// @(#)root/base:$Name: $:$Id: TMath.cxx,v 1.37 2003/05/07 16:25:29 brun Exp $
// Author: Fons Rademakers 29/07/95
/*************************************************************************
* Copyright (C) 1995-2000, Rene Brun and Fons Rademakers. *
* All rights reserved. *
* *
* For the licensing terms see $ROOTSYS/LICENSE. *
* For the list of contributors see $ROOTSYS/README/CREDITS. *
*************************************************************************/
//////////////////////////////////////////////////////////////////////////
// //
// TMath //
// //
// Encapsulate math routines (i.e. provide a kind of namespace). //
// For the time being avoid templates. //
// //
//////////////////////////////////////////////////////////////////////////
#include "TMath.h"
#include "TError.h"
#include <math.h>
#include <string.h>
//const Double_t
// TMath::Pi = 3.14159265358979323846,
// TMath::E = 2.7182818284590452354;
ClassImp(TMath)
//______________________________________________________________________________
#if defined(R__MAC) || defined(R__KCC)
Double_t hypot(Double_t x, Double_t y) {
return sqrt(x*x+y*y);
}
#endif
//______________________________________________________________________________
Long_t TMath::Sqrt(Long_t x)
{
return (Long_t) (sqrt((Double_t)x) + 0.5);
}
//______________________________________________________________________________
Long_t TMath::Hypot(Long_t x, Long_t y)
{
return (Long_t) (hypot((Double_t)x, (Double_t)y) + 0.5);
}
//______________________________________________________________________________
Double_t TMath::Hypot(Double_t x, Double_t y)
{
return hypot(x, y);
}
//______________________________________________________________________________
Double_t TMath::ASinH(Double_t x)
{
#if defined(WIN32) || defined(R__KCC)
return log(x+sqrt(x*x+1));
#else
return asinh(x);
#endif
}
//______________________________________________________________________________
Double_t TMath::ACosH(Double_t x)
{
#if defined(WIN32) || defined(R__KCC)
return log(x+sqrt(x*x-1));
#else
return acosh(x);
#endif
}
//______________________________________________________________________________
Double_t TMath::ATanH(Double_t x)
{
#if defined(WIN32) || defined(R__KCC)
return log((1+x)/(1-x))/2;
#else
return atanh(x);
#endif
}
//______________________________________________________________________________
Double_t TMath::Ceil(Double_t x)
{
return ceil(x);
}
//______________________________________________________________________________
Double_t TMath::Floor(Double_t x)
{
return floor(x);
}
//______________________________________________________________________________
Double_t TMath::Log2(Double_t x)
{
return log(x)/log(2.0);
}
//______________________________________________________________________________
Long_t TMath::NextPrime(Long_t x)
{
// Return next prime number after x, unless x is a prime in which case
// x is returned.
if (x < 2)
return 2;
if (x == 3)
return 3;
if (x % 2 == 0)
x++;
Long_t sqr = (Long_t) sqrt((Double_t)x) + 1;
for (;;) {
Long_t n;
for (n = 3; (n <= sqr) && ((x % n) != 0); n += 2)
;
if (n > sqr)
return x;
x += 2;
}
}
//______________________________________________________________________________
Int_t TMath::Nint(Float_t x)
{
// Round to nearest integer. Rounds half integers to the nearest
// even integer.
int i;
if (x >= 0) {
i = int(x + 0.5);
if (x + 0.5 == Float_t(i) && i & 1) i--;
} else {
i = int(x - 0.5);
if (x - 0.5 == Float_t(i) && i & 1) i++;
}
return i;
}
//______________________________________________________________________________
Int_t TMath::Nint(Double_t x)
{
// Round to nearest integer. Rounds half integers to the nearest
// even integer.
int i;
if (x >= 0) {
i = int(x + 0.5);
if (x + 0.5 == Double_t(i) && i & 1) i--;
} else {
i = int(x - 0.5);
if (x - 0.5 == Double_t(i) && i & 1) i++;
}
return i;
}
//______________________________________________________________________________
Float_t *TMath::Cross(Float_t v1[3],Float_t v2[3],Float_t out[3])
{
// Calculate the Cross Product of two vectors:
// out = [v1 x v2]
out[0] = v1[1] * v2[2] - v1[2] * v2[1];
out[1] = v1[2] * v2[0] - v1[0] * v2[2];
out[2] = v1[0] * v2[1] - v1[1] * v2[0];
return out;
}
//______________________________________________________________________________
Double_t *TMath::Cross(Double_t v1[3],Double_t v2[3],Double_t out[3])
{
// Calculate the Cross Product of two vectors:
// out = [v1 x v2]
out[0] = v1[1] * v2[2] - v1[2] * v2[1];
out[1] = v1[2] * v2[0] - v1[0] * v2[2];
out[2] = v1[0] * v2[1] - v1[1] * v2[0];
return out;
}
//______________________________________________________________________________
Double_t TMath::Erf(Double_t x)
{
// Computation of the error function erf(x).
// Erf(x) = (2/sqrt(pi)) Integral(exp(-t^2))dt between 0 and x
//--- NvE 14-nov-1998 UU-SAP Utrecht
return (1-Erfc(x));
}
//______________________________________________________________________________
Double_t TMath::Erfc(Double_t x)
{
// Compute the complementary error function erfc(x).
// Erfc(x) = (2/sqrt(pi)) Integral(exp(-t^2))dt between x and infinity
//
//--- 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 = Abs(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 TMath::Factorial(Int_t n)
{
// Compute factorial(n)
if(n <= 0) return 1.;
Double_t x=1;
Int_t b=0;
do {
b++;
x *= b;
} while(b!=n);
return x;
}
//______________________________________________________________________________
Double_t TMath::Freq(Double_t x)
{
// Computation of the normal frequency function freq(x).
// Freq(x) = (1/sqrt(2pi)) Integral(exp(-t^2/2))dt between -infinity and x.
//
// Translated from CERNLIB C300 by Rene Brun.
const Double_t C1 = 0.56418958354775629;
const Double_t W2 = 1.41421356237309505;
const Double_t p10 = 2.4266795523053175e+2, q10 = 2.1505887586986120e+2,
p11 = 2.1979261618294152e+1, q11 = 9.1164905404514901e+1,
p12 = 6.9963834886191355e+0, q12 = 1.5082797630407787e+1,
p13 =-3.5609843701815385e-2, q13 = 1;
const Double_t p20 = 3.00459261020161601e+2, q20 = 3.00459260956983293e+2,
p21 = 4.51918953711872942e+2, q21 = 7.90950925327898027e+2,
p22 = 3.39320816734343687e+2, q22 = 9.31354094850609621e+2,
p23 = 1.52989285046940404e+2, q23 = 6.38980264465631167e+2,
p24 = 4.31622272220567353e+1, q24 = 2.77585444743987643e+2,
p25 = 7.21175825088309366e+0, q25 = 7.70001529352294730e+1,
p26 = 5.64195517478973971e-1, q26 = 1.27827273196294235e+1,
p27 =-1.36864857382716707e-7, q27 = 1;
const Double_t p30 =-2.99610707703542174e-3, q30 = 1.06209230528467918e-2,
p31 =-4.94730910623250734e-2, q31 = 1.91308926107829841e-1,
p32 =-2.26956593539686930e-1, q32 = 1.05167510706793207e+0,
p33 =-2.78661308609647788e-1, q33 = 1.98733201817135256e+0,
p34 =-2.23192459734184686e-2, q34 = 1;
Double_t v = TMath::Abs(x)/W2;
Double_t vv = v*v;
Double_t ap, aq, h, hc, y;
if (v < 0.5) {
y=vv;
ap=p13;
aq=q13;
ap = p12 +y*ap;
ap = p11 +y*ap;
ap = p10 +y*ap;
aq = q12 +y*aq;
aq = q11 +y*aq;
aq = q10 +y*aq;
h = v*ap/aq;
hc = 1-h;
} else if (v < 4) {
ap = p27;
aq = q27;
ap = p26 +v*ap;
ap = p25 +v*ap;
ap = p24 +v*ap;
ap = p23 +v*ap;
ap = p22 +v*ap;
ap = p21 +v*ap;
ap = p20 +v*ap;
aq = q26 +v*aq;
aq = q25 +v*aq;
aq = q24 +v*aq;
aq = q23 +v*aq;
aq = q22 +v*aq;
aq = q21 +v*aq;
aq = q20 +v*aq;
hc = TMath::Exp(-vv)*ap/aq;
h = 1-hc;
} else {
y = 1/vv;
ap = p34;
aq = q34;
ap = p33 +y*ap;
ap = p32 +y*ap;
ap = p31 +y*ap;
ap = p30 +y*ap;
aq = q33 +y*aq;
aq = q32 +y*aq;
aq = q31 +y*aq;
aq = q30 +y*aq;
hc = TMath::Exp(-vv)*(C1+y*ap/aq)/v;
h = 1-hc;
}
if (x > 0) return 0.5 +0.5*h;
else return 0.5*hc;
}
//______________________________________________________________________________
Double_t TMath::Gamma(Double_t z)
{
// Computation of gamma(z) for all z>0.
//
// C.Lanczos, SIAM Journal of Numerical Analysis B1 (1964), 86.
//
//--- Nve 14-nov-1998 UU-SAP Utrecht
if (z<=0) return 0;
Double_t v = LnGamma(z);
return Exp(v);
}
//______________________________________________________________________________
Double_t TMath::Gamma(Double_t a,Double_t x)
{
// Computation of the incomplete gamma function P(a,x)
//
//--- Nve 14-nov-1998 UU-SAP Utrecht
if (a <= 0 || x <= 0) return 0;
if (x < (a+1)) return GamSer(a,x);
else return GamCf(a,x);
}
//______________________________________________________________________________
Double_t TMath::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
Double_t eps = 3.e-7; // Relative accuracy
Double_t fpmin = 1.e-30; // Smallest Double_t value allowed here
if (a <= 0 || x <= 0) 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_t(-i)*(Double_t(i)-a);
b += 2;
d = an*d+b;
if (Abs(d) < fpmin) d = fpmin;
c = b+an/c;
if (Abs(c) < fpmin) c = fpmin;
d = 1/d;
del = d*c;
h = h*del;
if (Abs(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 TMath::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
Double_t eps = 3.e-7; // Relative accuracy
if (a <= 0 || x <= 0) 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 (TMath::Abs(del) < Abs(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 TMath::BreitWigner(Double_t x, Double_t mean, Double_t gamma)
{
// Calculate a Breit Wigner function with mean and gamma.
Double_t bw = gamma/((x-mean)*(x-mean) + gamma*gamma/4);
return bw/(2*Pi());
}
//______________________________________________________________________________
Double_t TMath::Gaus(Double_t x, Double_t mean, Double_t sigma)
{
// Calculate a gaussian function with mean and sigma.
if (sigma == 0) return 1.e30;
Double_t arg = (x-mean)/sigma;
return TMath::Exp(-0.5*arg*arg);
}
//______________________________________________________________________________
Double_t TMath::Landau(Double_t x, Double_t mpv, Double_t sigma)
{
// The LANDAU function with mpv(most probable value) and sigma.
// This function has been adapted from the CERNLIB routine G110 denlan.
Double_t p1[5] = {0.4259894875,-0.1249762550, 0.03984243700, -0.006298287635, 0.001511162253};
Double_t q1[5] = {1.0 ,-0.3388260629, 0.09594393323, -0.01608042283, 0.003778942063};
Double_t p2[5] = {0.1788541609, 0.1173957403, 0.01488850518, -0.001394989411, 0.0001283617211};
Double_t q2[5] = {1.0 , 0.7428795082, 0.3153932961, 0.06694219548, 0.008790609714};
Double_t p3[5] = {0.1788544503, 0.09359161662,0.006325387654, 0.00006611667319,-0.000002031049101};
Double_t q3[5] = {1.0 , 0.6097809921, 0.2560616665, 0.04746722384, 0.006957301675};
Double_t p4[5] = {0.9874054407, 118.6723273, 849.2794360, -743.7792444, 427.0262186};
Double_t q4[5] = {1.0 , 106.8615961, 337.6496214, 2016.712389, 1597.063511};
Double_t p5[5] = {1.003675074, 167.5702434, 4789.711289, 21217.86767, -22324.94910};
Double_t q5[5] = {1.0 , 156.9424537, 3745.310488, 9834.698876, 66924.28357};
Double_t p6[5] = {1.000827619, 664.9143136, 62972.92665, 475554.6998, -5743609.109};
Double_t q6[5] = {1.0 , 651.4101098, 56974.73333, 165917.4725, -2815759.939};
Double_t a1[3] = {0.04166666667,-0.01996527778, 0.02709538966};
Double_t a2[2] = {-1.845568670,-4.284640743};
if (sigma <= 0) return 0;
Double_t v = (x-mpv)/sigma;
Double_t u, ue, us, den;
if (v < -5.5) {
u = TMath::Exp(v+1.0);
ue = TMath::Exp(-1/u);
us = TMath::Sqrt(u);
den = 0.3989422803*(ue/us)*(1+(a1[0]+(a1[1]+a1[2]*u)*u)*u);
} else if(v < -1) {
u = TMath::Exp(-v-1);
den = TMath::Exp(-u)*TMath::Sqrt(u)*
(p1[0]+(p1[1]+(p1[2]+(p1[3]+p1[4]*v)*v)*v)*v)/
(q1[0]+(q1[1]+(q1[2]+(q1[3]+q1[4]*v)*v)*v)*v);
} else if(v < 1) {
den = (p2[0]+(p2[1]+(p2[2]+(p2[3]+p2[4]*v)*v)*v)*v)/
(q2[0]+(q2[1]+(q2[2]+(q2[3]+q2[4]*v)*v)*v)*v);
} else if(v < 5) {
den = (p3[0]+(p3[1]+(p3[2]+(p3[3]+p3[4]*v)*v)*v)*v)/
(q3[0]+(q3[1]+(q3[2]+(q3[3]+q3[4]*v)*v)*v)*v);
} else if(v < 12) {
u = 1/v;
den = u*u*(p4[0]+(p4[1]+(p4[2]+(p4[3]+p4[4]*u)*u)*u)*u)/
(q4[0]+(q4[1]+(q4[2]+(q4[3]+q4[4]*u)*u)*u)*u);
} else if(v < 50) {
u = 1/v;
den = u*u*(p5[0]+(p5[1]+(p5[2]+(p5[3]+p5[4]*u)*u)*u)*u)/
(q5[0]+(q5[1]+(q5[2]+(q5[3]+q5[4]*u)*u)*u)*u);
} else if(v < 300) {
u = 1/v;
den = u*u*(p6[0]+(p6[1]+(p6[2]+(p6[3]+p6[4]*u)*u)*u)*u)/
(q6[0]+(q6[1]+(q6[2]+(q6[3]+q6[4]*u)*u)*u)*u);
} else {
u = 1/(v-v*TMath::Log(v)/(v+1));
den = u*u*(1+(a2[0]+a2[1]*u)*u);
}
return den;
}
//______________________________________________________________________________
Double_t TMath::LnGamma(Double_t z)
{
// Computation of ln[gamma(z)] for all z>0.
//
// 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) return 0;
// Coefficients for the series expansion
Double_t c[7] = { 2.5066282746310005, 76.18009172947146, -86.50532032941677
,24.01409824083091, -1.231739572450155, 0.1208650973866179e-2
,-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;
}
//______________________________________________________________________________
Float_t TMath::Normalize(Float_t v[3])
{
// Normalize a vector v in place.
// Returns the norm of the original vector.
Float_t d = Sqrt(v[0]*v[0]+v[1]*v[1]+v[2]*v[2]);
if (d != 0) {
v[0] /= d;
v[1] /= d;
v[2] /= d;
}
return d;
}
//______________________________________________________________________________
Double_t TMath::Normalize(Double_t v[3])
{
// Normalize a vector v in place.
// Returns the norm of the original vector.
Double_t d = Sqrt(v[0]*v[0]+v[1]*v[1]+v[2]*v[2]);
if (d != 0)
{
v[0] /= d;
v[1] /= d;
v[2] /= d;
}
return d;
}
//______________________________________________________________________________
Float_t *TMath::Normal2Plane(Float_t p1[3],Float_t p2[3],Float_t p3[3], Float_t normal[3])
{
// Calculate a normal vector of a plane.
//
// Input:
// Float_t *p1,*p2,*p3 - 3 3D points belonged the plane to define it.
//
// Return:
// Pointer to 3D normal vector (normalized)
Float_t v1[3], v2[3];
v1[0] = p2[0] - p1[0];
v1[1] = p2[1] - p1[1];
v1[2] = p2[2] - p1[2];
v2[0] = p3[0] - p1[0];
v2[1] = p3[1] - p1[1];
v2[2] = p3[2] - p1[2];
NormCross(v1,v2,normal);
return normal;
}
//______________________________________________________________________________
Double_t *TMath::Normal2Plane(Double_t p1[3],Double_t p2[3],Double_t p3[3], Double_t normal[3])
{
// Calculate a normal vector of a plane.
//
// Input:
// Float_t *p1,*p2,*p3 - 3 3D points belonged the plane to define it.
//
// Return:
// Pointer to 3D normal vector (normalized)
Double_t v1[3], v2[3];
v1[0] = p2[0] - p1[0];
v1[1] = p2[1] - p1[1];
v1[2] = p2[2] - p1[2];
v2[0] = p3[0] - p1[0];
v2[1] = p3[1] - p1[1];
v2[2] = p3[2] - p1[2];
NormCross(v1,v2,normal);
return normal;
}
//______________________________________________________________________________
Double_t TMath::Poisson(Double_t x, Double_t par)
{
// compute the Poisson distribution function for (x,par)
if (x<=0) return 0;
return TMath::Power(par,x)/TMath::Gamma(x+1)/TMath::Exp(par);
}
//______________________________________________________________________________
Double_t TMath::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;
}
if (ndf==1) {
Double_t v = 1.-Erf(Sqrt(chi2)/Sqrt(2.));
return v;
}
// Gaussian approximation for large ndf
Double_t q = Sqrt(2*chi2)-Sqrt(Double_t(2*ndf-1));
if (ndf > 30 && q > 5) {
Double_t v = 0.5*(1-Erf(q/Sqrt(2.)));
return v;
}
// Evaluate the incomplete gamma function
return (1-Gamma(0.5*ndf,0.5*chi2));
}
//______________________________________________________________________________
Double_t TMath::KolmogorovProb(Double_t z)
{
// Calculates the Kolmogorov distribution function,
//
/*
*/
//
// which gives the probability that Kolmogorov's test statistic will exceed
// the value z assuming the null hypothesis. This gives a very powerful
// test for comparing two one-dimensional distributions.
// see, for example, Eadie et al, "statistocal Methods in Experimental
// Physics', pp 269-270).
//
// This function returns the confidence level for the null hypothesis, where:
// z = dn*sqrt(n), and
// dn is the maximum deviation between a hypothetical distribution
// function and an experimental distribution with
// n events
//
// NOTE: To compare two experimental distributions with m and n events,
// use z = sqrt(m*n/(m+n))*dn
//
// Accuracy: The function is far too accurate for any imaginable application.
// Probabilities less than 10^-15 are returned as zero.
// However, remember that the formula is only valid for "large" n.
// Theta function inversion formula is used for z <= 1
//
// This function was translated by Rene Brun from PROBKL in CERNLIB.
// cons[j] = -0.5*(PI*2*j-1)/2(**2
const Double_t cons[3] = { -1.233700550136 , -11.10330496 , -30.84251376};
// jf2[j] = -2* j**2
const Double_t fj2[5] = {-2. , -8. , -18. , -32. , -50.};
const Double_t sqr2pi = 2.50662827463;
Double_t p = 0;
Int_t j;
if (z < 0.2) return 1;
if (z > 1) { // use series in exp(z**2)
Double_t c;
Double_t sig2 = -2;
Double_t z2 = z*z;
for (j=0;j<5;j++) {
sig2 = -sig2;
c = fj2[j] *z2;
if (c < -100) return p;
p += sig2*TMath::Exp(c);
}
return p;
}
// z< 1 use series in exp(1/z**2)
Double_t zinv = 1/z;
Double_t a = sqr2pi*zinv;
Double_t zinv2 = zinv*zinv;
Double_t arg;
for (j=0;j<3;j++) {
arg = cons[j]*zinv2;
if (arg < -30) continue;
p += TMath::Exp(arg);
}
p = 1 - a*p;
return p;
}
//______________________________________________________________________________
Double_t TMath::Voigt(Double_t x, Double_t sigma, Double_t lg, Int_t R)
{
// Computation of Voigt function (normalised).
// Voigt is a convolution of
// gauss(x) = 1/(sqrt(2*pi)*sigma) * exp(x*x/(2*sigma*sigma)
// and
// lorentz(x) = (1/pi) * (lg/2) / (x*x + g*g/4)
// functions.
//
// The Voigt function is known to be the real part of Faddeeva function also
// called complex error function [2].
//
// The algoritm was developed by J. Humlicek [1].
// This code is based on fortran code presented by R. J. Wells [2].
// Translated and adapted by Miha D. Puc
//
// To calculate the Faddeeva function with relative error less than 10^(-R).
// R can be set by the the user subject to the constraints 2 <= R <= 5.
//
// [1] J. Humlicek, JQSRT, 21, 437 (1982).
// [2] R.J. Wells "Rapid Approximation to the Voigt/Faddeeva Function and its
// Derivatives" JQSRT 62 (1999), pp 29-48.
// http://www-atm.physics.ox.ac.uk/user/wells/voigt.html
if ((sigma < 0 || lg < 0) || (sigma==0 && lg==0)) {
return 0; // Not meant to be for those who want to be thinner than 0
}
if (sigma == 0) {
return lg * 0.159154943 / (x*x + lg*lg /4); //pure Lorentz
}
if (lg == 0) { //pure gauss
return 0.39894228 / sigma * TMath::Exp(-x*x / (2*sigma*sigma));
}
Double_t X, Y, K;
X = x / sigma / 1.41421356;
Y = lg / 2 / sigma / 1.41421356;
Double_t R0, R1;
if (R < 2) R = 2;
if (R > 5) R = 5;
R0=1.51 * exp(1.144 * (Double_t)R);
R1=1.60 * exp(0.554 * (Double_t)R);
// Constants
const Double_t RRTPI = 0.56418958; // 1/SQRT(pi)
Double_t Y0, Y0PY0, Y0Q; // for CPF12 algorithm
Y0 = 1.5;
Y0PY0 = Y0 + Y0;
Y0Q = Y0 * Y0;
Double_t C[6] = { 1.0117281, -0.75197147, 0.012557727, 0.010022008, -0.00024206814, 0.00000050084806};
Double_t S[6] = { 1.393237, 0.23115241, -0.15535147, 0.0062183662, 0.000091908299, -0.00000062752596};
Double_t T[6] = { 0.31424038, 0.94778839, 1.5976826, 2.2795071, 3.0206370, 3.8897249};
// Local variables
int J; // Loop variables
int RG1, RG2, RG3; // y polynomial flags
Double_t ABX, XQ, YQ, YRRTPI; // --x--, x^2, y^2, y/SQRT(pi)
Double_t XLIM0, XLIM1, XLIM2, XLIM3, XLIM4; // --x-- on region boundaries
Double_t A0=0, D0=0, D2=0, E0=0, E2=0, E4=0, H0=0, H2=0, H4=0, H6=0;// W4 temporary variables
Double_t P0=0, P2=0, P4=0, P6=0, P8=0, Z0=0, Z2=0, Z4=0, Z6=0, Z8=0;
Double_t XP[6], XM[6], YP[6], YM[6]; // CPF12 temporary values
Double_t MQ[6], PQ[6], MF[6], PF[6];
Double_t D, YF, YPY0, YPY0Q;
//***** Start of executable code *****************************************
RG1 = 1; // Set flags
RG2 = 1;
RG3 = 1;
YQ = Y * Y; // y^2
YRRTPI = Y * RRTPI; // y/SQRT(pi)
// Region boundaries when both K and L are required or when R<>4
XLIM0 = R0 - Y;
XLIM1 = R1 - Y;
XLIM3 = 3.097 * Y - 0.45;
XLIM2 = 6.8 - Y;
XLIM4 = 18.1 * Y + 1.65;
if ( Y <= 1e-6 ) { // When y<10^-6 avoid W4 algorithm.
XLIM1 = XLIM0;
XLIM2 = XLIM0;
}
ABX = fabs(X); // |x|
XQ = ABX * ABX; // x^2
if ( ABX > XLIM0 ) { // Region 0 algorithm
K = YRRTPI / (XQ + YQ);
} else if ( ABX > XLIM1 ) { // Humlicek W4 Region 1
if ( RG1 != 0 ) { // First point in Region 1
RG1 = 0;
A0 = YQ + 0.5; // Region 1 y-dependents
D0 = A0*A0;
D2 = YQ + YQ - 1.0;
}
D = RRTPI / (D0 + XQ*(D2 + XQ));
K = D * Y * (A0 + XQ);
} else if ( ABX > XLIM2 ) { // Humlicek W4 Region 2
if ( RG2 != 0 ) { // First point in Region 2
RG2 = 0;
H0 = 0.5625 + YQ * (4.5 + YQ * (10.5 + YQ * (6.0 + YQ)));
// Region 2 y-dependents
H2 = -4.5 + YQ * (9.0 + YQ * ( 6.0 + YQ * 4.0));
H4 = 10.5 - YQ * (6.0 - YQ * 6.0);
H6 = -6.0 + YQ * 4.0;
E0 = 1.875 + YQ * (8.25 + YQ * (5.5 + YQ));
E2 = 5.25 + YQ * (1.0 + YQ * 3.0);
E4 = 0.75 * H6;
}
D = RRTPI / (H0 + XQ * (H2 + XQ * (H4 + XQ * (H6 + XQ))));
K = D * Y * (E0 + XQ * (E2 + XQ * (E4 + XQ)));
} else if ( ABX < XLIM3 ) { // Humlicek W4 Region 3
if ( RG3 != 0 ) { // First point in Region 3
RG3 = 0;
Z0 = 272.1014 + Y * (1280.829 + Y *
(2802.870 + Y *
(3764.966 + Y *
(3447.629 + Y *
(2256.981 + Y *
(1074.409 + Y *
(369.1989 + Y *
(88.26741 + Y *
(13.39880 + Y)
)))))))); // Region 3 y-dependents
Z2 = 211.678 + Y * (902.3066 + Y *
(1758.336 + Y *
(2037.310 + Y *
(1549.675 + Y *
(793.4273 + Y *
(266.2987 + Y *
(53.59518 + Y * 5.0)
))))));
Z4 = 78.86585 + Y * (308.1852 + Y *
(497.3014 + Y *
(479.2576 + Y *
(269.2916 + Y *
(80.39278 + Y * 10.0)
))));
Z6 = 22.03523 + Y * (55.02933 + Y *
(92.75679 + Y *
(53.59518 + Y * 10.0)
));
Z8 = 1.496460 + Y * (13.39880 + Y * 5.0);
P0 = 153.5168 + Y * (549.3954 + Y *
(919.4955 + Y *
(946.8970 + Y *
(662.8097 + Y *
(328.2151 + Y *
(115.3772 + Y *
(27.93941 + Y *
(4.264678 + Y * 0.3183291)
)))))));
P2 = -34.16955 + Y * (-1.322256+ Y *
(124.5975 + Y *
(189.7730 + Y *
(139.4665 + Y *
(56.81652 + Y *
(12.79458 + Y * 1.2733163)
)))));
P4 = 2.584042 + Y * (10.46332 + Y *
(24.01655 + Y *
(29.81482 + Y *
(12.79568 + Y * 1.9099744)
)));
P6 = -0.07272979 + Y * (0.9377051 + Y *
(4.266322 + Y * 1.273316));
P8 = 0.0005480304 + Y * 0.3183291;
}
D = 1.7724538 / (Z0 + XQ * (Z2 + XQ * (Z4 + XQ * (Z6 + XQ * (Z8 + XQ)))));
K = D * (P0 + XQ * (P2 + XQ * (P4 + XQ * (P6 + XQ * P8))));
} else { // Humlicek CPF12 algorithm
YPY0 = Y + Y0;
YPY0Q = YPY0 * YPY0;
K = 0.0;
for (J = 0; J <= 5; J++) {
D = X - T[J];
MQ[J] = D * D;
MF[J] = 1.0 / (MQ[J] + YPY0Q);
XM[J] = MF[J] * D;
YM[J] = MF[J] * YPY0;
D = X + T[J];
PQ[J] = D * D;
PF[J] = 1.0 / (PQ[J] + YPY0Q);
XP[J] = PF[J] * D;
YP[J] = PF[J] * YPY0;
}
if ( ABX <= XLIM4 ) { // Humlicek CPF12 Region I
for (J = 0; J <= 5; J++) {
K = K + C[J]*(YM[J]+YP[J]) - S[J]*(XM[J]-XP[J]) ;
}
} else { // Humlicek CPF12 Region II
YF = Y + Y0PY0;
for ( J = 0; J <= 5; J++) {
K = K + (C[J] *
(MQ[J] * MF[J] - Y0 * YM[J])
+ S[J] * YF * XM[J]) / (MQ[J]+Y0Q)
+ (C[J] * (PQ[J] * PF[J] - Y0 * YP[J])
- S[J] * YF * XP[J]) / (PQ[J]+Y0Q);
}
K = Y * K + exp( -XQ );
}
}
return K / 2.506628 / sigma; // Normalize by dividing by sqrt(2*pi)*sigma.
}
//______________________________________________________________________________
Int_t TMath::LocMin(Int_t n, const Short_t *a)
{
// Return index of array with the minimum element.
// If more than one element is minimum returns first found.
if (n <= 0) return -1;
Short_t xmin = a[0];
Int_t loc = 0;
for (Int_t i = 1; i < n; i++) {
if (xmin > a[i]) {
xmin = a[i];
loc = i;
}
}
return loc;
}
//______________________________________________________________________________
Int_t TMath::LocMin(Int_t n, const Int_t *a)
{
// Return index of array with the minimum element.
// If more than one element is minimum returns first found.
if (n <= 0) return -1;
Int_t xmin = a[0];
Int_t loc = 0;
for (Int_t i = 1; i < n; i++) {
if (xmin > a[i]) {
xmin = a[i];
loc = i;
}
}
return loc;
}
//______________________________________________________________________________
Int_t TMath::LocMin(Int_t n, const Float_t *a)
{
// Return index of array with the minimum element.
// If more than one element is minimum returns first found.
if (n <= 0) return -1;
Float_t xmin = a[0];
Int_t loc = 0;
for (Int_t i = 1; i < n; i++) {
if (xmin > a[i]) {
xmin = a[i];
loc = i;
}
}
return loc;
}
//______________________________________________________________________________
Int_t TMath::LocMin(Int_t n, const Double_t *a)
{
// Return index of array with the minimum element.
// If more than one element is minimum returns first found.
if (n <= 0) return -1;
Double_t xmin = a[0];
Int_t loc = 0;
for (Int_t i = 1; i < n; i++) {
if (xmin > a[i]) {
xmin = a[i];
loc = i;
}
}
return loc;
}
//______________________________________________________________________________
Int_t TMath::LocMin(Int_t n, const Long_t *a)
{
// Return index of array with the minimum element.
// If more than one element is minimum returns first found.
if (n <= 0) return -1;
Long_t xmin = a[0];
Int_t loc = 0;
for (Int_t i = 1; i < n; i++) {
if (xmin > a[i]) {
xmin = a[i];
loc = i;
}
}
return loc;
}
//______________________________________________________________________________
Int_t TMath::LocMax(Int_t n, const Short_t *a)
{
// Return index of array with the maximum element.
// If more than one element is maximum returns first found.
if (n <= 0) return -1;
Short_t xmax = a[0];
Int_t loc = 0;
for (Int_t i = 1; i < n; i++) {
if (xmax < a[i]) {
xmax = a[i];
loc = i;
}
}
return loc;
}
//______________________________________________________________________________
Int_t TMath::LocMax(Int_t n, const Int_t *a)
{
// Return index of array with the maximum element.
// If more than one element is maximum returns first found.
if (n <= 0) return -1;
Int_t xmax = a[0];
Int_t loc = 0;
for (Int_t i = 1; i < n; i++) {
if (xmax < a[i]) {
xmax = a[i];
loc = i;
}
}
return loc;
}
//______________________________________________________________________________
Int_t TMath::LocMax(Int_t n, const Float_t *a)
{
// Return index of array with the maximum element.
// If more than one element is maximum returns first found.
if (n <= 0) return -1;
Float_t xmax = a[0];
Int_t loc = 0;
for (Int_t i = 1; i < n; i++) {
if (xmax < a[i]) {
xmax = a[i];
loc = i;
}
}
return loc;
}
//______________________________________________________________________________
Int_t TMath::LocMax(Int_t n, const Double_t *a)
{
// Return index of array with the maximum element.
// If more than one element is maximum returns first found.
if (n <= 0) return -1;
Double_t xmax = a[0];
Int_t loc = 0;
for (Int_t i = 1; i < n; i++) {
if (xmax < a[i]) {
xmax = a[i];
loc = i;
}
}
return loc;
}
//______________________________________________________________________________
Int_t TMath::LocMax(Int_t n, const Long_t *a)
{
// Return index of array with the maximum element.
// If more than one element is maximum returns first found.
if (n <= 0) return -1;
Long_t xmax = a[0];
Int_t loc = 0;
for (Int_t i = 1; i < n; i++) {
if (xmax < a[i]) {
xmax = a[i];
loc = i;
}
}
return loc;
}
//______________________________________________________________________________
Int_t TMath::BinarySearch(Int_t n, const Short_t *array, Short_t value)
{
// Binary search in an array of n values to locate value.
//
// Array is supposed to be sorted prior to this call.
// If match is found, function returns position of element.
// If no match found, function gives nearest element smaller than value.
Int_t nabove, nbelow, middle;
nabove = n+1;
nbelow = 0;
while(nabove-nbelow > 1) {
middle = (nabove+nbelow)/2;
if (value == array[middle-1]) return middle-1;
if (value < array[middle-1]) nabove = middle;
else nbelow = middle;
}
return nbelow-1;
}
//______________________________________________________________________________
Int_t TMath::BinarySearch(Int_t n, const Short_t **array, Short_t value)
{
// Binary search in an array of n values to locate value.
//
// Array is supposed to be sorted prior to this call.
// If match is found, function returns position of element.
// If no match found, function gives nearest element smaller than value.
Int_t nabove, nbelow, middle;
nabove = n+1;
nbelow = 0;
while(nabove-nbelow > 1) {
middle = (nabove+nbelow)/2;
if (value == *array[middle-1]) return middle-1;
if (value < *array[middle-1]) nabove = middle;
else nbelow = middle;
}
return nbelow-1;
}
//______________________________________________________________________________
Int_t TMath::BinarySearch(Int_t n, const Int_t *array, Int_t value)
{
// Binary search in an array of n values to locate value.
//
// Array is supposed to be sorted prior to this call.
// If match is found, function returns position of element.
// If no match found, function gives nearest element smaller than value.
Int_t nabove, nbelow, middle;
nabove = n+1;
nbelow = 0;
while(nabove-nbelow > 1) {
middle = (nabove+nbelow)/2;
if (value == array[middle-1]) return middle-1;
if (value < array[middle-1]) nabove = middle;
else nbelow = middle;
}
return nbelow-1;
}
//______________________________________________________________________________
Int_t TMath::BinarySearch(Int_t n, const Int_t **array, Int_t value)
{
// Binary search in an array of n values to locate value.
//
// Array is supposed to be sorted prior to this call.
// If match is found, function returns position of element.
// If no match found, function gives nearest element smaller than value.
Int_t nabove, nbelow, middle;
nabove = n+1;
nbelow = 0;
while(nabove-nbelow > 1) {
middle = (nabove+nbelow)/2;
if (value == *array[middle-1]) return middle-1;
if (value < *array[middle-1]) nabove = middle;
else nbelow = middle;
}
return nbelow-1;
}
//______________________________________________________________________________
Int_t TMath::BinarySearch(Int_t n, const Float_t *array, Float_t value)
{
// Binary search in an array of n values to locate value.
//
// Array is supposed to be sorted prior to this call.
// If match is found, function returns position of element.
// If no match found, function gives nearest element smaller than value.
Int_t nabove, nbelow, middle;
nabove = n+1;
nbelow = 0;
while(nabove-nbelow > 1) {
middle = (nabove+nbelow)/2;
if (value == array[middle-1]) return middle-1;
if (value < array[middle-1]) nabove = middle;
else nbelow = middle;
}
return nbelow-1;
}
//______________________________________________________________________________
Int_t TMath::BinarySearch(Int_t n, const Float_t **array, Float_t value)
{
// Binary search in an array of n values to locate value.
//
// Array is supposed to be sorted prior to this call.
// If match is found, function returns position of element.
// If no match found, function gives nearest element smaller than value.
Int_t nabove, nbelow, middle;
nabove = n+1;
nbelow = 0;
while(nabove-nbelow > 1) {
middle = (nabove+nbelow)/2;
if (value == *array[middle-1]) return middle-1;
if (value < *array[middle-1]) nabove = middle;
else nbelow = middle;
}
return nbelow-1;
}
//______________________________________________________________________________
Int_t TMath::BinarySearch(Int_t n, const Double_t *array, Double_t value)
{
// Binary search in an array of n values to locate value.
//
// Array is supposed to be sorted prior to this call.
// If match is found, function returns position of element.
// If no match found, function gives nearest element smaller than value.
Int_t nabove, nbelow, middle;
nabove = n+1;
nbelow = 0;
while(nabove-nbelow > 1) {
middle = (nabove+nbelow)/2;
if (value == array[middle-1]) return middle-1;
if (value < array[middle-1]) nabove = middle;
else nbelow = middle;
}
return nbelow-1;
}
//______________________________________________________________________________
Int_t TMath::BinarySearch(Int_t n, const Double_t **array, Double_t value)
{
// Binary search in an array of n values to locate value.
//
// Array is supposed to be sorted prior to this call.
// If match is found, function returns position of element.
// If no match found, function gives nearest element smaller than value.
Int_t nabove, nbelow, middle;
nabove = n+1;
nbelow = 0;
while(nabove-nbelow > 1) {
middle = (nabove+nbelow)/2;
if (value == *array[middle-1]) return middle-1;
if (value < *array[middle-1]) nabove = middle;
else nbelow = middle;
}
return nbelow-1;
}
//______________________________________________________________________________
Int_t TMath::BinarySearch(Int_t n, const Long_t *array, Long_t value)
{
// Binary search in an array of n values to locate value.
//
// Array is supposed to be sorted prior to this call.
// If match is found, function returns position of element.
// If no match found, function gives nearest element smaller than value.
Int_t nabove, nbelow, middle;
nabove = n+1;
nbelow = 0;
while(nabove-nbelow > 1) {
middle = (nabove+nbelow)/2;
if (value == array[middle-1]) return middle-1;
if (value < array[middle-1]) nabove = middle;
else nbelow = middle;
}
return nbelow-1;
}
//______________________________________________________________________________
Int_t TMath::BinarySearch(Int_t n, const Long_t **array, Long_t value)
{
// Binary search in an array of n values to locate value.
//
// Array is supposed to be sorted prior to this call.
// If match is found, function returns position of element.
// If no match found, function gives nearest element smaller than value.
Int_t nabove, nbelow, middle;
nabove = n+1;
nbelow = 0;
while(nabove-nbelow > 1) {
middle = (nabove+nbelow)/2;
if (value == *array[middle-1]) return middle-1;
if (value < *array[middle-1]) nabove = middle;
else nbelow = middle;
}
return nbelow-1;
}
//_____________________________________________________________________________
Bool_t TMath::IsInside(Double_t xp, Double_t yp, Int_t np, Double_t *x, Double_t *y)
{
// Function which returns kTRUE if point xp,yp lies inside the
// polygon defined by the np points in arrays x and y, kFALSE otherwise
// NOTE that the polygon must be a closed polygon (1st and last point
// must be identical)
Double_t xint;
Int_t i;
Int_t inter = 0;
for (i=0;i<np-1;i++) {
if (y[i] == y[i+1]) continue;
if (yp <= y[i] && yp <= y[i+1]) continue;
if (y[i] < yp && y[i+1] < yp) continue;
xint = x[i] + (yp-y[i])*(x[i+1]-x[i])/(y[i+1]-y[i]);
if (xp < xint) inter++;
}
if (inter%2) return kTRUE;
return kFALSE;
}
//_____________________________________________________________________________
Bool_t TMath::IsInside(Float_t xp, Float_t yp, Int_t np, Float_t *x, Float_t *y)
{
// Function which returns kTRUE if point xp,yp lies inside the
// polygon defined by the np points in arrays x and y, kFALSE otherwise
// NOTE that the polygon must be a closed polygon (1st and last point
// must be identical)
Double_t xint;
Int_t i;
Int_t inter = 0;
for (i=0;i<np-1;i++) {
if (y[i] == y[i+1]) continue;
if (yp <= y[i] && yp <= y[i+1]) continue;
if (y[i] < yp && y[i+1] < yp) continue;
xint = x[i] + (yp-y[i])*(x[i+1]-x[i])/(y[i+1]-y[i]);
if ((Double_t)xp < xint) inter++;
}
if (inter%2) return kTRUE;
return kFALSE;
}
//_____________________________________________________________________________
Bool_t TMath::IsInside(Int_t xp, Int_t yp, Int_t np, Int_t *x, Int_t *y)
{
// Function which returns kTRUE if point xp,yp lies inside the
// polygon defined by the np points in arrays x and y, kFALSE otherwise
// NOTE that the polygon must be a closed polygon (1st and last point
// must be identical)
Double_t xint;
Int_t i;
Int_t inter = 0;
for (i=0;i<np-1;i++) {
if (y[i] == y[i+1]) continue;
if (yp <= y[i] && yp <= y[i+1]) continue;
if (y[i] < yp && y[i+1] < yp) continue;
xint = x[i] + (yp-y[i])*(x[i+1]-x[i])/(y[i+1]-y[i]);
if ((Double_t)xp < xint) inter++;
}
if (inter%2) return kTRUE;
return kFALSE;
}
//_____________________________________________________________________________
void TMath::Sort(Int_t n1, const Short_t *a, Int_t *index, Bool_t down)
{
// Sort the n1 elements of the Short_t array a.
// In output the array index contains the indices of the sorted array.
// If down is false sort in increasing order (default is decreasing order).
// This is a translation of the CERNLIB routine sortzv (M101)
// based on the quicksort algorithm.
// NOTE that the array index must be created with a length >= n1
// before calling this function.
Int_t i,i1,n,i2,i3,i33,i222,iswap,n2;
Int_t i22 = 0;
Short_t ai;
n = n1;
if (n <= 0) return;
if (n == 1) {index[0] = 0; return;}
for (i=0;i<n;i++) index[i] = i+1;
for (i1=2;i1<=n;i1++) {
i3 = i1;
i33 = index[i3-1];
ai = a[i33-1];
while(1) {
i2 = i3/2;
if (i2 <= 0) break;
i22 = index[i2-1];
if (ai <= a[i22-1]) break;
index[i3-1] = i22;
i3 = i2;
}
index[i3-1] = i33;
}
while(1) {
i3 = index[n-1];
index[n-1] = index[0];
ai = a[i3-1];
n--;
if(n-1 < 0) {index[0] = i3; break;}
i1 = 1;
while(2) {
i2 = i1+i1;
if (i2 <= n) i22 = index[i2-1];
if (i2-n > 0) {index[i1-1] = i3; break;}
if (i2-n < 0) {
i222 = index[i2];
if (a[i22-1] - a[i222-1] < 0) {
i2++;
i22 = i222;
}
}
if (ai - a[i22-1] > 0) {index[i1-1] = i3; break;}
index[i1-1] = i22;
i1 = i2;
}
}
for (i=0;i<n1;i++) index[i]--;
if (!down) return;
n2 = n1/2;
for (i=0;i<n2;i++) {
iswap = index[i];
index[i] = index[n1-i-1];
index[n1-i-1] = iswap;
}
}
//_____________________________________________________________________________
void TMath::Sort(Int_t n1, const Int_t *a, Int_t *index, Bool_t down)
{
// Sort the n1 elements of the Int_t array a.
// In output the array index contains the indices of the sorted array.
// If down is false sort in increasing order (default is decreasing order).
// This is a translation of the CERNLIB routine sortzv (M101)
// based on the quicksort algorithm.
// NOTE that the array index must be created with a length >= n1
// before calling this function.
Int_t i,i1,n,i2,i3,i33,i222,iswap,n2;
Int_t i22 = 0;
Int_t ai;
n = n1;
if (n <= 0) return;
if (n == 1) {index[0] = 0; return;}
for (i=0;i<n;i++) index[i] = i+1;
for (i1=2;i1<=n;i1++) {
i3 = i1;
i33 = index[i3-1];
ai = a[i33-1];
while(1) {
i2 = i3/2;
if (i2 <= 0) break;
i22 = index[i2-1];
if (ai <= a[i22-1]) break;
index[i3-1] = i22;
i3 = i2;
}
index[i3-1] = i33;
}
while(1) {
i3 = index[n-1];
index[n-1] = index[0];
ai = a[i3-1];
n--;
if(n-1 < 0) {index[0] = i3; break;}
i1 = 1;
while(2) {
i2 = i1+i1;
if (i2 <= n) i22 = index[i2-1];
if (i2-n > 0) {index[i1-1] = i3; break;}
if (i2-n < 0) {
i222 = index[i2];
if (a[i22-1] - a[i222-1] < 0) {
i2++;
i22 = i222;
}
}
if (ai - a[i22-1] > 0) {index[i1-1] = i3; break;}
index[i1-1] = i22;
i1 = i2;
}
}
for (i=0;i<n1;i++) index[i]--;
if (!down) return;
n2 = n1/2;
for (i=0;i<n2;i++) {
iswap = index[i];
index[i] = index[n1-i-1];
index[n1-i-1] = iswap;
}
}
//_____________________________________________________________________________
void TMath::Sort(Int_t n1, const Float_t *a, Int_t *index, Bool_t down)
{
// Sort the n1 elements of the Float_t array a.
// In output the array index contains the indices of the sorted array.
// If down is false sort in increasing order (default is decreasing order).
// This is a translation of the CERNLIB routine sortzv (M101)
// based on the quicksort algorithm.
// NOTE that the array index must be created with a length >= n1
// before calling this function.
Int_t i,i1,n,i2,i3,i33,i222,iswap,n2;
Int_t i22 = 0;
Float_t ai;
n = n1;
if (n <= 0) return;
if (n == 1) {index[0] = 0; return;}
for (i=0;i<n;i++) index[i] = i+1;
for (i1=2;i1<=n;i1++) {
i3 = i1;
i33 = index[i3-1];
ai = a[i33-1];
while(1) {
i2 = i3/2;
if (i2 <= 0) break;
i22 = index[i2-1];
if (ai <= a[i22-1]) break;
index[i3-1] = i22;
i3 = i2;
}
index[i3-1] = i33;
}
while(1) {
i3 = index[n-1];
index[n-1] = index[0];
ai = a[i3-1];
n--;
if(n-1 < 0) {index[0] = i3; break;}
i1 = 1;
while(2) {
i2 = i1+i1;
if (i2 <= n) i22 = index[i2-1];
if (i2-n > 0) {index[i1-1] = i3; break;}
if (i2-n < 0) {
i222 = index[i2];
if (a[i22-1] - a[i222-1] < 0) {
i2++;
i22 = i222;
}
}
if (ai - a[i22-1] > 0) {index[i1-1] = i3; break;}
index[i1-1] = i22;
i1 = i2;
}
}
for (i=0;i<n1;i++) index[i]--;
if (!down) return;
n2 = n1/2;
for (i=0;i<n2;i++) {
iswap = index[i];
index[i] = index[n1-i-1];
index[n1-i-1] = iswap;
}
}
//_____________________________________________________________________________
void TMath::Sort(Int_t n1, const Double_t *a, Int_t *index, Bool_t down)
{
// Sort the n1 elements of the Double_t array a.
// In output the array index contains the indices of the sorted array.
// If down is false sort in increasing order (default is decreasing order).
// This is a translation of the CERNLIB routine sortzv (M101)
// based on the quicksort algorithm.
// NOTE that the array index must be created with a length >= n1
// before calling this function.
Int_t i,i1,n,i2,i3,i33,i222,iswap,n2;
Int_t i22 = 0;
Double_t ai;
n = n1;
if (n <= 0) return;
if (n == 1) {index[0] = 0; return;}
for (i=0;i<n;i++) index[i] = i+1;
for (i1=2;i1<=n;i1++) {
i3 = i1;
i33 = index[i3-1];
ai = a[i33-1];
while(1) {
i2 = i3/2;
if (i2 <= 0) break;
i22 = index[i2-1];
if (ai <= a[i22-1]) break;
index[i3-1] = i22;
i3 = i2;
}
index[i3-1] = i33;
}
while(1) {
i3 = index[n-1];
index[n-1] = index[0];
ai = a[i3-1];
n--;
if(n-1 < 0) {index[0] = i3; break;}
i1 = 1;
while(2) {
i2 = i1+i1;
if (i2 <= n) i22 = index[i2-1];
if (i2-n > 0) {index[i1-1] = i3; break;}
if (i2-n < 0) {
i222 = index[i2];
if (a[i22-1] - a[i222-1] < 0) {
i2++;
i22 = i222;
}
}
if (ai - a[i22-1] > 0) {index[i1-1] = i3; break;}
index[i1-1] = i22;
i1 = i2;
}
}
for (i=0;i<n1;i++) index[i]--;
if (!down) return;
n2 = n1/2;
for (i=0;i<n2;i++) {
iswap = index[i];
index[i] = index[n1-i-1];
index[n1-i-1] = iswap;
}
}
//_____________________________________________________________________________
void TMath::Sort(Int_t n1, const Long_t *a, Int_t *index, Bool_t down)
{
// Sort the n1 elements of the Long_t array a.
// In output the array index contains the indices of the sorted array.
// If down is false sort in increasing order (default is decreasing order).
// This is a translation of the CERNLIB routine sortzv (M101)
// based on the quicksort algorithm.
// NOTE that the array index must be created with a length >= n1
// before calling this function.
Int_t i,i1,n,i2,i3,i33,i222,iswap,n2;
Int_t i22 = 0;
Long_t ai;
n = n1;
if (n <= 0) return;
if (n == 1) {index[0] = 0; return;}
for (i=0;i<n;i++) index[i] = i+1;
for (i1=2;i1<=n;i1++) {
i3 = i1;
i33 = index[i3-1];
ai = a[i33-1];
while(1) {
i2 = i3/2;
if (i2 <= 0) break;
i22 = index[i2-1];
if (ai <= a[i22-1]) break;
index[i3-1] = i22;
i3 = i2;
}
index[i3-1] = i33;
}
while(1) {
i3 = index[n-1];
index[n-1] = index[0];
ai = a[i3-1];
n--;
if(n-1 < 0) {index[0] = i3; break;}
i1 = 1;
while(2) {
i2 = i1+i1;
if (i2 <= n) i22 = index[i2-1];
if (i2-n > 0) {index[i1-1] = i3; break;}
if (i2-n < 0) {
i222 = index[i2];
if (a[i22-1] - a[i222-1] < 0) {
i2++;
i22 = i222;
}
}
if (ai - a[i22-1] > 0) {index[i1-1] = i3; break;}
index[i1-1] = i22;
i1 = i2;
}
}
for (i=0;i<n1;i++) index[i]--;
if (!down) return;
n2 = n1/2;
for (i=0;i<n2;i++) {
iswap = index[i];
index[i] = index[n1-i-1];
index[n1-i-1] = iswap;
}
}
//______________________________________________________________________________
void TMath::BubbleHigh(Int_t Narr, Double_t *arr1, Int_t *arr2)
{
// Bubble sort variant to obtain the order of an array's elements into
// an index in order to do more useful things than the standard built
// in functions.
// *arr1 is unchanged;
// *arr2 is the array of indicies corresponding to the decending value
// of arr1 with arr2[0] corresponding to the largest arr1 value and
// arr2[Narr] the smallest.
//
// Author: Adrian Bevan (bevan@slac.stanford.edu)
// Copyright: Liverpool University, July 2001
if (Narr <= 0) return;
double *localArr1 = new double[Narr];
int *localArr2 = new int[Narr];
int iEl;
int iEl2;
for(iEl = 0; iEl < Narr; iEl++) {
localArr1[iEl] = arr1[iEl];
localArr2[iEl] = iEl;
}
for (iEl = 0; iEl < Narr; iEl++) {
for (iEl2 = Narr-1; iEl2 > iEl; --iEl2) {
if (localArr1[iEl2-1] < localArr1[iEl2]) {
double tmp = localArr1[iEl2-1];
localArr1[iEl2-1] = localArr1[iEl2];
localArr1[iEl2] = tmp;
int tmp2 = localArr2[iEl2-1];
localArr2[iEl2-1] = localArr2[iEl2];
localArr2[iEl2] = tmp2;
}
}
}
for (iEl = 0; iEl < Narr; iEl++) {
arr2[iEl] = localArr2[iEl];
}
delete [] localArr2;
delete [] localArr1;
}
//______________________________________________________________________________
void TMath::BubbleLow(Int_t Narr, Double_t *arr1, Int_t *arr2)
{
// Opposite ordering of the array arr2[] to that of BubbleHigh.
//
// Author: Adrian Bevan (bevan@slac.stanford.edu)
// Copyright: Liverpool University, July 2001
if (Narr <= 0) return;
double *localArr1 = new double[Narr];
int *localArr2 = new int[Narr];
int iEl;
int iEl2;
for (iEl = 0; iEl < Narr; iEl++) {
localArr1[iEl] = arr1[iEl];
localArr2[iEl] = iEl;
}
for (iEl = 0; iEl < Narr; iEl++) {
for (iEl2 = Narr-1; iEl2 > iEl; --iEl2) {
if (localArr1[iEl2-1] > localArr1[iEl2]) {
double tmp = localArr1[iEl2-1];
localArr1[iEl2-1] = localArr1[iEl2];
localArr1[iEl2] = tmp;
int tmp2 = localArr2[iEl2-1];
localArr2[iEl2-1] = localArr2[iEl2];
localArr2[iEl2] = tmp2;
}
}
}
for (iEl = 0; iEl < Narr; iEl++) {
arr2[iEl] = localArr2[iEl];
}
delete [] localArr2;
delete [] localArr1;
}
#ifdef OLD_HASH
//______________________________________________________________________________
ULong_t TMath::Hash(const void *txt, Int_t ntxt)
{
// Calculates hash index from any char string.
// Based on precalculated table of 256 specially selected random numbers.
//
// For string: i = TMath::Hash(string,nstring);
// For int: i = TMath::Hash(&intword,sizeof(int));
// For pointer: i = TMath::Hash(&pointer,sizeof(void*));
//
// Limitation: for ntxt>256 calculates hash only from first 256 bytes
//
// V.Perev
const UChar_t *uc = (const UChar_t*) txt;
ULong_t u = 0, uu = 0;
static ULong_t utab[256] =
{0xb93f6fc0,0x553dfc51,0xb22c1e8c,0x638462c0,0x13e81418,0x2836e171,0x7c4abb90,0xda1a4f39
,0x38f211d1,0x8c804829,0x95a6602d,0x4c590993,0x1810580a,0x721057d4,0x0f587215,0x9f49ce2a
,0xcd5ab255,0xab923a99,0x80890f39,0xbcfa2290,0x16587b52,0x6b6d3f0d,0xea8ff307,0x51542d5c
,0x189bf223,0x39643037,0x0e4a326a,0x214eca01,0x47645a9b,0x0f364260,0x8e9b2da4,0x5563ebd9
,0x57a31c1c,0xab365854,0xdd63ab1f,0x0b89acbd,0x23d57d33,0x1800a0fd,0x225ac60a,0xd0e51943
,0x6c65f669,0xcb966ea0,0xcbafda95,0x2e5c0c5f,0x2988e87e,0xc781cbab,0x3add3dc7,0x693a2c30
,0x42d6c23c,0xebf85f26,0x2544987e,0x2e315e3f,0xac88b5b5,0x7ebd2bbb,0xda07c87b,0x20d460f1
,0xc61c3f40,0x182046e7,0x3b6c3b66,0x2fc10d4a,0x0780dfbb,0xc437280c,0x0988dd07,0xe1498606
,0x8e61d728,0x4f1f3909,0x040a9682,0x49411b29,0x391b0e1c,0xd7905241,0xdd77d95b,0x88426c13
,0x33033e58,0xe158e30e,0x7e342647,0x1e09544b,0x4637353d,0x18ea0924,0x39212b08,0x12580ae8
,0x269a6f06,0x3e10b73b,0x123db33b,0x085412da,0x3bb5f464,0xd9b2d442,0x103d26bb,0xd0038bab
,0x45b6177f,0xfb48f1fe,0x99074c55,0xb545e82e,0x5f79fd0d,0x570f3ae4,0x57e43255,0x037a12ae
,0x4357bdb2,0x337c2c4d,0x2982499d,0x2ab72793,0x3900e7d1,0x57a6bb81,0x7503609b,0x3f39c0d0
,0x717b389d,0x5748034f,0x4698162b,0x5801b97c,0x1dfd5d7e,0xc1386d1c,0xa387a72a,0x084547e4
,0x2e54d8e9,0x2e2f384c,0xe09ccc20,0x8904b71e,0x3e24edc5,0x06a22e16,0x8a2be1df,0x9e5058b2
,0xe01a2f16,0x03325eed,0x587ecfe6,0x584d9cd3,0x32926930,0xe943d68c,0xa9442da8,0xf9650560
,0xf003871e,0x1109c663,0x7a2f2f89,0x1c2210bb,0x37335787,0xb92b382f,0xea605cb5,0x336bbe38
,0x08126bd3,0x1f8c2bd6,0xba6c46f2,0x1a4d1b83,0xc988180d,0xe2582505,0xa8a1b375,0x59a08c49
,0x3db54b48,0x44400f35,0x272d4e7f,0x5579f733,0x98eb590e,0x8ee09813,0x12cc9301,0xc85c402d
,0x135c1039,0x22318128,0x4063c705,0x87a8a3fa,0xfc14431f,0x6e27bf47,0x2d080a19,0x01dba174
,0xe343530b,0xaa1bfced,0x283bb2c8,0x5df250c8,0x4ff9140b,0x045039c1,0xa377780d,0x750f2661
,0x2b108918,0x0b152120,0x3cbc251f,0x5e87b350,0x060625bb,0xe068ba3b,0xdb73ebd7,0x66014ff3
,0xdb003000,0x161a3a0b,0xdc24e142,0x97ea5575,0x635a3cab,0xa719100a,0x256084db,0xc1f4a1e7
,0xe13388f2,0xb8199fc9,0x50c70dc9,0x08154211,0xd60e5220,0xe52c6592,0x584c5fe1,0xfe5e0875
,0x21072b30,0x3370d773,0x92608fe2,0x2d013d93,0x53414b3c,0x2c066142,0x64676644,0x0420887c
,0x35c01187,0x6822119b,0xf9bfe6df,0x273f4ee4,0x87973149,0x7b41282d,0x635d0d1f,0x5f7ecc1e
,0x14c3608a,0x462dfdab,0xc33d8808,0x1dcd995e,0x0fcb11ba,0x11755914,0x5a62044b,0x37f76755
,0x345bd058,0x8831c2b5,0x204a8468,0x3b0b1cd2,0x444e56f4,0x97a93e2c,0xd5f15067,0x266a95fa
,0xff4f8036,0x6160060d,0x930c472f,0xed922184,0x37120251,0xc0add74f,0x1c0bc89d,0x018d47f2
,0xff59ef66,0xd1901a17,0x91f6701b,0x0960082f,0x86f6a8f3,0x1154fecd,0x9867d1de,0x0945482f
,0x790ffcac,0xe5610011,0x4765637e,0xa745dbff,0x841fdcb3,0x4f7372a0,0x3c05013d,0xf1ac4ab7
,0x3bc5b5cc,0x49a73349,0x356a7f67,0x1174f031,0x11d32634,0x4413d301,0x1dd285c4,0x3fae4800
};
if (ntxt > 255) ntxt = 255;
for ( ; ntxt--; uc++) {
uu = uu<<1 ^ utab[(*uc) ^ ntxt];
u ^= uu;
}
return u;
}
#else
//______________________________________________________________________________
ULong_t TMath::Hash(const void *txt, Int_t ntxt)
{
// Calculates hash index from any char string.
// Based on precalculated table of 256 specially selected numbers.
// These numbers are selected in such a way, that for string
// length == 4 (integer number) the hash is unambigous, i.e.
// from hash value we can recalculate input (no degeneration).
//
// The quality of hash method is good enough, that
// "random" numbers made as R = Hash(1), Hash(2), ...Hash(N)
// tested by <R>, <R*R>, <Ri*Ri+1> gives the same result
// as for libc rand().
//
// For string: i = TMath::Hash(string,nstring);
// For int: i = TMath::Hash(&intword,sizeof(int));
// For pointer: i = TMath::Hash(&pointer,sizeof(void*));
//
// V.Perev
static const ULong_t utab[] = {
0xdd367647,0x9caf993f,0x3f3cc5ff,0xfde25082,0x4c764b21,0x89affca7,0x5431965c,0xce22eeec
,0xc61ab4dc,0x59cc93bd,0xed3107e3,0x0b0a287a,0x4712475a,0xce4a4c71,0x352c8403,0x94cb3cee
,0xc3ac509b,0x09f827a2,0xce02e37e,0x7b20bbba,0x76adcedc,0x18c52663,0x19f74103,0x6f30e47b
,0x132ea5a1,0xfdd279e0,0xa3d57d00,0xcff9cb40,0x9617f384,0x6411acfa,0xff908678,0x5c796b2c
,0x4471b62d,0xd38e3275,0xdb57912d,0x26bf953f,0xfc41b2a5,0xe64bcebd,0x190b7839,0x7e8e6a56
,0x9ca22311,0xef28aa60,0xe6b9208e,0xd257fb65,0x45781c2c,0x9a558ac3,0x2743e74d,0x839417a8
,0x06b54d5d,0x1a82bcb4,0x06e97a66,0x70abdd03,0xd163f30d,0x222ed322,0x777bfeda,0xab7a2e83
,0x8494e0cf,0x2dca2d4f,0x78f94278,0x33f04a09,0x402b6452,0x0cd8b709,0xdb72a39e,0x170e00a2
,0x26354faa,0x80e57453,0xcfe8d4e1,0x19e45254,0x04c291c3,0xeb503738,0x425af3bc,0x67836f2a
,0xfac22add,0xfafc2b8c,0x59b8c2a0,0x03e806f9,0xcb4938b9,0xccc942af,0xcee3ae2e,0xfbe748fa
,0xb223a075,0x85c49b5d,0xe4576ac9,0x0fbd46e2,0xb49f9cf5,0xf3e1e86a,0x7d7927fb,0x711afe12
,0xbf61c346,0x157c9956,0x86b6b046,0x2e402146,0xb2a57d8a,0x0d064bb1,0x30ce390c,0x3a3e1eb1
,0xbe7f6f8f,0xd8e30f87,0x5be2813c,0x73a3a901,0xa3aaf967,0x59ff092c,0x1705c798,0xf610dd66
,0xb17da91e,0x8e59534e,0x2211ea5b,0xa804ba03,0xd890efbb,0xb8b48110,0xff390068,0xc8c325b4
,0xf7289c07,0x787e104f,0x3d0df3d0,0x3526796d,0x10548055,0x1d59a42b,0xed1cc5a3,0xdd45372a
,0x31c50d57,0x65757cb7,0x3cfb85be,0xa329910d,0x6ad8ce39,0xa2de44de,0x0dd32432,0xd4a5b617
,0x8f3107fc,0x96485175,0x7f94d4f3,0x35097634,0xdb3ca782,0x2c0290b8,0x2045300b,0xe0f5d15a
,0x0e8cbffa,0xaa1cc38a,0x84008d6f,0xe9a9e794,0x5c602c25,0xfa3658fa,0x98d9d82b,0x3f1497e7
,0x84b6f031,0xe381eff9,0xfc7ae252,0xb239e05d,0xe3723d1f,0xcc3bda82,0xe21b1ad3,0x9104f7c8
,0x4bb2dfcd,0x4d14a8bc,0x6ba7f28c,0x8f89886c,0xad44c97e,0xb30fd975,0x633cdab1,0xf6c2d514
,0x067a49d2,0xdc461ad9,0xebaf9f3f,0x8dc6cac3,0x7a060f16,0xbab063ad,0xf42e25e6,0x60724ca6
,0xc7245c2e,0x4e48ea3c,0x9f89a609,0xa1c49890,0x4bb7f116,0xd722865c,0xa8ee3995,0x0ee070b1
,0xd9bffcc2,0xe55b64f9,0x25507a5a,0xc7a3e2b5,0x5f395f7e,0xe7957652,0x7381ba6a,0xde3d21f1
,0xdf1708dd,0xad0c9d0c,0x00cbc9e5,0x1160e833,0x6779582c,0x29d5d393,0x3f11d7d7,0x826a6b9b
,0xe73ff12f,0x8bad3d86,0xee41d3e5,0x7f0c8917,0x8089ef24,0x90c5cb28,0x2f7f8e6b,0x6966418a
,0x345453fb,0x7a2f8a68,0xf198593d,0xc079a532,0xc1971e81,0x1ab74e26,0x329ef347,0x7423d3d0
,0x942c510b,0x7f6c6382,0x14ae6acc,0x64b59da7,0x2356fa47,0xb6749d9c,0x499de1bb,0x92ffd191
,0xe8f2fb75,0x848dc913,0x3e8727d3,0x1dcffe61,0xb6e45245,0x49055738,0x827a6b55,0xb4788887
,0x7e680125,0xd19ce7ed,0x6b4b8e30,0xa8cadea2,0x216035d8,0x1c63bc3c,0xe1299056,0x1ad3dff4
,0x0aefd13c,0x0e7b921c,0xca0173c6,0x9995782d,0xcccfd494,0xd4b0ac88,0x53d552b1,0x630dae8b
,0xa8332dad,0x7139d9a2,0x5d76f2c4,0x7a4f8f1e,0x8d1aef97,0xd1cf285d,0xc8239153,0xce2608a9
,0x7b562475,0xe4b4bc83,0xf3db0c3a,0x70a65e48,0x6016b302,0xdebd5046,0x707e786a,0x6f10200c
};
static const ULong_t msk[] = { 0x11111111, 0x33333333, 0x77777777, 0xffffffff };
const UChar_t *uc = (const UChar_t *) txt;
ULong_t uu = 0;
union {
ULong_t u;
UShort_t s[2];
} u;
u.u = 0;
Int_t i, idx;
for (i = 0; i < ntxt; i++) {
idx = (uc[i] ^ i) & 255;
uu = (uu << 1) ^ (utab[idx] & msk[i & 3]);
if (i & 3 == 3) u.u ^= uu;
}
if (i & 3) u.u ^= uu;
u.u *= 1879048201; // prime number
u.s[0] += u.s[1];
u.u *= 1979048191; // prime number
u.s[1] ^= u.s[0];
u.u *= 2079048197; // prime number
return u.u;
}
#endif
//______________________________________________________________________________
ULong_t TMath::Hash(const char *txt)
{
return Hash(txt, Int_t(strlen(txt)));
}
//______________________________________________________________________________
Double_t TMath::BesselI0(Double_t x)
{
// Compute the modified Bessel function I_0(x) for any real x.
//
//--- 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;
const Double_t k1 = 3.75;
Double_t ax = TMath::Abs(x);
Double_t y=0, result=0;
if (ax < k1) {
Double_t xx = x/k1;
y = xx*xx;
result = p1+y*(p2+y*(p3+y*(p4+y*(p5+y*(p6+y*p7)))));
} else {
y = k1/ax;
result = (TMath::Exp(ax)/TMath::Sqrt(ax))*(q1+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*(q7+y*(q8+y*q9))))))));
}
return result;
}
//______________________________________________________________________________
Double_t TMath::BesselK0(Double_t x)
{
// Compute the modified Bessel function K_0(x) for positive real x.
//
// 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) {
Error("TMath::BesselK0", "*K0* Invalid argument x = %gn",x);
return 0;
}
Double_t y=0, result=0;
if (x <= 2) {
y = x*x/4;
result = (-log(x/2.)*TMath::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 TMath::BesselI1(Double_t x)
{
// Compute the modified Bessel function I_1(x) for any real x.
//
// 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;
const Double_t k1 = 3.75;
Double_t ax = TMath::Abs(x);
Double_t y=0, result=0;
if (ax < k1) {
Double_t xx = x/k1;
y = xx*xx;
result = x*(p1+y*(p2+y*(p3+y*(p4+y*(p5+y*(p6+y*p7))))));
} else {
y = k1/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 TMath::BesselK1(Double_t x)
{
// Compute the modified Bessel function K_1(x) for positive real x.
//
// 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) {
Error("TMath::BesselK1", "*K1* Invalid argument x = %gn",x);
return 0;
}
Double_t y=0,result=0;
if (x <= 2) {
y = x*x/4;
result = (log(x/2.)*TMath::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 TMath::BesselK(Int_t n,Double_t x)
{
// Compute the Integer Order Modified Bessel function K_n(x)
// for n=0,1,2,... and positive real x.
//
//--- NvE 12-mar-2000 UU-SAP Utrecht
if (x <= 0 || n < 0) {
Error("TMath::BesselK", "*K* Invalid argument(s) (n,x) = (%d, %g)n",n,x);
return 0;
}
if (n==0) return TMath::BesselK0(x);
if (n==1) return TMath::BesselK1(x);
// Perform upward recurrence for all x
Double_t tox = 2/x;
Double_t bkm = TMath::BesselK0(x);
Double_t bk = TMath::BesselK1(x);
Double_t bkp = 0;
for (Int_t j=1; j<n; j++) {
bkp = bkm+Double_t(j)*tox*bk;
bkm = bk;
bk = bkp;
}
return bk;
}
//______________________________________________________________________________
Double_t TMath::BesselI(Int_t n,Double_t x)
{
// Compute the Integer Order Modified Bessel function I_n(x)
// for n=0,1,2,... and any real x.
//
//--- NvE 12-mar-2000 UU-SAP Utrecht
Int_t iacc = 40; // Increase to enhance accuracy
const Double_t kBigPositive = 1.e10;
const Double_t kBigNegative = 1.e-10;
if (n < 0) {
Error("TMath::BesselI", "*I* Invalid argument (n,x) = (%d, %g)n",n,x);
return 0;
}
if (n==0) return TMath::BesselI0(x);
if (n==1) return TMath::BesselI1(x);
if (TMath::Abs(x) > kBigPositive) return 0;
Double_t tox = 2/TMath::Abs(x);
Double_t bip = 0, bim = 0;
Double_t bi = 1;
Double_t result = 0;
Int_t m = 2*((n+Int_t(sqrt(Float_t(iacc*n)))));
for (Int_t j=m; j>=1; j--) {
bim = bip+Double_t(j)*tox*bi;
bip = bi;
bi = bim;
// Renormalise to prevent overflows
if (TMath::Abs(bi) > kBigPositive) {
result *= kBigNegative;
bi *= kBigNegative;
bip *= kBigNegative;
}
if (j==n) result=bip;
}
result *= TMath::BesselI0(x)/bi; // Normalise with BesselI0(x)
if ((x < 0) && (n%2 == 1)) result = -result;
return result;
}
//______________________________________________________________________________
Double_t TMath::BesselJ0(Double_t x)
{
// Returns the Bessel function J0(x) for any real x.
Double_t ax,z;
Double_t xx,y,result,result1,result2;
const Double_t p1 = 57568490574.0, p2 = -13362590354.0, p3 = 651619640.7;
const Double_t p4 = -11214424.18, p5 = 77392.33017, p6 = -184.9052456;
const Double_t p7 = 57568490411.0, p8 = 1029532985.0, p9 = 9494680.718;
const Double_t p10 = 59272.64853, p11 = 267.8532712;
const Double_t q1 = 0.785398164;
const Double_t q2 = -0.1098628627e-2, q3 = 0.2734510407e-4;
const Double_t q4 = -0.2073370639e-5, q5 = 0.2093887211e-6;
const Double_t q6 = -0.1562499995e-1, q7 = 0.1430488765e-3;
const Double_t q8 = -0.6911147651e-5, q9 = 0.7621095161e-6;
const Double_t q10 = 0.934935152e-7, q11 = 0.636619772;
if ((ax=fabs(x)) < 8) {
y=x*x;
result1 = p1 + y*(p2 + y*(p3 + y*(p4 + y*(p5 + y*p6))));
result2 = p7 + y*(p8 + y*(p9 + y*(p10 + y*(p11 + y))));
result = result1/result2;
} else {
z = 8/ax;
y = z*z;
xx = ax-q1;
result1 = 1 + y*(q2 + y*(q3 + y*(q4 + y*q5)));
result2 = q6 + y*(q7 + y*(q8 + y*(q9 - y*q10)));
result = sqrt(q11/ax)*(cos(xx)*result1-z*sin(xx)*result2);
}
return result;
}
//______________________________________________________________________________
Double_t TMath::BesselJ1(Double_t x)
{
// Returns the Bessel function J1(x) for any real x.
Double_t ax,z;
Double_t xx,y,result,result1,result2;
const Double_t p1 = 72362614232.0, p2 = -7895059235.0, p3 = 242396853.1;
const Double_t p4 = -2972611.439, p5 = 15704.48260, p6 = -30.16036606;
const Double_t p7 = 144725228442.0, p8 = 2300535178.0, p9 = 18583304.74;
const Double_t p10 = 99447.43394, p11 = 376.9991397;
const Double_t q1 = 2.356194491;
const Double_t q2 = 0.183105e-2, q3 = -0.3516396496e-4;
const Double_t q4 = 0.2457520174e-5, q5 = -0.240337019e-6;
const Double_t q6 = 0.04687499995, q7 = -0.2002690873e-3;
const Double_t q8 = 0.8449199096e-5, q9 = -0.88228987e-6;
const Double_t q10 = 0.105787412e-6, q11 = 0.636619772;
if ((ax=fabs(x)) < 8) {
y=x*x;
result1 = x*(p1 + y*(p2 + y*(p3 + y*(p4 + y*(p5 + y*p6)))));
result2 = p7 + y*(p8 + y*(p9 + y*(p10 + y*(p11 + y))));
result = result1/result2;
} else {
z = 8/ax;
y = z*z;
xx = ax-q1;
result1 = 1 + y*(q2 + y*(q3 + y*(q4 + y*q5)));
result2 = q6 + y*(q7 + y*(q8 + y*(q9 + y*q10)));
result = sqrt(q11/ax)*(cos(xx)*result1-z*sin(xx)*result2);
if (x < 0) result = -result;
}
return result;
}
//______________________________________________________________________________
Double_t TMath::BesselY0(Double_t x)
{
// Returns the Bessel function Y0(x) for positive x.
Double_t z,xx,y,result,result1,result2;
const Double_t p1 = -2957821389., p2 = 7062834065.0, p3 = -512359803.6;
const Double_t p4 = 10879881.29, p5 = -86327.92757, p6 = 228.4622733;
const Double_t p7 = 40076544269., p8 = 745249964.8, p9 = 7189466.438;
const Double_t p10 = 47447.26470, p11 = 226.1030244, p12 = 0.636619772;
const Double_t q1 = 0.785398164;
const Double_t q2 = -0.1098628627e-2, q3 = 0.2734510407e-4;
const Double_t q4 = -0.2073370639e-5, q5 = 0.2093887211e-6;
const Double_t q6 = -0.1562499995e-1, q7 = 0.1430488765e-3;
const Double_t q8 = -0.6911147651e-5, q9 = 0.7621095161e-6;
const Double_t q10 = -0.934945152e-7, q11 = 0.636619772;
if (x < 8) {
y = x*x;
result1 = p1 + y*(p2 + y*(p3 + y*(p4 + y*(p5 + y*p6))));
result2 = p7 + y*(p8 + y*(p9 + y*(p10 + y*(p11 + y))));
result = (result1/result2) + p12*TMath::BesselJ0(x)*log(x);
} else {
z = 8/x;
y = z*z;
xx = x-q1;
result1 = 1 + y*(q2 + y*(q3 + y*(q4 + y*q5)));
result2 = q6 + y*(q7 + y*(q8 + y*(q9 + y*q10)));
result = sqrt(q11/x)*(sin(xx)*result1+z*cos(xx)*result2);
}
return result;
}
//______________________________________________________________________________
Double_t TMath::BesselY1(Double_t x)
{
// Returns the Bessel function Y1(x) for positive x.
Double_t z,xx,y,result,result1,result2;
const Double_t p1 = -0.4900604943e13, p2 = 0.1275274390e13;
const Double_t p3 = -0.5153438139e11, p4 = 0.7349264551e9;
const Double_t p5 = -0.4237922726e7, p6 = 0.8511937935e4;
const Double_t p7 = 0.2499580570e14, p8 = 0.4244419664e12;
const Double_t p9 = 0.3733650367e10, p10 = 0.2245904002e8;
const Double_t p11 = 0.1020426050e6, p12 = 0.3549632885e3;
const Double_t p13 = 0.636619772;
const Double_t q1 = 2.356194491;
const Double_t q2 = 0.183105e-2, q3 = -0.3516396496e-4;
const Double_t q4 = 0.2457520174e-5, q5 = -0.240337019e-6;
const Double_t q6 = 0.04687499995, q7 = -0.2002690873e-3;
const Double_t q8 = 0.8449199096e-5, q9 = -0.88228987e-6;
const Double_t q10 = 0.105787412e-6, q11 = 0.636619772;
if (x < 8) {
y=x*x;
result1 = x*(p1 + y*(p2 + y*(p3 + y*(p4 + y*(p5 + y*p6)))));
result2 = p7 + y*(p8 + y*(p9 + y*(p10 + y*(p11 + y*(p12+y)))));
result = (result1/result2) + p13*(TMath::BesselJ1(x)*log(x)-1/x);
} else {
z = 8/x;
y = z*z;
xx = x-q1;
result1 = 1 + y*(q2 + y*(q3 + y*(q4 + y*q5)));
result2 = q6 + y*(q7 + y*(q8 + y*(q9 + y*q10)));
result = sqrt(q11/x)*(sin(xx)*result1+z*cos(xx)*result2);
}
return result;
}
//______________________________________________________________________________
Double_t TMath::StruveH0(Double_t x)
{
// Struve Functions of Order 0
//
// Converted from CERNLIB M342 by Rene Brun.
const Int_t n1 = 15;
const Int_t n2 = 25;
const Double_t c1[16] = { 1.00215845609911981, -1.63969292681309147,
1.50236939618292819, -.72485115302121872,
.18955327371093136, -.03067052022988,
.00337561447375194, -2.6965014312602e-4,
1.637461692612e-5, -7.8244408508e-7,
3.021593188e-8, -9.6326645e-10,
2.579337e-11, -5.8854e-13,
1.158e-14, -2e-16 };
const Double_t c2[26] = { .99283727576423943, -.00696891281138625,
1.8205103787037e-4, -1.063258252844e-5,
9.8198294287e-7, -1.2250645445e-7,
1.894083312e-8, -3.44358226e-9,
7.1119102e-10, -1.6288744e-10,
4.065681e-11, -1.091505e-11,
3.12005e-12, -9.4202e-13,
2.9848e-13, -9.872e-14,
3.394e-14, -1.208e-14,
4.44e-15, -1.68e-15,
6.5e-16, -2.6e-16,
1.1e-16, -4e-17,
2e-17, -1e-17 };
const Double_t c0 = 2/TMath::Pi();
Int_t i;
Double_t alfa, h, r, y, b0, b1, b2;
Double_t v = TMath::Abs(x);
v = TMath::Abs(x);
if (v < 8) {
y = v/8;
h = 2*y*y -1;
alfa = h + h;
b0 = 0;
b1 = 0;
b2 = 0;
for (i = n1; i >= 0; --i) {
b0 = c1[i] + alfa*b1 - b2;
b2 = b1;
b1 = b0;
}
h = y*(b0 - h*b2);
} else {
r = 1/v;
h = 128*r*r -1;
alfa = h + h;
b0 = 0;
b1 = 0;
b2 = 0;
for (i = n2; i >= 0; --i) {
b0 = c2[i] + alfa*b1 - b2;
b2 = b1;
b1 = b0;
}
h = TMath::BesselY0(v) + r*c0*(b0 - h*b2);
}
if (x < 0) h = -h;
return h;
}
//______________________________________________________________________________
Double_t TMath::StruveH1(Double_t x)
{
// Struve Functions of Order 1
//
// Converted from CERNLIB M342 by Rene Brun.
const Int_t n3 = 16;
const Int_t n4 = 22;
const Double_t c3[17] = { .5578891446481605, -.11188325726569816,
-.16337958125200939, .32256932072405902,
-.14581632367244242, .03292677399374035,
-.00460372142093573, 4.434706163314e-4,
-3.142099529341e-5, 1.7123719938e-6,
-7.416987005e-8, 2.61837671e-9,
-7.685839e-11, 1.9067e-12,
-4.052e-14, 7.5e-16,
-1e-17 };
const Double_t c4[23] = { 1.00757647293865641, .00750316051248257,
-7.043933264519e-5, 2.66205393382e-6,
-1.8841157753e-7, 1.949014958e-8,
-2.6126199e-9, 4.236269e-10,
-7.955156e-11, 1.679973e-11,
-3.9072e-12, 9.8543e-13,
-2.6636e-13, 7.645e-14,
-2.313e-14, 7.33e-15,
-2.42e-15, 8.3e-16,
-3e-16, 1.1e-16,
-4e-17, 2e-17,-1e-17 };
const Double_t c0 = 2/TMath::Pi();
const Double_t cc = 2/(3*TMath::Pi());
Int_t i, i1;
Double_t alfa, h, r, y, b0, b1, b2;
Double_t v = TMath::Abs(x);
if (v == 0) {
h = 0;
} else if (v <= 0.3) {
y = v*v;
r = 1;
h = 1;
i1 = (Int_t)(-8. / TMath::Log10(v));
for (i = 1; i <= i1; ++i) {
h = -h*y / ((2*i+ 1)*(2*i + 3));
r += h;
}
h = cc*y*r;
} else if (v < 8) {
h = v*v/32 -1;
alfa = h + h;
b0 = 0;
b1 = 0;
b2 = 0;
for (i = n3; i >= 0; --i) {
b0 = c3[i] + alfa*b1 - b2;
b2 = b1;
b1 = b0;
}
h = b0 - h*b2;
} else {
h = 128/(v*v) -1;
alfa = h + h;
b0 = 0;
b1 = 0;
b2 = 0;
for (i = n4; i >= 0; --i) {
b0 = c4[i] + alfa*b1 - b2;
b2 = b1;
b1 = b0;
}
h = TMath::BesselY1(v) + c0*(b0 - h*b2);
}
return h;
}
//______________________________________________________________________________
Double_t TMath::StruveL0(Double_t x)
{
// Modified Struve Function of Order 0
// (from Kirill Filimonov)
const Double_t pi=TMath::Pi();
Double_t s=1.0;
Double_t r=1.0;
Double_t a0,sl0,a1,bi0;
Int_t km,i;
if (x<=20.) {
a0=2.0*x/pi;
for (int i=1; i<=60;i++) {
r*=(x/(2*i+1))*(x/(2*i+1));
s+=r;
if(TMath::Abs(r/s)<1.e-12) break;
}
sl0=a0*s;
} else {
km=int(5*(x+1.0));
if(x>=50.0)km=25;
for (i=1; i<=km; i++) {
r*=(2*i-1)*(2*i-1)/x/x;
s+=r;
if(TMath::Abs(r/s)<1.0e-12) break;
}
a1=TMath::Exp(x)/TMath::Sqrt(2*pi*x);
r=1.0;
bi0=1.0;
for (i=1; i<=16; i++) {
r=0.125*r*(2.0*i-1.0)*(2.0*i-1.0)/(i*x);
bi0+=r;
if(TMath::Abs(r/bi0)<1.0e-12) break;
}
bi0=a1*bi0;
sl0=-2.0/(pi*x)*s+bi0;
}
return sl0;
}
//______________________________________________________________________________
Double_t TMath::StruveL1(Double_t x)
{
// Modified Struve Function of Order 1
// (from Kirill Filimonov)
const Double_t pi=TMath::Pi();
Double_t a1,sl1,bi1,s;
Double_t r=1.0;
Int_t km,i;
if (x<=20.) {
s=0.0;
for (i=1; i<=60;i++) {
r*=x*x/(4.0*i*i-1.0);
s+=r;
if(TMath::Abs(r)<TMath::Abs(s)*1.e-12) break;
}
sl1=2.0/pi*s;
} else {
s=1.0;
km=int(0.5*x);
if(x>50.0)km=25;
for (i=1; i<=km; i++) {
r*=(2*i+3)*(2*i+1)/x/x;
s+=r;
if(TMath::Abs(r/s)<1.0e-12) break;
}
sl1=2.0/pi*(-1.0+1.0/(x*x)+3.0*s/(x*x*x*x));
a1=TMath::Exp(x)/TMath::Sqrt(2*pi*x);
r=1.0;
bi1=1.0;
for (i=1; i<=16; i++) {
r=-0.125*r*(4.0-(2.0*i-1.0)*(2.0*i-1.0))/(i*x);
bi1+=r;
if(TMath::Abs(r/bi1)<1.0e-12) break;
}
sl1+=a1*bi1;
}
return sl1;
}
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