[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"
 
ClassImp(AliMath) // Class implementation to enable ROOT I/O
 
AliMath::AliMath()
{
// Default constructor
}
///////////////////////////////////////////////////////////////////////////
AliMath::~AliMath()
{
// Destructor
}
///////////////////////////////////////////////////////////////////////////
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 a1=-1.26551223,  a2=1.00002368,
                a3= 0.37409196,  a4=0.09678418,
                a5=-0.18628806,  a6=0.27886807,
                a7=-1.13520398,  a8=1.48851587,
                a9=-0.82215223, a10=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)
         +a1+t*(a2+t*(a3+t*(a4+t*(a5+t*(a6+t*(a7+t*(a8+t*(a9+t*a10)))))))));
 
 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 p1=1.0,          p2=3.5156229,    p3=3.0899424,
                p4=1.2067492,    p5=0.2659732,    p6=3.60768e-2,  p7=4.5813e-3;

 const Double_t q1= 0.39894228,  q2= 1.328592e-2, q3= 2.25319e-3,
                q4=-1.57565e-3,  q5= 9.16281e-3,  q6=-2.057706e-2,
                q7= 2.635537e-2, q8=-1.647633e-2, q9= 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=p1+y*(p2+y*(p3+y*(p4+y*(p5+y*(p6+y*p7)))));
 }
 else
 {
  y=3.75/ax;
  result=(exp(ax)/sqrt(ax))*(q1+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*(q7+y*(q8+y*q9))))))));
 }

 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 p1=-0.57721566,  p2=0.42278420,   p3=0.23069756,
                p4= 3.488590e-2, p5=2.62698e-3,   p6=1.0750e-4,    p7=7.4e-5;

 const Double_t q1= 1.25331414,  q2=-7.832358e-2, q3= 2.189568e-2,
                q4=-1.062446e-2, q5= 5.87872e-3,  q6=-2.51540e-3,  q7=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))+(p1+y*(p2+y*(p3+y*(p4+y*(p5+y*(p6+y*p7))))));
 }
 else
 {
  y=2./x;
  result=(exp(-x)/sqrt(x))*(q1+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*q7))))));
 }

 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 p1=0.5,          p2=0.87890594,   p3=0.51498869,
                p4=0.15084934,   p5=2.658733e-2,  p6=3.01532e-3,  p7=3.2411e-4;

 const Double_t q1= 0.39894228,  q2=-3.988024e-2, q3=-3.62018e-3,
                q4= 1.63801e-3,  q5=-1.031555e-2, q6= 2.282967e-2,
                q7=-2.895312e-2, q8= 1.787654e-2, q9=-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*(p1+y*(p2+y*(p3+y*(p4+y*(p5+y*(p6+y*p7))))));
 }
 else
 {
  y=3.75/ax;
  result=(exp(ax)/sqrt(ax))*(q1+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*(q7+y*(q8+y*q9))))))));
  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 p1= 1.,          p2= 0.15443144,  p3=-0.67278579,
                p4=-0.18156897,  p5=-1.919402e-2, p6=-1.10404e-3,  p7=-4.686e-5;

 const Double_t q1= 1.25331414,  q2= 0.23498619,  q3=-3.655620e-2,
                q4= 1.504268e-2, q5=-7.80353e-3,  q6= 3.25614e-3,  q7=-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)*(p1+y*(p2+y*(p3+y*(p4+y*(p5+y*(p6+y*p7))))));
 }
 else
 {
  y=2./x;
  result=(exp(-x)/sqrt(x))*(q1+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*q7))))));
 }

 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"
	38
	39	ClassImp(AliMath) // Class implementation to enable ROOT I/O
	40
	41	AliMath::AliMath()
	42	{
	43	// Default constructor
	44	}
	45	///////////////////////////////////////////////////////////////////////////
	46	AliMath::~AliMath()
	47	{
	48	// Destructor
	49	}
	50	///////////////////////////////////////////////////////////////////////////
29beb80d	51	Double_t AliMath::Gamma(Double_t z)
d88f97cc	52	{
	53	// Computation of gamma(z) for all z>0.
	54	//
	55	// The algorithm is based on the article by C.Lanczos [1] as denoted in
	56	// Numerical Recipes 2nd ed. on p. 207 (W.H.Press et al.).
	57	//
	58	// [1] C.Lanczos, SIAM Journal of Numerical Analysis B1 (1964), 86.
	59	//
	60	//--- Nve 14-nov-1998 UU-SAP Utrecht
	61
	62	if (z<=0.)
	63	{
	64	cout << "Gamma(z) Wrong argument z = " << z << endl;
	65	return 0;
	66	}
	67
29beb80d	68	Double_t v=LnGamma(z);
d88f97cc	69	return exp(v);
	70	}
	71	///////////////////////////////////////////////////////////////////////////
29beb80d	72	Double_t AliMath::Gamma(Double_t a,Double_t x)
d88f97cc	73	{
	74	// Computation of the incomplete gamma function P(a,x)
	75	//
	76	// The algorithm is based on the formulas and code as denoted in
	77	// Numerical Recipes 2nd ed. on p. 210-212 (W.H.Press et al.).
	78	//
	79	//--- Nve 14-nov-1998 UU-SAP Utrecht
	80
	81	if (a<=0.)
	82	{
	83	cout << "Gamma(a,x) Invalid argument a = " << a << endl;
	84	return 0;
	85	}
	86
	87	if (x<=0.)
	88	{
	89	if (x<0) cout << "Gamma(a,x) Invalid argument x = " << x << endl;
	90	return 0;
	91	}
	92
	93	if (x<(a+1.))
	94	{
	95	return GamSer(a,x);
	96	}
	97	else
	98	{
	99	return GamCf(a,x);
	100	}
	101	}
	102	///////////////////////////////////////////////////////////////////////////
29beb80d	103	Double_t AliMath::LnGamma(Double_t z)
d88f97cc	104	{
	105	// Computation of ln[gamma(z)] for all z>0.
	106	//
	107	// The algorithm is based on the article by C.Lanczos [1] as denoted in
	108	// Numerical Recipes 2nd ed. on p. 207 (W.H.Press et al.).
	109	//
	110	// [1] C.Lanczos, SIAM Journal of Numerical Analysis B1 (1964), 86.
	111	//
	112	// The accuracy of the result is better than 2e-10.
	113	//
	114	//--- Nve 14-nov-1998 UU-SAP Utrecht
	115
	116	if (z<=0.)
	117	{
	118	cout << "LnGamma(z) Wrong argument z = " << z << endl;
	119	return 0;
	120	}
	121
	122	// Coefficients for the series expansion
	123	Double_t c[7];
	124	c[0]= 2.5066282746310005;
	125	c[1]= 76.18009172947146;
	126	c[2]=-86.50532032941677;
	127	c[3]= 24.01409824083091;
	128	c[4]= -1.231739572450155;
	129	c[5]= 0.1208650973866179e-2;
	130	c[6]= -0.5395239384953e-5;
	131
	132	Double_t x=z;
	133	Double_t y=x;
	134	Double_t tmp=x+5.5;
	135	tmp=(x+0.5)*log(tmp)-tmp;
	136	Double_t ser=1.000000000190015;
	137	for (Int_t i=1; i<7; i++)
	138	{
	139	y+=1.;
	140	ser+=c[i]/y;
	141	}
29beb80d	142	Double_t v=tmp+log(c[0]*ser/x);
d88f97cc	143	return v;
	144	}
	145	///////////////////////////////////////////////////////////////////////////
29beb80d	146	Double_t AliMath::GamSer(Double_t a,Double_t x)
d88f97cc	147	{
	148	// Computation of the incomplete gamma function P(a,x)
	149	// via its series representation.
	150	//
	151	// The algorithm is based on the formulas and code as denoted in
	152	// Numerical Recipes 2nd ed. on p. 210-212 (W.H.Press et al.).
	153	//
	154	//--- Nve 14-nov-1998 UU-SAP Utrecht
	155
	156	Int_t itmax=100; // Maximum number of iterations
29beb80d	157	Double_t eps=3.e-7; // Relative accuracy
d88f97cc	158
	159	if (a<=0.)
	160	{
	161	cout << "GamSer(a,x) Invalid argument a = " << a << endl;
	162	return 0;
	163	}
	164
	165	if (x<=0.)
	166	{
	167	if (x<0) cout << "GamSer(a,x) Invalid argument x = " << x << endl;
	168	return 0;
	169	}
	170
29beb80d	171	Double_t gln=LnGamma(a);
	172	Double_t ap=a;
	173	Double_t sum=1./a;
	174	Double_t del=sum;
d88f97cc	175	for (Int_t n=1; n<=itmax; n++)
	176	{
	177	ap+=1.;
	178	del=del*x/ap;
	179	sum+=del;
	180	if (fabs(del)<fabs(sum*eps)) break;
	181	if (n==itmax) cout << "GamSer(a,x) a too large or itmax too small" << endl;
	182	}
29beb80d	183	Double_t v=sumexp(-x+alog(x)-gln);
d88f97cc	184	return v;
	185	}
	186	///////////////////////////////////////////////////////////////////////////
29beb80d	187	Double_t AliMath::GamCf(Double_t a,Double_t x)
d88f97cc	188	{
	189	// Computation of the incomplete gamma function P(a,x)
	190	// via its continued fraction representation.
	191	//
	192	// The algorithm is based on the formulas and code as denoted in
	193	// Numerical Recipes 2nd ed. on p. 210-212 (W.H.Press et al.).
	194	//
	195	//--- Nve 14-nov-1998 UU-SAP Utrecht
	196
	197	Int_t itmax=100; // Maximum number of iterations
29beb80d	198	Double_t eps=3.e-7; // Relative accuracy
29beb80d	199	Double_t fpmin=1.e-30; // Smallest Double_t value allowed here
d88f97cc	200
	201	if (a<=0.)
	202	{
	203	cout << "GamCf(a,x) Invalid argument a = " << a << endl;
	204	return 0;
	205	}
	206
	207	if (x<=0.)
	208	{
	209	if (x<0) cout << "GamCf(a,x) Invalid argument x = " << x << endl;
	210	return 0;
	211	}
	212
29beb80d	213	Double_t gln=LnGamma(a);
	214	Double_t b=x+1.-a;
	215	Double_t c=1./fpmin;
	216	Double_t d=1./b;
	217	Double_t h=d;
	218	Double_t an,del;
d88f97cc	219	for (Int_t i=1; i<=itmax; i++)
d88f97cc	220	{
29beb80d	221	an=double(-i)*(double(i)-a);
d88f97cc	222	b+=2.;
	223	d=an*d+b;
	224	if (fabs(d)<fpmin) d=fpmin;
	225	c=b+an/c;
	226	if (fabs(c)<fpmin) c=fpmin;
	227	d=1./d;
	228	del=d*c;
	229	h=h*del;
	230	if (fabs(del-1.)<eps) break;
	231	if (i==itmax) cout << "GamCf(a,x) a too large or itmax too small" << endl;
	232	}
29beb80d	233	Double_t v=exp(-x+alog(x)-gln)h;
d88f97cc	234	return (1.-v);
	235	}
	236	///////////////////////////////////////////////////////////////////////////
29beb80d	237	Double_t AliMath::Erf(Double_t x)
d88f97cc	238	{
	239	// Computation of the error function erf(x).
	240	//
	241	//--- NvE 14-nov-1998 UU-SAP Utrecht
	242
	243	return (1.-Erfc(x));
	244	}
	245	///////////////////////////////////////////////////////////////////////////
29beb80d	246	Double_t AliMath::Erfc(Double_t x)
d88f97cc	247	{
	248	// Computation of the complementary error function erfc(x).
	249	//
	250	// The algorithm is based on a Chebyshev fit as denoted in
	251	// Numerical Recipes 2nd ed. on p. 214 (W.H.Press et al.).
	252	//
	253	// The fractional error is always less than 1.2e-7.
	254	//
	255	//--- Nve 14-nov-1998 UU-SAP Utrecht
	256
	257	// The parameters of the Chebyshev fit
29beb80d	258	const Double_t a1=-1.26551223, a2=1.00002368,
	259	a3= 0.37409196, a4=0.09678418,
	260	a5=-0.18628806, a6=0.27886807,
	261	a7=-1.13520398, a8=1.48851587,
	262	a9=-0.82215223, a10=0.17087277;
d88f97cc	263
29beb80d	264	Double_t v=1.; // The return value
d88f97cc	265
29beb80d	266	Double_t z=fabs(x);
d88f97cc	267
	268	if (z <= 0.) return v; // erfc(0)=1
	269
29beb80d	270	Double_t t=1./(1.+0.5*z);
d88f97cc	271
	272	v=texp((-zz)
	273	+a1+t(a2+t(a3+t(a4+t(a5+t(a6+t(a7+t(a8+t(a9+t*a10)))))))));
	274
	275	if (x < 0.) v=2.-v; // erfc(-x)=2-erfc(x)
	276
	277	return v;
	278	}
	279	///////////////////////////////////////////////////////////////////////////
29beb80d	280	Double_t AliMath::Prob(Double_t chi2,Int_t ndf)
d88f97cc	281	{
	282	// Computation of the probability for a certain Chi-squared (chi2)
	283	// and number of degrees of freedom (ndf).
	284	//
	285	// Calculations are based on the incomplete gamma function P(a,x),
	286	// where a=ndf/2 and x=chi2/2.
	287	//
	288	// P(a,x) represents the probability that the observed Chi-squared
	289	// for a correct model should be less than the value chi2.
	290	//
	291	// The returned probability corresponds to 1-P(a,x),
	292	// which denotes the probability that an observed Chi-squared exceeds
	293	// the value chi2 by chance, even for a correct model.
	294	//
	295	//--- NvE 14-nov-1998 UU-SAP Utrecht
	296
	297	if (ndf <= 0) return 0; // Set CL to zero in case ndf<=0
	298
	299	if (chi2 <= 0.)
	300	{
	301	if (chi2 < 0.)
	302	{
	303	return 0;
	304	}
	305	else
	306	{
	307	return 1;
	308	}
	309	}
	310
	311	// Alternative which is exact
	312	// This code may be activated in case the gamma function gives problems
	313	// if (ndf==1)
	314	// {
29beb80d	315	// Double_t v=1.-Erf(sqrt(chi2)/sqrt(2.));
d88f97cc	316	// return v;
	317	// }
	318
	319	// Gaussian approximation for large ndf
	320	// This code may be activated in case the gamma function shows a problem
29beb80d	321	// Double_t q=sqrt(2.chi2)-sqrt(double(2ndf-1));
d88f97cc	322	// if (n>30 && q>0.)
d88f97cc	323	// {
29beb80d	324	// Double_t v=0.5*(1.-Erf(q/sqrt(2.)));
d88f97cc	325	// return v;
	326	// }
	327
	328	// Evaluate the incomplete gamma function
29beb80d	329	Double_t a=double(ndf)/2.;
29beb80d	330	Double_t x=chi2/2.;
d88f97cc	331	return (1.-Gamma(a,x));
	332	}
	333	///////////////////////////////////////////////////////////////////////////
29beb80d	334	Double_t AliMath::BesselI0(Double_t x)
	335	{
	336	// Computation of the modified Bessel function I_0(x) for any real x.
	337	//
	338	// The algorithm is based on the article by Abramowitz and Stegun [1]
	339	// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
	340	//
	341	// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
	342	// Applied Mathematics Series vol. 55 (1964), Washington.
	343	//
	344	//--- NvE 12-mar-2000 UU-SAP Utrecht
	345
	346	// Parameters of the polynomial approximation
	347	const Double_t p1=1.0, p2=3.5156229, p3=3.0899424,
	348	p4=1.2067492, p5=0.2659732, p6=3.60768e-2, p7=4.5813e-3;
	349
	350	const Double_t q1= 0.39894228, q2= 1.328592e-2, q3= 2.25319e-3,
	351	q4=-1.57565e-3, q5= 9.16281e-3, q6=-2.057706e-2,
	352	q7= 2.635537e-2, q8=-1.647633e-2, q9= 3.92377e-3;
	353
	354	Double_t ax=fabs(x);
	355
	356	Double_t y=0,result=0;
	357
	358	if (ax < 3.75)
	359	{
	360	y=pow(x/3.75,2);
	361	result=p1+y(p2+y(p3+y(p4+y(p5+y(p6+yp7)))));
	362	}
	363	else
	364	{
	365	y=3.75/ax;
	366	result=(exp(ax)/sqrt(ax))(q1+y(q2+y(q3+y(q4+y(q5+y(q6+y(q7+y(q8+y*q9))))))));
	367	}
	368
	369	return result;
	370	}
	371	///////////////////////////////////////////////////////////////////////////
	372	Double_t AliMath::BesselK0(Double_t x)
	373	{
	374	// Computation of the modified Bessel function K_0(x) for positive real x.
	375	//
	376	// The algorithm is based on the article by Abramowitz and Stegun [1]
	377	// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
	378	//
	379	// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
	380	// Applied Mathematics Series vol. 55 (1964), Washington.
	381	//
	382	//--- NvE 12-mar-2000 UU-SAP Utrecht
	383
	384	// Parameters of the polynomial approximation
	385	const Double_t p1=-0.57721566, p2=0.42278420, p3=0.23069756,
	386	p4= 3.488590e-2, p5=2.62698e-3, p6=1.0750e-4, p7=7.4e-5;
	387
	388	const Double_t q1= 1.25331414, q2=-7.832358e-2, q3= 2.189568e-2,
	389	q4=-1.062446e-2, q5= 5.87872e-3, q6=-2.51540e-3, q7=5.3208e-4;
	390
	391	if (x <= 0)
	392	{
	393	cout << " BesselK0 Invalid argument x = " << x << endl;
	394	return 0;
	395	}
	396
	397	Double_t y=0,result=0;
398
399	if (x <= 2)
400	{
401	y=x*x/4.;
402	result=(-log(x/2.)BesselI0(x))+(p1+y(p2+y(p3+y(p4+y(p5+y(p6+y*p7))))));
403	}
404	else
405	{
406	y=2./x;
407	result=(exp(-x)/sqrt(x))(q1+y(q2+y(q3+y(q4+y(q5+y(q6+y*q7))))));
408	}
409
410	return result;
411	}
412	///////////////////////////////////////////////////////////////////////////
413	Double_t AliMath::BesselI1(Double_t x)
414	{
415	// Computation of the modified Bessel function I_1(x) for any real x.
416	//
417	// The algorithm is based on the article by Abramowitz and Stegun [1]
418	// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
419	//
420	// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
421	// Applied Mathematics Series vol. 55 (1964), Washington.
422	//
423	//--- NvE 12-mar-2000 UU-SAP Utrecht
424
425	// Parameters of the polynomial approximation
426	const Double_t p1=0.5, p2=0.87890594, p3=0.51498869,
427	p4=0.15084934, p5=2.658733e-2, p6=3.01532e-3, p7=3.2411e-4;
428
429	const Double_t q1= 0.39894228, q2=-3.988024e-2, q3=-3.62018e-3,
430	q4= 1.63801e-3, q5=-1.031555e-2, q6= 2.282967e-2,
431	q7=-2.895312e-2, q8= 1.787654e-2, q9=-4.20059e-3;
432
433	Double_t ax=fabs(x);
434
435	Double_t y=0,result=0;
436
437	if (ax < 3.75)
438	{
439	y=pow(x/3.75,2);
440	result=x(p1+y(p2+y(p3+y(p4+y(p5+y(p6+y*p7))))));
441	}
442	else
443	{
444	y=3.75/ax;
445	result=(exp(ax)/sqrt(ax))(q1+y(q2+y(q3+y(q4+y(q5+y(q6+y(q7+y(q8+y*q9))))))));
446	if (x < 0) result=-result;
447	}
448
449	return result;
450	}
451	///////////////////////////////////////////////////////////////////////////
452	Double_t AliMath::BesselK1(Double_t x)
453	{
454	// Computation of the modified Bessel function K_1(x) for positive real x.
455	//
456	// The algorithm is based on the article by Abramowitz and Stegun [1]
457	// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
458	//
459	// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
460	// Applied Mathematics Series vol. 55 (1964), Washington.
461	//
462	//--- NvE 12-mar-2000 UU-SAP Utrecht
463
464	// Parameters of the polynomial approximation
465	const Double_t p1= 1., p2= 0.15443144, p3=-0.67278579,
466	p4=-0.18156897, p5=-1.919402e-2, p6=-1.10404e-3, p7=-4.686e-5;
467
468	const Double_t q1= 1.25331414, q2= 0.23498619, q3=-3.655620e-2,
469	q4= 1.504268e-2, q5=-7.80353e-3, q6= 3.25614e-3, q7=-6.8245e-4;
470
471	if (x <= 0)
472	{
473	cout << " BesselK1 Invalid argument x = " << x << endl;
474	return 0;
475	}
476
477	Double_t y=0,result=0;
478
479	if (x <= 2)
480	{
481	y=x*x/4.;
482	result=(log(x/2.)BesselI1(x))+(1./x)(p1+y(p2+y(p3+y(p4+y(p5+y(p6+yp7))))));
483	}
484	else
485	{
486	y=2./x;
487	result=(exp(-x)/sqrt(x))(q1+y(q2+y(q3+y(q4+y(q5+y(q6+y*q7))))));
488	}
489
490	return result;
491	}
492	///////////////////////////////////////////////////////////////////////////
493	Double_t AliMath::BesselK(Int_t n,Double_t x)
494	{
495	// Computation of the Integer Order Modified Bessel function K_n(x)
496	// for n=0,1,2,... and positive real x.
497	//
498	// The algorithm uses the recurrence relation
499	//
500	// K_n+1(x) = (2n/x)*K_n(x) + K_n-1(x)
501	//
502	// as denoted in Numerical Recipes 2nd ed. on p. 232 (W.H.Press et al.).
503	//
504	//--- NvE 12-mar-2000 UU-SAP Utrecht
505
506	if (x <= 0 \|\| n < 0)
507	{
508	cout << " BesselK Invalid argument(s) (n,x) = (" << n << " , " << x << ")" << endl;
509	return 0;
510	}
511
512	if (n==0) return BesselK0(x);
513
514	if (n==1) return BesselK1(x);
515
516	// Perform upward recurrence for all x
517	Double_t tox=2./x;
518	Double_t bkm=BesselK0(x);
519	Double_t bk=BesselK1(x);
520	Double_t bkp=0;
521	for (Int_t j=1; j<n; j++)
522	{
523	bkp=bkm+double(j)toxbk;
524	bkm=bk;
525	bk=bkp;
526	}
527
528	return bk;
529	}
530	///////////////////////////////////////////////////////////////////////////
531	Double_t AliMath::BesselI(Int_t n,Double_t x)
532	{
533	// Computation of the Integer Order Modified Bessel function I_n(x)
534	// for n=0,1,2,... and any real x.
535	//
536	// The algorithm uses the recurrence relation
537	//
538	// I_n+1(x) = (-2n/x)*I_n(x) + I_n-1(x)
539	//
540	// as denoted in Numerical Recipes 2nd ed. on p. 232 (W.H.Press et al.).
541	//
542	//--- NvE 12-mar-2000 UU-SAP Utrecht
543
544	Int_t iacc=40; // Increase to enhance accuracy
545	Double_t bigno=1.e10, bigni=1.e-10;
546
547	if (n < 0)
548	{
549	cout << " BesselI Invalid argument (n,x) = (" << n << " , " << x << ")" << endl;
550	return 0;
551	}
552
553	if (n==0) return BesselI0(x);
554
555	if (n==1) return BesselI1(x);
556
557	if (fabs(x) < 1.e-10) return 0;
558
559	Double_t tox=2./fabs(x);
560	Double_t bip=0,bim=0;
561	Double_t bi=1;
562	Double_t result=0;
563	Int_t m=2((n+int(sqrt(float(iaccn))))); // Downward recurrence from even m
564	for (Int_t j=m; j<=1; j--)
565	{
566	bim=bip+double(j)toxbi;
567	bip=bi;
568	bi=bim;
569	if (fabs(bi) > bigno) // Renormalise to prevent overflows
570	{
571	result*=bigni;
572	bi*=bigni;
573	bip*=bigni;
574	}
575	if (j==n) result=bip;
576	}
577
578	result*=BesselI0(x)/bi; // Normalise with I0(x)
579	if ((x < 0) && (n%2 == 1)) result=-result;
580
581	return result;
582	}
583	///////////////////////////////////////////////////////////////////////////