear in the supporting documentation. The authors make no claims     *
 * about the suitability of this software for any purpose. It is          *
 * provided "as is" without express or implied warranty.                  *
 **************************************************************************/

// $Id$

///////////////////////////////////////////////////////////////////////////
// Class AliMath
// Various mathematical tools which may be very convenient while
// performing physics analysis.
//
// Example : Probability of a Chi-squared value
// =========
//
// AliMath M;
// Float_t chi2=20;            // The chi-squared value
// Int_t ndf=12;               // The number of degrees of freedom
// Float_t p=M.Prob(chi2,ndf); // The probability that at least a Chi-squared
//                             // value of chi2 will be observed, even for a
//                             // correct model
//
//--- Author: Nick van Eijndhoven 14-nov-1998 UU-SAP Utrecht
//- Modified: NvE $Date$ UU-SAP Utrecht
///////////////////////////////////////////////////////////////////////////

#include "AliMath.h"
#include "Riostream.h"
 
ClassImp(AliMath) // Class implementation to enable ROOT I/O
 
AliMath::AliMath() : TObject()
{
// Default constructor
}
///////////////////////////////////////////////////////////////////////////
AliMath::~AliMath()
{
// Destructor
}
///////////////////////////////////////////////////////////////////////////
AliMath::AliMath(const AliMath& m) : TObject(m)
{
// Copy constructor
}
///////////////////////////////////////////////////////////////////////////
Double_t AliMath::Gamma(Double_t z) const
{
// Computation of gamma(z) for all z>0.
//
// The algorithm is based on the article by C.Lanczos [1] as denoted in
// Numerical Recipes 2nd ed. on p. 207 (W.H.Press et al.).
//
// [1] C.Lanczos, SIAM Journal of Numerical Analysis B1 (1964), 86.
//
//--- Nve 14-nov-1998 UU-SAP Utrecht
 
 if (z<=0.)
 {
  cout << "*Gamma(z)* Wrong argument z = " << z << endl;
  return 0;
 }
 
 Double_t v=LnGamma(z);
 return exp(v);
}
///////////////////////////////////////////////////////////////////////////
Double_t AliMath::Gamma(Double_t a,Double_t x) const
{
// Computation of the incomplete gamma function P(a,x)
//
// The algorithm is based on the formulas and code as denoted in
// Numerical Recipes 2nd ed. on p. 210-212 (W.H.Press et al.).
//
//--- Nve 14-nov-1998 UU-SAP Utrecht
 
 if (a<=0.)
 {
  cout << "*Gamma(a,x)* Invalid argument a = " << a << endl;
  return 0;
 }
 
 if (x<=0.)
 {
  if (x<0) cout << "*Gamma(a,x)* Invalid argument x = " << x << endl;
  return 0;
 }
 
 if (x<(a+1.))
 {
  return GamSer(a,x);
 }
 else
 {
  return GamCf(a,x);
 }
}
///////////////////////////////////////////////////////////////////////////
Double_t AliMath::LnGamma(Double_t z) const
{
// Computation of ln[gamma(z)] for all z>0.
//
// The algorithm is based on the article by C.Lanczos [1] as denoted in
// Numerical Recipes 2nd ed. on p. 207 (W.H.Press et al.).
//
// [1] C.Lanczos, SIAM Journal of Numerical Analysis B1 (1964), 86.
//
// The accuracy of the result is better than 2e-10.
//
//--- Nve 14-nov-1998 UU-SAP Utrecht
 
 if (z<=0.)
 {
  cout << "*LnGamma(z)* Wrong argument z = " << z << endl;
  return 0;
 }
 
 // Coefficients for the series expansion
 Double_t c[7];
 c[0]=  2.5066282746310005;
 c[1]= 76.18009172947146;
 c[2]=-86.50532032941677;
 c[3]= 24.01409824083091;
 c[4]= -1.231739572450155;
 c[5]=  0.1208650973866179e-2;
 c[6]= -0.5395239384953e-5;
 
 Double_t x=z;
 Double_t y=x;
 Double_t tmp=x+5.5;
 tmp=(x+0.5)*log(tmp)-tmp;
 Double_t ser=1.000000000190015;
 for (Int_t i=1; i<7; i++)
 {
  y+=1.;
  ser+=c[i]/y;
 }
 Double_t v=tmp+log(c[0]*ser/x);
 return v;
}
///////////////////////////////////////////////////////////////////////////
Double_t AliMath::GamSer(Double_t a,Double_t x) const
{
// Computation of the incomplete gamma function P(a,x)
// via its series representation.
//
// The algorithm is based on the formulas and code as denoted in
// Numerical Recipes 2nd ed. on p. 210-212 (W.H.Press et al.).
//
//--- Nve 14-nov-1998 UU-SAP Utrecht
 
 Int_t itmax=100;   // Maximum number of iterations
 Double_t eps=3.e-7; // Relative accuracy
 
 if (a<=0.)
 {
  cout << "*GamSer(a,x)* Invalid argument a = " << a << endl;
  return 0;
 }
 
 if (x<=0.)
 {
  if (x<0) cout << "*GamSer(a,x)* Invalid argument x = " << x << endl;
  return 0;
 }
 
 Double_t gln=LnGamma(a);
 Double_t ap=a;
 Double_t sum=1./a;
 Double_t del=sum;
 for (Int_t n=1; n<=itmax; n++)
 {
  ap+=1.;
  del=del*x/ap;
  sum+=del;
  if (fabs(del)<fabs(sum*eps)) break;
  if (n==itmax) cout << "*GamSer(a,x)* a too large or itmax too small" << endl;
 }
 Double_t v=sum*exp(-x+a*log(x)-gln);
 return v;
}
///////////////////////////////////////////////////////////////////////////
Double_t AliMath::GamCf(Double_t a,Double_t x) const
{
// Computation of the incomplete gamma function P(a,x)
// via its continued fraction representation.
//
// The algorithm is based on the formulas and code as denoted in
// Numerical Recipes 2nd ed. on p. 210-212 (W.H.Press et al.).
//
//--- Nve 14-nov-1998 UU-SAP Utrecht
 
 Int_t itmax=100;      // Maximum number of iterations
 Double_t eps=3.e-7;    // Relative accuracy
 Double_t fpmin=1.e-30; // Smallest Double_t value allowed here
 
 if (a<=0.)
 {
  cout << "*GamCf(a,x)* Invalid argument a = " << a << endl;
  return 0;
 }
 
 if (x<=0.)
 {
  if (x<0) cout << "*GamCf(a,x)* Invalid argument x = " << x << endl;
  return 0;
 }
 
 Double_t gln=LnGamma(a);
 Double_t b=x+1.-a;
 Double_t c=1./fpmin;
 Double_t d=1./b;
 Double_t h=d;
 Double_t an,del;
 for (Int_t i=1; i<=itmax; i++)
 {
  an=double(-i)*(double(i)-a);
  b+=2.;
  d=an*d+b;
  if (fabs(d)<fpmin) d=fpmin;
  c=b+an/c;
  if (fabs(c)<fpmin) c=fpmin;
  d=1./d;
  del=d*c;
  h=h*del;
  if (fabs(del-1.)<eps) break;
  if (i==itmax) cout << "*GamCf(a,x)* a too large or itmax too small" << endl;
 }
 Double_t v=exp(-x+a*log(x)-gln)*h;
 return (1.-v);
}
///////////////////////////////////////////////////////////////////////////
Double_t AliMath::Erf(Double_t x) const
{
// Computation of the error function erf(x).
//
//--- NvE 14-nov-1998 UU-SAP Utrecht
 
 return (1.-Erfc(x));
}
///////////////////////////////////////////////////////////////////////////
Double_t AliMath::Erfc(Double_t x) const
{
// Computation of the complementary error function erfc(x).
//
// The algorithm is based on a Chebyshev fit as denoted in
// Numerical Recipes 2nd ed. on p. 214 (W.H.Press et al.).
//
// The fractional error is always less than 1.2e-7.
//
//--- Nve 14-nov-1998 UU-SAP Utrecht
 
 // The parameters of the Chebyshev fit
 const Double_t ka1=-1.26551223,  ka2=1.00002368,
                ka3= 0.37409196,  ka4=0.09678418,
                ka5=-0.18628806,  ka6=0.27886807,
                ka7=-1.13520398,  ka8=1.48851587,
                ka9=-0.82215223, ka10=0.17087277;
 
 Double_t v=1.; // The return value
 
 Double_t z=fabs(x);
 
 if (z <= 0.) return v; // erfc(0)=1
 
 Double_t t=1./(1.+0.5*z);
 
 v=t*exp((-z*z)
   +ka1+t*(ka2+t*(ka3+t*(ka4+t*(ka5+t*(ka6+t*(ka7+t*(ka8+t*(ka9+t*ka10)))))))));
 
 if (x < 0.) v=2.-v; // erfc(-x)=2-erfc(x)
 
 return v;
}
///////////////////////////////////////////////////////////////////////////
Double_t AliMath::Prob(Double_t chi2,Int_t ndf,Int_t mode) const
{
// Computation of the probability for a certain Chi-squared (chi2)
// and number of degrees of freedom (ndf).
//
// According to the value of the parameter "mode" various algorithms
// can be selected.
//
// mode = 0 : Calculations are based on the incomplete gamma function P(a,x),
//            where a=ndf/2 and x=chi2/2.
//
//        1 : Same as for mode=0. However, in case ndf=1 an exact expression
//            based on the error function Erf() is used.
//
//        2 : Same as for mode=0. However, in case ndf>30 a Gaussian approximation
//            is used instead of the gamma function.
//
// When invoked as Prob(chi2,ndf) the default mode=1 is used.
//
// P(a,x) represents the probability that the observed Chi-squared
// for a correct model should be less than the value chi2.
//
// The returned probability corresponds to 1-P(a,x),
// which denotes the probability that an observed Chi-squared exceeds
// the value chi2 by chance, even for a correct model.
//
//--- NvE 14-nov-1998 UU-SAP Utrecht
 
 if (ndf <= 0) return 0; // Set CL to zero in case ndf<=0
 
 if (chi2 <= 0.)
 {
  if (chi2 < 0.)
  {
    return 0;
  }
  else
  {
   return 1;
  }
 }

 Double_t v=-1.;

 switch (mode)
 {
  case 1: // Exact expression for ndf=1 as alternative for the gamma function
   if (ndf==1) v=1.-Erf(sqrt(chi2)/sqrt(2.));
   break;
 
  case 2: // Gaussian approximation for large ndf (i.e. ndf>30) as alternative for the gamma function
   if (ndf>30)
   {
    Double_t q=sqrt(2.*chi2)-sqrt(double(2*ndf-1));
    if (q>0.) v=0.5*(1.-Erf(q/sqrt(2.)));
   }
   break;
 }
 
 if (v<0.)
 {
  // Evaluate the incomplete gamma function
  Double_t a=double(ndf)/2.;
  Double_t x=chi2/2.;
  v=1.-Gamma(a,x);
 }

 return v;
}
///////////////////////////////////////////////////////////////////////////
Double_t AliMath::BesselI0(Double_t x) const
{
// Computation of the modified Bessel function I_0(x) for any real x.
//
// The algorithm is based on the article by Abramowitz and Stegun [1]
// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
//
// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
//     Applied Mathematics Series vol. 55 (1964), Washington.  
//
//--- NvE 12-mar-2000 UU-SAP Utrecht

 // Parameters of the polynomial approximation  
 const Double_t kp1=1.0,          kp2=3.5156229,    kp3=3.0899424,
                kp4=1.2067492,    kp5=0.2659732,    kp6=3.60768e-2,  kp7=4.5813e-3;

 const Double_t kq1= 0.39894228,  kq2= 1.328592e-2, kq3= 2.25319e-3,
                kq4=-1.57565e-3,  kq5= 9.16281e-3,  kq6=-2.057706e-2,
                kq7= 2.635537e-2, kq8=-1.647633e-2, kq9= 3.92377e-3; 

 Double_t ax=fabs(x);

 Double_t y=0,result=0;

 if (ax < 3.75)
 {
  y=pow(x/3.75,2);
  result=kp1+y*(kp2+y*(kp3+y*(kp4+y*(kp5+y*(kp6+y*kp7)))));
 }
 else
 {
  y=3.75/ax;
  result=(exp(ax)/sqrt(ax))
         *(kq1+y*(kq2+y*(kq3+y*(kq4+y*(kq5+y*(kq6+y*(kq7+y*(kq8+y*kq9))))))));
 }

 return result;
}
///////////////////////////////////////////////////////////////////////////
Double_t AliMath::BesselK0(Double_t x) const
{
// Computation of the modified Bessel function K_0(x) for positive real x.
//
// The algorithm is based on the article by Abramowitz and Stegun [1]
// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
//
// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
//     Applied Mathematics Series vol. 55 (1964), Washington.  
//
//--- NvE 12-mar-2000 UU-SAP Utrecht

 // Parameters of the polynomial approximation  
 const Double_t kp1=-0.57721566,  kp2=0.42278420,   kp3=0.23069756,
                kp4= 3.488590e-2, kp5=2.62698e-3,   kp6=1.0750e-4,    kp7=7.4e-6;

 const Double_t kq1= 1.25331414,  kq2=-7.832358e-2, kq3= 2.189568e-2,
                kq4=-1.062446e-2, kq5= 5.87872e-3,  kq6=-2.51540e-3,  kq7=5.3208e-4;

 if (x <= 0)
 {
  cout << " *BesselK0* Invalid argument x = " << x << endl;
  return 0;
 }

 Double_t y=0,result=0;

 if (x <= 2)
 {
  y=x*x/4.;
  result=(-log(x/2.)*BesselI0(x))
         +(kp1+y*(kp2+y*(kp3+y*(kp4+y*(kp5+y*(kp6+y*kp7))))));
 }
 else
 {
  y=2./x;
  result=(exp(-x)/sqrt(x))
         *(kq1+y*(kq2+y*(kq3+y*(kq4+y*(kq5+y*(kq6+y*kq7))))));
 }

 return result;
}
///////////////////////////////////////////////////////////////////////////
Double_t AliMath::BesselI1(Double_t x) const
{
// Computation of the modified Bessel function I_1(x) for any real x.
//
// The algorithm is based on the article by Abramowitz and Stegun [1]
// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
//
// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
//     Applied Mathematics Series vol. 55 (1964), Washington.  
//
//--- NvE 12-mar-2000 UU-SAP Utrecht

 // Parameters of the polynomial approximation  
 const Double_t kp1=0.5,          kp2=0.87890594,   kp3=0.51498869,
                kp4=0.15084934,   kp5=2.658733e-2,  kp6=3.01532e-3,  kp7=3.2411e-4;

 const Double_t kq1= 0.39894228,  kq2=-3.988024e-2, kq3=-3.62018e-3,
                kq4= 1.63801e-3,  kq5=-1.031555e-2, kq6= 2.282967e-2,
                kq7=-2.895312e-2, kq8= 1.787654e-2, kq9=-4.20059e-3; 

 Double_t ax=fabs(x);

 Double_t y=0,result=0;

 if (ax < 3.75)
 {
  y=pow(x/3.75,2);
  result=x*(kp1+y*(kp2+y*(kp3+y*(kp4+y*(kp5+y*(kp6+y*kp7))))));
 }
 else
 {
  y=3.75/ax;
  result=(exp(ax)/sqrt(ax))
         *(kq1+y*(kq2+y*(kq3+y*(kq4+y*(kq5+y*(kq6+y*(kq7+y*(kq8+y*kq9))))))));
  if (x < 0) result=-result;
 }

 return result;
}
///////////////////////////////////////////////////////////////////////////
Double_t AliMath::BesselK1(Double_t x) const
{
// Computation of the modified Bessel function K_1(x) for positive real x.
//
// The algorithm is based on the article by Abramowitz and Stegun [1]
// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
//
// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
//     Applied Mathematics Series vol. 55 (1964), Washington.  
//
//--- NvE 12-mar-2000 UU-SAP Utrecht

 // Parameters of the polynomial approximation  
 const Double_t kp1= 1.,          kp2= 0.15443144,  kp3=-0.67278579,
                kp4=-0.18156897,  kp5=-1.919402e-2, kp6=-1.10404e-3,  kp7=-4.686e-5;

 const Double_t kq1= 1.25331414,  kq2= 0.23498619,  kq3=-3.655620e-2,
                kq4= 1.504268e-2, kq5=-7.80353e-3,  kq6= 3.25614e-3,  kq7=-6.8245e-4;

 if (x <= 0)
 {
  cout << " *BesselK1* Invalid argument x = " << x << endl;
  return 0;
 }

 Double_t y=0,result=0;

 if (x <= 2)
 {
  y=x*x/4.;
  result=(log(x/2.)*BesselI1(x))
         +(1./x)*(kp1+y*(kp2+y*(kp3+y*(kp4+y*(kp5+y*(kp6+y*kp7))))));
 }
 else
 {
  y=2./x;
  result=(exp(-x)/sqrt(x))
         *(kq1+y*(kq2+y*(kq3+y*(kq4+y*(kq5+y*(kq6+y*kq7))))));
 }

 return result;
}
///////////////////////////////////////////////////////////////////////////
Double_t AliMath::BesselK(Int_t n,Double_t x) const
{
// Computation of the Integer Order Modified Bessel function K_n(x)
// for n=0,1,2,... and positive real x.
//
// The algorithm uses the recurrence relation
//
//               K_n+1(x) = (2n/x)*K_n(x) + K_n-1(x) 
//
// as denoted in Numerical Recipes 2nd ed. on p. 232 (W.H.Press et al.).
//
//--- NvE 12-mar-2000 UU-SAP Utrecht

 if (x <= 0 || n < 0)
 {
  cout << " *BesselK* Invalid argument(s) (n,x) = (" << n << " , " << x << ")" << endl;
  return 0;
 }

 if (n==0) return BesselK0(x);

 if (n==1) return BesselK1(x);

 // Perform upward recurrence for all x
 Double_t tox=2./x;
 Double_t bkm=BesselK0(x);
 Double_t bk=BesselK1(x);
 Double_t bkp=0;
 for (Int_t j=1; j<n; j++)
 {
  bkp=bkm+double(j)*tox*bk;
  bkm=bk;
  bk=bkp;
 }

 return bk;
}
///////////////////////////////////////////////////////////////////////////
Double_t AliMath::BesselI(Int_t n,Double_t x) const
{
// Computation of the Integer Order Modified Bessel function I_n(x)
// for n=0,1,2,... and any real x.
//
// The algorithm uses the recurrence relation
//
//               I_n+1(x) = (-2n/x)*I_n(x) + I_n-1(x) 
//
// as denoted in Numerical Recipes 2nd ed. on p. 232 (W.H.Press et al.).
//
//--- NvE 12-mar-2000 UU-SAP Utrecht

 Int_t iacc=40; // Increase to enhance accuracy
 Double_t bigno=1.e10, bigni=1.e-10;

 if (n < 0)
 {
  cout << " *BesselI* Invalid argument (n,x) = (" << n << " , " << x << ")" << endl;
  return 0;
 }

 if (n==0) return BesselI0(x);

 if (n==1) return BesselI1(x);

 if (fabs(x) < 1.e-10) return 0;

 Double_t tox=2./fabs(x);
 Double_t bip=0,bim=0;
 Double_t bi=1;
 Double_t result=0;
 Int_t m=2*((n+int(sqrt(float(iacc*n))))); // Downward recurrence from even m
 for (Int_t j=m; j<=1; j--)
 {
  bim=bip+double(j)*tox*bi;
  bip=bi;
  bi=bim;
  if (fabs(bi) > bigno) // Renormalise to prevent overflows
  {
   result*=bigni;
   bi*=bigni;
   bip*=bigni;
  }
  if (j==n) result=bip;
 }

 result*=BesselI0(x)/bi; // Normalise with I0(x)
 if ((x < 0) && (n%2 == 1)) result=-result;

 return result;
}
///////////////////////////////////////////////////////////////////////////
Commit	Line	Data
4c039060	1	/**************************************************************************
	2	* Copyright(c) 1998-1999, ALICE Experiment at CERN, All rights reserved. *
	3	* *
	4	* Author: The ALICE Off-line Project. *
	5	* Contributors are mentioned in the code where appropriate. *
	6	* *
	7	* Permission to use, copy, modify and distribute this software and its *
	8	* documentation strictly for non-commercial purposes is hereby granted *
	9	* without fee, provided that the above copyright notice appears in all *
	10	* copies and that both the copyright notice and this permission notice *
	11	* appear in the supporting documentation. The authors make no claims *
	12	* about the suitability of this software for any purpose. It is *
	13	* provided "as is" without express or implied warranty. *
	14	**************************************************************************/
	15
f531a546	16	// $Id$
4c039060	17
959fbac5	18	///////////////////////////////////////////////////////////////////////////
	19	// Class AliMath
	20	// Various mathematical tools which may be very convenient while
	21	// performing physics analysis.
	22	//
	23	// Example : Probability of a Chi-squared value
	24	// =========
	25	//
	26	// AliMath M;
	27	// Float_t chi2=20; // The chi-squared value
	28	// Int_t ndf=12; // The number of degrees of freedom
	29	// Float_t p=M.Prob(chi2,ndf); // The probability that at least a Chi-squared
	30	// // value of chi2 will be observed, even for a
	31	// // correct model
	32	//
	33	//--- Author: Nick van Eijndhoven 14-nov-1998 UU-SAP Utrecht
f531a546	34	//- Modified: NvE $Date$ UU-SAP Utrecht
959fbac5	35	///////////////////////////////////////////////////////////////////////////
959fbac5	36
d88f97cc	37	#include "AliMath.h"
c72198f1	38	#include "Riostream.h"
d88f97cc	39
	40	ClassImp(AliMath) // Class implementation to enable ROOT I/O
	41
c72198f1	42	AliMath::AliMath() : TObject()
d88f97cc	43	{
	44	// Default constructor
	45	}
	46	///////////////////////////////////////////////////////////////////////////
	47	AliMath::~AliMath()
	48	{
	49	// Destructor
	50	}
	51	///////////////////////////////////////////////////////////////////////////
261c0caf	52	AliMath::AliMath(const AliMath& m) : TObject(m)
c72198f1	53	{
	54	// Copy constructor
	55	}
	56	///////////////////////////////////////////////////////////////////////////
261c0caf	57	Double_t AliMath::Gamma(Double_t z) const
d88f97cc	58	{
	59	// Computation of gamma(z) for all z>0.
	60	//
	61	// The algorithm is based on the article by C.Lanczos [1] as denoted in
	62	// Numerical Recipes 2nd ed. on p. 207 (W.H.Press et al.).
	63	//
	64	// [1] C.Lanczos, SIAM Journal of Numerical Analysis B1 (1964), 86.
	65	//
	66	//--- Nve 14-nov-1998 UU-SAP Utrecht
	67
	68	if (z<=0.)
	69	{
	70	cout << "Gamma(z) Wrong argument z = " << z << endl;
	71	return 0;
	72	}
	73
29beb80d	74	Double_t v=LnGamma(z);
d88f97cc	75	return exp(v);
	76	}
	77	///////////////////////////////////////////////////////////////////////////
261c0caf	78	Double_t AliMath::Gamma(Double_t a,Double_t x) const
d88f97cc	79	{
	80	// Computation of the incomplete gamma function P(a,x)
	81	//
	82	// The algorithm is based on the formulas and code as denoted in
	83	// Numerical Recipes 2nd ed. on p. 210-212 (W.H.Press et al.).
	84	//
	85	//--- Nve 14-nov-1998 UU-SAP Utrecht
	86
	87	if (a<=0.)
	88	{
	89	cout << "Gamma(a,x) Invalid argument a = " << a << endl;
	90	return 0;
	91	}
	92
	93	if (x<=0.)
	94	{
	95	if (x<0) cout << "Gamma(a,x) Invalid argument x = " << x << endl;
	96	return 0;
	97	}
	98
	99	if (x<(a+1.))
	100	{
	101	return GamSer(a,x);
	102	}
	103	else
	104	{
	105	return GamCf(a,x);
	106	}
	107	}
	108	///////////////////////////////////////////////////////////////////////////
261c0caf	109	Double_t AliMath::LnGamma(Double_t z) const
d88f97cc	110	{
	111	// Computation of ln[gamma(z)] for all z>0.
	112	//
	113	// The algorithm is based on the article by C.Lanczos [1] as denoted in
	114	// Numerical Recipes 2nd ed. on p. 207 (W.H.Press et al.).
	115	//
	116	// [1] C.Lanczos, SIAM Journal of Numerical Analysis B1 (1964), 86.
	117	//
	118	// The accuracy of the result is better than 2e-10.
	119	//
	120	//--- Nve 14-nov-1998 UU-SAP Utrecht
	121
	122	if (z<=0.)
	123	{
	124	cout << "LnGamma(z) Wrong argument z = " << z << endl;
	125	return 0;
	126	}
	127
	128	// Coefficients for the series expansion
	129	Double_t c[7];
	130	c[0]= 2.5066282746310005;
	131	c[1]= 76.18009172947146;
	132	c[2]=-86.50532032941677;
	133	c[3]= 24.01409824083091;
	134	c[4]= -1.231739572450155;
	135	c[5]= 0.1208650973866179e-2;
	136	c[6]= -0.5395239384953e-5;
	137
	138	Double_t x=z;
	139	Double_t y=x;
	140	Double_t tmp=x+5.5;
	141	tmp=(x+0.5)*log(tmp)-tmp;
	142	Double_t ser=1.000000000190015;
	143	for (Int_t i=1; i<7; i++)
	144	{
	145	y+=1.;
	146	ser+=c[i]/y;
	147	}
29beb80d	148	Double_t v=tmp+log(c[0]*ser/x);
d88f97cc	149	return v;
	150	}
	151	///////////////////////////////////////////////////////////////////////////
261c0caf	152	Double_t AliMath::GamSer(Double_t a,Double_t x) const
d88f97cc	153	{
	154	// Computation of the incomplete gamma function P(a,x)
	155	// via its series representation.
	156	//
	157	// The algorithm is based on the formulas and code as denoted in
	158	// Numerical Recipes 2nd ed. on p. 210-212 (W.H.Press et al.).
	159	//
	160	//--- Nve 14-nov-1998 UU-SAP Utrecht
	161
	162	Int_t itmax=100; // Maximum number of iterations
29beb80d	163	Double_t eps=3.e-7; // Relative accuracy
d88f97cc	164
	165	if (a<=0.)
	166	{
	167	cout << "GamSer(a,x) Invalid argument a = " << a << endl;
	168	return 0;
	169	}
	170
	171	if (x<=0.)
	172	{
	173	if (x<0) cout << "GamSer(a,x) Invalid argument x = " << x << endl;
	174	return 0;
	175	}
	176
29beb80d	177	Double_t gln=LnGamma(a);
	178	Double_t ap=a;
	179	Double_t sum=1./a;
	180	Double_t del=sum;
d88f97cc	181	for (Int_t n=1; n<=itmax; n++)
	182	{
	183	ap+=1.;
	184	del=del*x/ap;
	185	sum+=del;
	186	if (fabs(del)<fabs(sum*eps)) break;
	187	if (n==itmax) cout << "GamSer(a,x) a too large or itmax too small" << endl;
	188	}
29beb80d	189	Double_t v=sumexp(-x+alog(x)-gln);
d88f97cc	190	return v;
	191	}
	192	///////////////////////////////////////////////////////////////////////////
261c0caf	193	Double_t AliMath::GamCf(Double_t a,Double_t x) const
d88f97cc	194	{
	195	// Computation of the incomplete gamma function P(a,x)
	196	// via its continued fraction representation.
	197	//
	198	// The algorithm is based on the formulas and code as denoted in
	199	// Numerical Recipes 2nd ed. on p. 210-212 (W.H.Press et al.).
	200	//
	201	//--- Nve 14-nov-1998 UU-SAP Utrecht
	202
	203	Int_t itmax=100; // Maximum number of iterations
29beb80d	204	Double_t eps=3.e-7; // Relative accuracy
29beb80d	205	Double_t fpmin=1.e-30; // Smallest Double_t value allowed here
d88f97cc	206
	207	if (a<=0.)
	208	{
	209	cout << "GamCf(a,x) Invalid argument a = " << a << endl;
	210	return 0;
	211	}
	212
	213	if (x<=0.)
	214	{
	215	if (x<0) cout << "GamCf(a,x) Invalid argument x = " << x << endl;
	216	return 0;
	217	}
	218
29beb80d	219	Double_t gln=LnGamma(a);
	220	Double_t b=x+1.-a;
	221	Double_t c=1./fpmin;
	222	Double_t d=1./b;
	223	Double_t h=d;
	224	Double_t an,del;
d88f97cc	225	for (Int_t i=1; i<=itmax; i++)
d88f97cc	226	{
29beb80d	227	an=double(-i)*(double(i)-a);
d88f97cc	228	b+=2.;
	229	d=an*d+b;
	230	if (fabs(d)<fpmin) d=fpmin;
	231	c=b+an/c;
	232	if (fabs(c)<fpmin) c=fpmin;
	233	d=1./d;
	234	del=d*c;
	235	h=h*del;
	236	if (fabs(del-1.)<eps) break;
	237	if (i==itmax) cout << "GamCf(a,x) a too large or itmax too small" << endl;
	238	}
29beb80d	239	Double_t v=exp(-x+alog(x)-gln)h;
d88f97cc	240	return (1.-v);
	241	}
	242	///////////////////////////////////////////////////////////////////////////
261c0caf	243	Double_t AliMath::Erf(Double_t x) const
d88f97cc	244	{
	245	// Computation of the error function erf(x).
	246	//
	247	//--- NvE 14-nov-1998 UU-SAP Utrecht
	248
	249	return (1.-Erfc(x));
	250	}
	251	///////////////////////////////////////////////////////////////////////////
261c0caf	252	Double_t AliMath::Erfc(Double_t x) const
d88f97cc	253	{
	254	// Computation of the complementary error function erfc(x).
	255	//
	256	// The algorithm is based on a Chebyshev fit as denoted in
	257	// Numerical Recipes 2nd ed. on p. 214 (W.H.Press et al.).
	258	//
	259	// The fractional error is always less than 1.2e-7.
	260	//
	261	//--- Nve 14-nov-1998 UU-SAP Utrecht
	262
	263	// The parameters of the Chebyshev fit
387a745b	264	const Double_t ka1=-1.26551223, ka2=1.00002368,
	265	ka3= 0.37409196, ka4=0.09678418,
	266	ka5=-0.18628806, ka6=0.27886807,
	267	ka7=-1.13520398, ka8=1.48851587,
	268	ka9=-0.82215223, ka10=0.17087277;
d88f97cc	269
29beb80d	270	Double_t v=1.; // The return value
d88f97cc	271
29beb80d	272	Double_t z=fabs(x);
d88f97cc	273
	274	if (z <= 0.) return v; // erfc(0)=1
	275
29beb80d	276	Double_t t=1./(1.+0.5*z);
d88f97cc	277
d88f97cc	278	v=texp((-zz)
387a745b	279	+ka1+t(ka2+t(ka3+t(ka4+t(ka5+t(ka6+t(ka7+t(ka8+t(ka9+t*ka10)))))))));
d88f97cc	280
	281	if (x < 0.) v=2.-v; // erfc(-x)=2-erfc(x)
	282
	283	return v;
	284	}
	285	///////////////////////////////////////////////////////////////////////////
261c0caf	286	Double_t AliMath::Prob(Double_t chi2,Int_t ndf,Int_t mode) const
d88f97cc	287	{
	288	// Computation of the probability for a certain Chi-squared (chi2)
	289	// and number of degrees of freedom (ndf).
	290	//
176f88c0	291	// According to the value of the parameter "mode" various algorithms
	292	// can be selected.
	293	//
	294	// mode = 0 : Calculations are based on the incomplete gamma function P(a,x),
	295	// where a=ndf/2 and x=chi2/2.
	296	//
	297	// 1 : Same as for mode=0. However, in case ndf=1 an exact expression
	298	// based on the error function Erf() is used.
	299	//
	300	// 2 : Same as for mode=0. However, in case ndf>30 a Gaussian approximation
	301	// is used instead of the gamma function.
	302	//
	303	// When invoked as Prob(chi2,ndf) the default mode=1 is used.
d88f97cc	304	//
	305	// P(a,x) represents the probability that the observed Chi-squared
	306	// for a correct model should be less than the value chi2.
	307	//
	308	// The returned probability corresponds to 1-P(a,x),
	309	// which denotes the probability that an observed Chi-squared exceeds
	310	// the value chi2 by chance, even for a correct model.
	311	//
	312	//--- NvE 14-nov-1998 UU-SAP Utrecht
	313
	314	if (ndf <= 0) return 0; // Set CL to zero in case ndf<=0
	315
	316	if (chi2 <= 0.)
	317	{
	318	if (chi2 < 0.)
	319	{
	320	return 0;
	321	}
	322	else
	323	{
	324	return 1;
	325	}
	326	}
176f88c0	327
	328	Double_t v=-1.;
	329
	330	switch (mode)
	331	{
	332	case 1: // Exact expression for ndf=1 as alternative for the gamma function
	333	if (ndf==1) v=1.-Erf(sqrt(chi2)/sqrt(2.));
	334	break;
	335
	336	case 2: // Gaussian approximation for large ndf (i.e. ndf>30) as alternative for the gamma function
	337	if (ndf>30)
	338	{
	339	Double_t q=sqrt(2.chi2)-sqrt(double(2ndf-1));
	340	if (q>0.) v=0.5*(1.-Erf(q/sqrt(2.)));
	341	}
	342	break;
	343	}
d88f97cc	344
176f88c0	345	if (v<0.)
	346	{
	347	// Evaluate the incomplete gamma function
	348	Double_t a=double(ndf)/2.;
	349	Double_t x=chi2/2.;
	350	v=1.-Gamma(a,x);
	351	}
	352
	353	return v;
d88f97cc	354	}
d88f97cc	355	///////////////////////////////////////////////////////////////////////////
261c0caf	356	Double_t AliMath::BesselI0(Double_t x) const
29beb80d	357	{
	358	// Computation of the modified Bessel function I_0(x) for any real x.
	359	//
	360	// The algorithm is based on the article by Abramowitz and Stegun [1]
	361	// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
	362	//
	363	// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
	364	// Applied Mathematics Series vol. 55 (1964), Washington.
	365	//
	366	//--- NvE 12-mar-2000 UU-SAP Utrecht
	367
	368	// Parameters of the polynomial approximation
387a745b	369	const Double_t kp1=1.0, kp2=3.5156229, kp3=3.0899424,
387a745b	370	kp4=1.2067492, kp5=0.2659732, kp6=3.60768e-2, kp7=4.5813e-3;
29beb80d	371
387a745b	372	const Double_t kq1= 0.39894228, kq2= 1.328592e-2, kq3= 2.25319e-3,
	373	kq4=-1.57565e-3, kq5= 9.16281e-3, kq6=-2.057706e-2,
	374	kq7= 2.635537e-2, kq8=-1.647633e-2, kq9= 3.92377e-3;
29beb80d	375
	376	Double_t ax=fabs(x);
	377
	378	Double_t y=0,result=0;
	379
	380	if (ax < 3.75)
	381	{
	382	y=pow(x/3.75,2);
387a745b	383	result=kp1+y(kp2+y(kp3+y(kp4+y(kp5+y(kp6+ykp7)))));
29beb80d	384	}
	385	else
	386	{
	387	y=3.75/ax;
387a745b	388	result=(exp(ax)/sqrt(ax))
387a745b	389	(kq1+y(kq2+y(kq3+y(kq4+y(kq5+y(kq6+y(kq7+y(kq8+y*kq9))))))));
29beb80d	390	}
	391
	392	return result;
	393	}
	394	///////////////////////////////////////////////////////////////////////////
261c0caf	395	Double_t AliMath::BesselK0(Double_t x) const
29beb80d	396	{
	397	// Computation of the modified Bessel function K_0(x) for positive real x.
	398	//
	399	// The algorithm is based on the article by Abramowitz and Stegun [1]
	400	// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
	401	//
	402	// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
	403	// Applied Mathematics Series vol. 55 (1964), Washington.
	404	//
	405	//--- NvE 12-mar-2000 UU-SAP Utrecht
	406
	407	// Parameters of the polynomial approximation
387a745b	408	const Double_t kp1=-0.57721566, kp2=0.42278420, kp3=0.23069756,
7a086578	409	kp4= 3.488590e-2, kp5=2.62698e-3, kp6=1.0750e-4, kp7=7.4e-6;
29beb80d	410
387a745b	411	const Double_t kq1= 1.25331414, kq2=-7.832358e-2, kq3= 2.189568e-2,
387a745b	412	kq4=-1.062446e-2, kq5= 5.87872e-3, kq6=-2.51540e-3, kq7=5.3208e-4;
29beb80d	413
	414	if (x <= 0)
	415	{
	416	cout << " BesselK0 Invalid argument x = " << x << endl;
	417	return 0;
	418	}
	419
	420	Double_t y=0,result=0;
	421
	422	if (x <= 2)
	423	{
	424	y=x*x/4.;
387a745b	425	result=(-log(x/2.)*BesselI0(x))
387a745b	426	+(kp1+y(kp2+y(kp3+y(kp4+y(kp5+y(kp6+ykp7))))));
29beb80d	427	}
	428	else
	429	{
	430	y=2./x;
387a745b	431	result=(exp(-x)/sqrt(x))
387a745b	432	(kq1+y(kq2+y(kq3+y(kq4+y(kq5+y(kq6+y*kq7))))));
29beb80d	433	}
	434
	435	return result;
	436	}
	437	///////////////////////////////////////////////////////////////////////////
261c0caf	438	Double_t AliMath::BesselI1(Double_t x) const
29beb80d	439	{
	440	// Computation of the modified Bessel function I_1(x) for any real x.
	441	//
	442	// The algorithm is based on the article by Abramowitz and Stegun [1]
	443	// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
	444	//
	445	// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
	446	// Applied Mathematics Series vol. 55 (1964), Washington.
	447	//
	448	//--- NvE 12-mar-2000 UU-SAP Utrecht
	449
	450	// Parameters of the polynomial approximation
387a745b	451	const Double_t kp1=0.5, kp2=0.87890594, kp3=0.51498869,
387a745b	452	kp4=0.15084934, kp5=2.658733e-2, kp6=3.01532e-3, kp7=3.2411e-4;
29beb80d	453
387a745b	454	const Double_t kq1= 0.39894228, kq2=-3.988024e-2, kq3=-3.62018e-3,
	455	kq4= 1.63801e-3, kq5=-1.031555e-2, kq6= 2.282967e-2,
	456	kq7=-2.895312e-2, kq8= 1.787654e-2, kq9=-4.20059e-3;
29beb80d	457
	458	Double_t ax=fabs(x);
	459
	460	Double_t y=0,result=0;
	461
	462	if (ax < 3.75)
	463	{
	464	y=pow(x/3.75,2);
387a745b	465	result=x(kp1+y(kp2+y(kp3+y(kp4+y(kp5+y(kp6+y*kp7))))));
29beb80d	466	}
	467	else
	468	{
	469	y=3.75/ax;
387a745b	470	result=(exp(ax)/sqrt(ax))
387a745b	471	(kq1+y(kq2+y(kq3+y(kq4+y(kq5+y(kq6+y(kq7+y(kq8+y*kq9))))))));
29beb80d	472	if (x < 0) result=-result;
	473	}
	474
	475	return result;
	476	}
	477	///////////////////////////////////////////////////////////////////////////
261c0caf	478	Double_t AliMath::BesselK1(Double_t x) const
29beb80d	479	{
	480	// Computation of the modified Bessel function K_1(x) for positive real x.
	481	//
	482	// The algorithm is based on the article by Abramowitz and Stegun [1]
	483	// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
	484	//
	485	// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
	486	// Applied Mathematics Series vol. 55 (1964), Washington.
	487	//
	488	//--- NvE 12-mar-2000 UU-SAP Utrecht
	489
	490	// Parameters of the polynomial approximation
387a745b	491	const Double_t kp1= 1., kp2= 0.15443144, kp3=-0.67278579,
387a745b	492	kp4=-0.18156897, kp5=-1.919402e-2, kp6=-1.10404e-3, kp7=-4.686e-5;
29beb80d	493
387a745b	494	const Double_t kq1= 1.25331414, kq2= 0.23498619, kq3=-3.655620e-2,
387a745b	495	kq4= 1.504268e-2, kq5=-7.80353e-3, kq6= 3.25614e-3, kq7=-6.8245e-4;
29beb80d	496
	497	if (x <= 0)
	498	{
	499	cout << " BesselK1 Invalid argument x = " << x << endl;
	500	return 0;
	501	}
	502
	503	Double_t y=0,result=0;
	504
	505	if (x <= 2)
	506	{
	507	y=x*x/4.;
387a745b	508	result=(log(x/2.)*BesselI1(x))
387a745b	509	+(1./x)(kp1+y(kp2+y(kp3+y(kp4+y(kp5+y(kp6+y*kp7))))));
29beb80d	510	}
	511	else
	512	{
	513	y=2./x;
387a745b	514	result=(exp(-x)/sqrt(x))
387a745b	515	(kq1+y(kq2+y(kq3+y(kq4+y(kq5+y(kq6+y*kq7))))));
29beb80d	516	}
	517
	518	return result;
	519	}
	520	///////////////////////////////////////////////////////////////////////////
261c0caf	521	Double_t AliMath::BesselK(Int_t n,Double_t x) const
29beb80d	522	{
	523	// Computation of the Integer Order Modified Bessel function K_n(x)
	524	// for n=0,1,2,... and positive real x.
	525	//
	526	// The algorithm uses the recurrence relation
	527	//
	528	// K_n+1(x) = (2n/x)*K_n(x) + K_n-1(x)
	529	//
	530	// as denoted in Numerical Recipes 2nd ed. on p. 232 (W.H.Press et al.).
	531	//
	532	//--- NvE 12-mar-2000 UU-SAP Utrecht
	533
	534	if (x <= 0 \|\| n < 0)
	535	{
	536	cout << " BesselK Invalid argument(s) (n,x) = (" << n << " , " << x << ")" << endl;
	537	return 0;
	538	}
	539
	540	if (n==0) return BesselK0(x);
	541
	542	if (n==1) return BesselK1(x);
	543
	544	// Perform upward recurrence for all x
	545	Double_t tox=2./x;
	546	Double_t bkm=BesselK0(x);
	547	Double_t bk=BesselK1(x);
	548	Double_t bkp=0;
	549	for (Int_t j=1; j<n; j++)
	550	{
	551	bkp=bkm+double(j)toxbk;
	552	bkm=bk;
	553	bk=bkp;
	554	}
	555
	556	return bk;
	557	}
	558	///////////////////////////////////////////////////////////////////////////
261c0caf	559	Double_t AliMath::BesselI(Int_t n,Double_t x) const
29beb80d	560	{
	561	// Computation of the Integer Order Modified Bessel function I_n(x)
	562	// for n=0,1,2,... and any real x.
	563	//
	564	// The algorithm uses the recurrence relation
	565	//
	566	// I_n+1(x) = (-2n/x)*I_n(x) + I_n-1(x)
	567	//
	568	// as denoted in Numerical Recipes 2nd ed. on p. 232 (W.H.Press et al.).
	569	//
	570	//--- NvE 12-mar-2000 UU-SAP Utrecht
	571
	572	Int_t iacc=40; // Increase to enhance accuracy
	573	Double_t bigno=1.e10, bigni=1.e-10;
	574
	575	if (n < 0)
	576	{
	577	cout << " BesselI Invalid argument (n,x) = (" << n << " , " << x << ")" << endl;
	578	return 0;
	579	}
	580
	581	if (n==0) return BesselI0(x);
	582
	583	if (n==1) return BesselI1(x);
	584
	585	if (fabs(x) < 1.e-10) return 0;
	586
	587	Double_t tox=2./fabs(x);
	588	Double_t bip=0,bim=0;
	589	Double_t bi=1;
	590	Double_t result=0;
	591	Int_t m=2((n+int(sqrt(float(iaccn))))); // Downward recurrence from even m
	592	for (Int_t j=m; j<=1; j--)
	593	{
	594	bim=bip+double(j)toxbi;
	595	bip=bi;
	596	bi=bim;
	597	if (fabs(bi) > bigno) // Renormalise to prevent overflows
	598	{
	599	result*=bigni;
	600	bi*=bigni;
	601	bip*=bigni;
	602	}
	603	if (j==n) result=bip;
	604	}
	605
	606	result*=BesselI0(x)/bi; // Normalise with I0(x)
	607	if ((x < 0) && (n%2 == 1)) result=-result;
	608
	609	return result;
	610	}
	611	///////////////////////////////////////////////////////////////////////////