[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(AliMath& m) : TObject(m)
{
// Copy constructor
}
///////////////////////////////////////////////////////////////////////////
Double_t AliMath::Gamma(Double_t z)
{
// 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)
{
// 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)
{
// 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)
{
// 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)
{
// 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)
{
// 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)
{
// 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)
{
// Computation of the probability for a certain Chi-squared (chi2)
// and number of degrees of freedom (ndf).
//
// Calculations are based on the incomplete gamma function P(a,x),
// where a=ndf/2 and x=chi2/2.
//
// 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;
  }
 }
 
// Alternative which is exact
// This code may be activated in case the gamma function gives problems
// if (ndf==1)
// {
//  Double_t v=1.-Erf(sqrt(chi2)/sqrt(2.));
//  return v;
// }
 
// Gaussian approximation for large ndf
// This code may be activated in case the gamma function shows a problem
// Double_t q=sqrt(2.*chi2)-sqrt(double(2*ndf-1));
// if (n>30 && q>0.)
// {
//  Double_t v=0.5*(1.-Erf(q/sqrt(2.)));
//  return v;
// }
 
 // Evaluate the incomplete gamma function
 Double_t a=double(ndf)/2.;
 Double_t x=chi2/2.;
 return (1.-Gamma(a,x));
}
///////////////////////////////////////////////////////////////////////////
Double_t AliMath::BesselI0(Double_t x)
{
// 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)
{
// 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-5;

 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)
{
// 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)
{
// 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)
{
// 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)
{
// 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	///////////////////////////////////////////////////////////////////////////
c72198f1	52	AliMath::AliMath(AliMath& m) : TObject(m)
	53	{
	54	// Copy constructor
	55	}
	56	///////////////////////////////////////////////////////////////////////////
29beb80d	57	Double_t AliMath::Gamma(Double_t z)
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	///////////////////////////////////////////////////////////////////////////
29beb80d	78	Double_t AliMath::Gamma(Double_t a,Double_t x)
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	///////////////////////////////////////////////////////////////////////////
29beb80d	109	Double_t AliMath::LnGamma(Double_t z)
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	///////////////////////////////////////////////////////////////////////////
29beb80d	152	Double_t AliMath::GamSer(Double_t a,Double_t x)
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	///////////////////////////////////////////////////////////////////////////
29beb80d	193	Double_t AliMath::GamCf(Double_t a,Double_t x)
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	///////////////////////////////////////////////////////////////////////////
29beb80d	243	Double_t AliMath::Erf(Double_t x)
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	///////////////////////////////////////////////////////////////////////////
29beb80d	252	Double_t AliMath::Erfc(Double_t x)
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	///////////////////////////////////////////////////////////////////////////
29beb80d	286	Double_t AliMath::Prob(Double_t chi2,Int_t ndf)
d88f97cc	287	{
	288	// Computation of the probability for a certain Chi-squared (chi2)
	289	// and number of degrees of freedom (ndf).
	290	//
	291	// Calculations are based on the incomplete gamma function P(a,x),
	292	// where a=ndf/2 and x=chi2/2.
	293	//
	294	// P(a,x) represents the probability that the observed Chi-squared
	295	// for a correct model should be less than the value chi2.
	296	//
	297	// The returned probability corresponds to 1-P(a,x),
	298	// which denotes the probability that an observed Chi-squared exceeds
	299	// the value chi2 by chance, even for a correct model.
	300	//
	301	//--- NvE 14-nov-1998 UU-SAP Utrecht
	302
	303	if (ndf <= 0) return 0; // Set CL to zero in case ndf<=0
	304
	305	if (chi2 <= 0.)
	306	{
	307	if (chi2 < 0.)
	308	{
	309	return 0;
	310	}
	311	else
	312	{
	313	return 1;
	314	}
	315	}
	316
	317	// Alternative which is exact
	318	// This code may be activated in case the gamma function gives problems
	319	// if (ndf==1)
	320	// {
29beb80d	321	// Double_t v=1.-Erf(sqrt(chi2)/sqrt(2.));
d88f97cc	322	// return v;
	323	// }
	324
	325	// Gaussian approximation for large ndf
	326	// This code may be activated in case the gamma function shows a problem
29beb80d	327	// Double_t q=sqrt(2.chi2)-sqrt(double(2ndf-1));
d88f97cc	328	// if (n>30 && q>0.)
d88f97cc	329	// {
29beb80d	330	// Double_t v=0.5*(1.-Erf(q/sqrt(2.)));
d88f97cc	331	// return v;
	332	// }
	333
	334	// Evaluate the incomplete gamma function
29beb80d	335	Double_t a=double(ndf)/2.;
29beb80d	336	Double_t x=chi2/2.;
d88f97cc	337	return (1.-Gamma(a,x));
	338	}
	339	///////////////////////////////////////////////////////////////////////////
29beb80d	340	Double_t AliMath::BesselI0(Double_t x)
	341	{
	342	// Computation of the modified Bessel function I_0(x) for any real x.
	343	//
	344	// The algorithm is based on the article by Abramowitz and Stegun [1]
	345	// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
	346	//
	347	// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
	348	// Applied Mathematics Series vol. 55 (1964), Washington.
	349	//
	350	//--- NvE 12-mar-2000 UU-SAP Utrecht
	351
	352	// Parameters of the polynomial approximation
387a745b	353	const Double_t kp1=1.0, kp2=3.5156229, kp3=3.0899424,
387a745b	354	kp4=1.2067492, kp5=0.2659732, kp6=3.60768e-2, kp7=4.5813e-3;
29beb80d	355
387a745b	356	const Double_t kq1= 0.39894228, kq2= 1.328592e-2, kq3= 2.25319e-3,
	357	kq4=-1.57565e-3, kq5= 9.16281e-3, kq6=-2.057706e-2,
	358	kq7= 2.635537e-2, kq8=-1.647633e-2, kq9= 3.92377e-3;
29beb80d	359
	360	Double_t ax=fabs(x);
	361
	362	Double_t y=0,result=0;
	363
	364	if (ax < 3.75)
	365	{
	366	y=pow(x/3.75,2);
387a745b	367	result=kp1+y(kp2+y(kp3+y(kp4+y(kp5+y(kp6+ykp7)))));
29beb80d	368	}
	369	else
	370	{
	371	y=3.75/ax;
387a745b	372	result=(exp(ax)/sqrt(ax))
387a745b	373	(kq1+y(kq2+y(kq3+y(kq4+y(kq5+y(kq6+y(kq7+y(kq8+y*kq9))))))));
29beb80d	374	}
	375
	376	return result;
	377	}
	378	///////////////////////////////////////////////////////////////////////////
	379	Double_t AliMath::BesselK0(Double_t x)
	380	{
	381	// Computation of the modified Bessel function K_0(x) for positive real x.
	382	//
	383	// The algorithm is based on the article by Abramowitz and Stegun [1]
	384	// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
	385	//
	386	// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
	387	// Applied Mathematics Series vol. 55 (1964), Washington.
	388	//
	389	//--- NvE 12-mar-2000 UU-SAP Utrecht
	390
	391	// Parameters of the polynomial approximation
387a745b	392	const Double_t kp1=-0.57721566, kp2=0.42278420, kp3=0.23069756,
387a745b	393	kp4= 3.488590e-2, kp5=2.62698e-3, kp6=1.0750e-4, kp7=7.4e-5;
29beb80d	394
387a745b	395	const Double_t kq1= 1.25331414, kq2=-7.832358e-2, kq3= 2.189568e-2,
387a745b	396	kq4=-1.062446e-2, kq5= 5.87872e-3, kq6=-2.51540e-3, kq7=5.3208e-4;
29beb80d	397
	398	if (x <= 0)
	399	{
	400	cout << " BesselK0 Invalid argument x = " << x << endl;
	401	return 0;
	402	}
	403
	404	Double_t y=0,result=0;
	405
	406	if (x <= 2)
	407	{
	408	y=x*x/4.;
387a745b	409	result=(-log(x/2.)*BesselI0(x))
387a745b	410	+(kp1+y(kp2+y(kp3+y(kp4+y(kp5+y(kp6+ykp7))))));
29beb80d	411	}
	412	else
	413	{
	414	y=2./x;
387a745b	415	result=(exp(-x)/sqrt(x))
387a745b	416	(kq1+y(kq2+y(kq3+y(kq4+y(kq5+y(kq6+y*kq7))))));
29beb80d	417	}
	418
	419	return result;
	420	}
	421	///////////////////////////////////////////////////////////////////////////
	422	Double_t AliMath::BesselI1(Double_t x)
	423	{
	424	// Computation of the modified Bessel function I_1(x) for any real x.
	425	//
	426	// The algorithm is based on the article by Abramowitz and Stegun [1]
	427	// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
	428	//
	429	// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
	430	// Applied Mathematics Series vol. 55 (1964), Washington.
	431	//
	432	//--- NvE 12-mar-2000 UU-SAP Utrecht
	433
	434	// Parameters of the polynomial approximation
387a745b	435	const Double_t kp1=0.5, kp2=0.87890594, kp3=0.51498869,
387a745b	436	kp4=0.15084934, kp5=2.658733e-2, kp6=3.01532e-3, kp7=3.2411e-4;
29beb80d	437
387a745b	438	const Double_t kq1= 0.39894228, kq2=-3.988024e-2, kq3=-3.62018e-3,
	439	kq4= 1.63801e-3, kq5=-1.031555e-2, kq6= 2.282967e-2,
	440	kq7=-2.895312e-2, kq8= 1.787654e-2, kq9=-4.20059e-3;
29beb80d	441
	442	Double_t ax=fabs(x);
	443
	444	Double_t y=0,result=0;
	445
	446	if (ax < 3.75)
	447	{
	448	y=pow(x/3.75,2);
387a745b	449	result=x(kp1+y(kp2+y(kp3+y(kp4+y(kp5+y(kp6+y*kp7))))));
29beb80d	450	}
	451	else
	452	{
	453	y=3.75/ax;
387a745b	454	result=(exp(ax)/sqrt(ax))
387a745b	455	(kq1+y(kq2+y(kq3+y(kq4+y(kq5+y(kq6+y(kq7+y(kq8+y*kq9))))))));
29beb80d	456	if (x < 0) result=-result;
	457	}
	458
	459	return result;
	460	}
	461	///////////////////////////////////////////////////////////////////////////
	462	Double_t AliMath::BesselK1(Double_t x)
	463	{
	464	// Computation of the modified Bessel function K_1(x) for positive real x.
	465	//
	466	// The algorithm is based on the article by Abramowitz and Stegun [1]
	467	// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
	468	//
	469	// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
	470	// Applied Mathematics Series vol. 55 (1964), Washington.
	471	//
	472	//--- NvE 12-mar-2000 UU-SAP Utrecht
	473
	474	// Parameters of the polynomial approximation
387a745b	475	const Double_t kp1= 1., kp2= 0.15443144, kp3=-0.67278579,
387a745b	476	kp4=-0.18156897, kp5=-1.919402e-2, kp6=-1.10404e-3, kp7=-4.686e-5;
29beb80d	477
387a745b	478	const Double_t kq1= 1.25331414, kq2= 0.23498619, kq3=-3.655620e-2,
387a745b	479	kq4= 1.504268e-2, kq5=-7.80353e-3, kq6= 3.25614e-3, kq7=-6.8245e-4;
29beb80d	480
	481	if (x <= 0)
	482	{
	483	cout << " BesselK1 Invalid argument x = " << x << endl;
	484	return 0;
	485	}
	486
	487	Double_t y=0,result=0;
	488
	489	if (x <= 2)
	490	{
	491	y=x*x/4.;
387a745b	492	result=(log(x/2.)*BesselI1(x))
387a745b	493	+(1./x)(kp1+y(kp2+y(kp3+y(kp4+y(kp5+y(kp6+y*kp7))))));
29beb80d	494	}
	495	else
	496	{
	497	y=2./x;
387a745b	498	result=(exp(-x)/sqrt(x))
387a745b	499	(kq1+y(kq2+y(kq3+y(kq4+y(kq5+y(kq6+y*kq7))))));
29beb80d	500	}
	501
	502	return result;
	503	}
	504	///////////////////////////////////////////////////////////////////////////
	505	Double_t AliMath::BesselK(Int_t n,Double_t x)
	506	{
	507	// Computation of the Integer Order Modified Bessel function K_n(x)
	508	// for n=0,1,2,... and positive real x.
	509	//
	510	// The algorithm uses the recurrence relation
	511	//
	512	// K_n+1(x) = (2n/x)*K_n(x) + K_n-1(x)
	513	//
	514	// as denoted in Numerical Recipes 2nd ed. on p. 232 (W.H.Press et al.).
	515	//
	516	//--- NvE 12-mar-2000 UU-SAP Utrecht
	517
	518	if (x <= 0 \|\| n < 0)
	519	{
	520	cout << " BesselK Invalid argument(s) (n,x) = (" << n << " , " << x << ")" << endl;
	521	return 0;
	522	}
	523
	524	if (n==0) return BesselK0(x);
	525
	526	if (n==1) return BesselK1(x);
	527
	528	// Perform upward recurrence for all x
	529	Double_t tox=2./x;
	530	Double_t bkm=BesselK0(x);
	531	Double_t bk=BesselK1(x);
	532	Double_t bkp=0;
	533	for (Int_t j=1; j<n; j++)
	534	{
	535	bkp=bkm+double(j)toxbk;
	536	bkm=bk;
	537	bk=bkp;
	538	}
	539
	540	return bk;
	541	}
	542	///////////////////////////////////////////////////////////////////////////
	543	Double_t AliMath::BesselI(Int_t n,Double_t x)
	544	{
	545	// Computation of the Integer Order Modified Bessel function I_n(x)
	546	// for n=0,1,2,... and any real x.
	547	//
	548	// The algorithm uses the recurrence relation
	549	//
	550	// I_n+1(x) = (-2n/x)*I_n(x) + I_n-1(x)
	551	//
	552	// as denoted in Numerical Recipes 2nd ed. on p. 232 (W.H.Press et al.).
	553	//
	554	//--- NvE 12-mar-2000 UU-SAP Utrecht
	555
	556	Int_t iacc=40; // Increase to enhance accuracy
	557	Double_t bigno=1.e10, bigni=1.e-10;
	558
	559	if (n < 0)
	560	{
	561	cout << " BesselI Invalid argument (n,x) = (" << n << " , " << x << ")" << endl;
	562	return 0;
	563	}
564
565	if (n==0) return BesselI0(x);
566
567	if (n==1) return BesselI1(x);
568
569	if (fabs(x) < 1.e-10) return 0;
570
571	Double_t tox=2./fabs(x);
572	Double_t bip=0,bim=0;
573	Double_t bi=1;
574	Double_t result=0;
575	Int_t m=2((n+int(sqrt(float(iaccn))))); // Downward recurrence from even m
576	for (Int_t j=m; j<=1; j--)
577	{
578	bim=bip+double(j)toxbi;
579	bip=bi;
580	bi=bim;
581	if (fabs(bi) > bigno) // Renormalise to prevent overflows
582	{
583	result*=bigni;
584	bi*=bigni;
585	bip*=bigni;
586	}
587	if (j==n) result=bip;
588	}
589
590	result*=BesselI0(x)/bi; // Normalise with I0(x)
591	if ((x < 0) && (n%2 == 1)) result=-result;
592
593	return result;
594	}
595	///////////////////////////////////////////////////////////////////////////