* provided "as is" without express or implied warranty. *
**************************************************************************/
-/*
-$Log$
-*/
+// $Id$
+
+///////////////////////////////////////////////////////////////////////////
+// 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()
+AliMath::AliMath() : TObject()
{
// Default constructor
}
// Destructor
}
///////////////////////////////////////////////////////////////////////////
-Float_t AliMath::Gamma(Float_t z)
+AliMath::AliMath(AliMath& m) : TObject(m)
+{
+// Copy constructor
+}
+///////////////////////////////////////////////////////////////////////////
+Double_t AliMath::Gamma(Double_t z)
{
// Computation of gamma(z) for all z>0.
//
return 0;
}
- Float_t v=LnGamma(z);
+ Double_t v=LnGamma(z);
return exp(v);
}
///////////////////////////////////////////////////////////////////////////
-Float_t AliMath::Gamma(Float_t a,Float_t x)
+Double_t AliMath::Gamma(Double_t a,Double_t x)
{
// Computation of the incomplete gamma function P(a,x)
//
}
}
///////////////////////////////////////////////////////////////////////////
-Float_t AliMath::LnGamma(Float_t z)
+Double_t AliMath::LnGamma(Double_t z)
{
// Computation of ln[gamma(z)] for all z>0.
//
y+=1.;
ser+=c[i]/y;
}
- Float_t v=tmp+log(c[0]*ser/x);
+ Double_t v=tmp+log(c[0]*ser/x);
return v;
}
///////////////////////////////////////////////////////////////////////////
-Float_t AliMath::GamSer(Float_t a,Float_t x)
+Double_t AliMath::GamSer(Double_t a,Double_t x)
{
// Computation of the incomplete gamma function P(a,x)
// via its series representation.
//--- Nve 14-nov-1998 UU-SAP Utrecht
Int_t itmax=100; // Maximum number of iterations
- Float_t eps=3.e-7; // Relative accuracy
+ Double_t eps=3.e-7; // Relative accuracy
if (a<=0.)
{
return 0;
}
- Float_t gln=LnGamma(a);
- Float_t ap=a;
- Float_t sum=1./a;
- Float_t del=sum;
+ 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.;
if (fabs(del)<fabs(sum*eps)) break;
if (n==itmax) cout << "*GamSer(a,x)* a too large or itmax too small" << endl;
}
- Float_t v=sum*exp(-x+a*log(x)-gln);
+ Double_t v=sum*exp(-x+a*log(x)-gln);
return v;
}
///////////////////////////////////////////////////////////////////////////
-Float_t AliMath::GamCf(Float_t a,Float_t x)
+Double_t AliMath::GamCf(Double_t a,Double_t x)
{
// Computation of the incomplete gamma function P(a,x)
// via its continued fraction representation.
//--- Nve 14-nov-1998 UU-SAP Utrecht
Int_t itmax=100; // Maximum number of iterations
- Float_t eps=3.e-7; // Relative accuracy
- Float_t fpmin=1.e-30; // Smallest Float_t value allowed here
+ Double_t eps=3.e-7; // Relative accuracy
+ Double_t fpmin=1.e-30; // Smallest Double_t value allowed here
if (a<=0.)
{
return 0;
}
- Float_t gln=LnGamma(a);
- Float_t b=x+1.-a;
- Float_t c=1./fpmin;
- Float_t d=1./b;
- Float_t h=d;
- Float_t an,del;
+ 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=float(-i)*(float(i)-a);
+ an=double(-i)*(double(i)-a);
b+=2.;
d=an*d+b;
if (fabs(d)<fpmin) d=fpmin;
if (fabs(del-1.)<eps) break;
if (i==itmax) cout << "*GamCf(a,x)* a too large or itmax too small" << endl;
}
- Float_t v=exp(-x+a*log(x)-gln)*h;
+ Double_t v=exp(-x+a*log(x)-gln)*h;
return (1.-v);
}
///////////////////////////////////////////////////////////////////////////
-Float_t AliMath::Erf(Float_t x)
+Double_t AliMath::Erf(Double_t x)
{
// Computation of the error function erf(x).
//
return (1.-Erfc(x));
}
///////////////////////////////////////////////////////////////////////////
-Float_t AliMath::Erfc(Float_t x)
+Double_t AliMath::Erfc(Double_t x)
{
// Computation of the complementary error function erfc(x).
//
//--- Nve 14-nov-1998 UU-SAP Utrecht
// The parameters of the Chebyshev fit
- const Float_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;
+ 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;
- Float_t v=1.; // The return value
+ Double_t v=1.; // The return value
- Float_t z=fabs(x);
+ Double_t z=fabs(x);
if (z <= 0.) return v; // erfc(0)=1
- Float_t t=1./(1.+0.5*z);
+ 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)))))))));
return v;
}
///////////////////////////////////////////////////////////////////////////
-Float_t AliMath::Prob(Float_t chi2,Int_t ndf)
+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).
// This code may be activated in case the gamma function gives problems
// if (ndf==1)
// {
-// Float_t v=1.-Erf(sqrt(chi2)/sqrt(2.));
+// 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
-// Float_t q=sqrt(2.*chi2)-sqrt(float(2*ndf-1));
+// Double_t q=sqrt(2.*chi2)-sqrt(double(2*ndf-1));
// if (n>30 && q>0.)
// {
-// Float_t v=0.5*(1.-Erf(q/sqrt(2.)));
+// Double_t v=0.5*(1.-Erf(q/sqrt(2.)));
// return v;
// }
// Evaluate the incomplete gamma function
- Float_t a=float(ndf)/2.;
- Float_t x=chi2/2.;
+ 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;
+}
+///////////////////////////////////////////////////////////////////////////
git://git.uio.no / u / mrichter / AliRoot.git / blobdiff