// @(#)root/mathcore:$Id$ // Authors: Rene Brun, Anna Kreshuk, Eddy Offermann, Fons Rademakers 29/07/95 /************************************************************************* * Copyright (C) 1995-2004, 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. // // // ////////////////////////////////////////////////////////////////////////// #include "TMath.h" #include "TError.h" #include <math.h> #include <string.h> #include <algorithm> #include "Riostream.h" #include "TString.h" #include <Math/SpecFuncMathCore.h> #include <Math/PdfFuncMathCore.h> #include <Math/ProbFuncMathCore.h> //const Double_t // TMath::Pi = 3.14159265358979323846, // TMath::E = 2.7182818284590452354; // Without this macro the THtml doc for TMath can not be generated #if !defined(R__ALPHA) && !defined(R__SOLARIS) && !defined(R__ACC) && !defined(R__FBSD) NamespaceImp(TMath) #endif namespace TMath { Double_t GamCf(Double_t a,Double_t x); Double_t GamSer(Double_t a,Double_t x); Double_t VavilovDenEval(Double_t rlam, Double_t *AC, Double_t *HC, Int_t itype); void VavilovSet(Double_t rkappa, Double_t beta2, Bool_t mode, Double_t *WCM, Double_t *AC, Double_t *HC, Int_t &itype, Int_t &npt); } //______________________________________________________________________________ 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) if(x==0.0) return 0.0; Double_t ax = Abs(x); return log(x+ax*sqrt(1.+1./(ax*ax))); #else return asinh(x); #endif } //______________________________________________________________________________ Double_t TMath::ACosH(Double_t x) { #if defined(WIN32) if(x==0.0) return 0.0; Double_t ax = Abs(x); return log(x+ax*sqrt(1.-1./(ax*ax))); #else return acosh(x); #endif } //______________________________________________________________________________ Double_t TMath::ATanH(Double_t x) { #if defined(WIN32) return log((1+x)/(1-x))/2; #else return atanh(x); #endif } //______________________________________________________________________________ Double_t TMath::Log2(Double_t x) { return log(x)/log(2.0); } //______________________________________________________________________________ Double_t TMath::DiLog(Double_t x) { // The DiLogarithm function // Code translated by R.Brun from CERNLIB DILOG function C332 const Double_t hf = 0.5; const Double_t pi = TMath::Pi(); const Double_t pi2 = pi*pi; const Double_t pi3 = pi2/3; const Double_t pi6 = pi2/6; const Double_t pi12 = pi2/12; const Double_t c[20] = {0.42996693560813697, 0.40975987533077105, -0.01858843665014592, 0.00145751084062268,-0.00014304184442340, 0.00001588415541880,-0.00000190784959387, 0.00000024195180854, -0.00000003193341274, 0.00000000434545063,-0.00000000060578480, 0.00000000008612098,-0.00000000001244332, 0.00000000000182256, -0.00000000000027007, 0.00000000000004042,-0.00000000000000610, 0.00000000000000093,-0.00000000000000014, 0.00000000000000002}; Double_t t,h,y,s,a,alfa,b1,b2,b0; if (x == 1) { h = pi6; } else if (x == -1) { h = -pi12; } else { t = -x; if (t <= -2) { y = -1/(1+t); s = 1; b1= TMath::Log(-t); b2= TMath::Log(1+1/t); a = -pi3+hf*(b1*b1-b2*b2); } else if (t < -1) { y = -1-t; s = -1; a = TMath::Log(-t); a = -pi6+a*(a+TMath::Log(1+1/t)); } else if (t <= -0.5) { y = -(1+t)/t; s = 1; a = TMath::Log(-t); a = -pi6+a*(-hf*a+TMath::Log(1+t)); } else if (t < 0) { y = -t/(1+t); s = -1; b1= TMath::Log(1+t); a = hf*b1*b1; } else if (t <= 1) { y = t; s = 1; a = 0; } else { y = 1/t; s = -1; b1= TMath::Log(t); a = pi6+hf*b1*b1; } h = y+y-1; alfa = h+h; b1 = 0; b2 = 0; for (Int_t i=19;i>=0;i--){ b0 = c[i] + alfa*b1-b2; b2 = b1; b1 = b0; } h = -(s*(b0-h*b2)+a); } return h; } //______________________________________________________________________________ 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 return ::ROOT::Math::erf(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 // return ::ROOT::Math::erfc(x); } //______________________________________________________________________________ Double_t TMath::ErfInverse(Double_t x) { // returns the inverse error function // x must be <-1<x<1 Int_t kMaxit = 50; Double_t kEps = 1e-14; Double_t kConst = 0.8862269254527579; // sqrt(pi)/2.0 if(TMath::Abs(x) <= kEps) return kConst*x; // Newton iterations Double_t erfi, derfi, y0,y1,dy0,dy1; if(TMath::Abs(x) < 1.0) { erfi = kConst*TMath::Abs(x); y0 = TMath::Erf(0.9*erfi); derfi = 0.1*erfi; for (Int_t iter=0; iter<kMaxit; iter++) { y1 = 1. - TMath::Erfc(erfi); dy1 = TMath::Abs(x) - y1; if (TMath::Abs(dy1) < kEps) {if (x < 0) return -erfi; else return erfi;} dy0 = y1 - y0; derfi *= dy1/dy0; y0 = y1; erfi += derfi; if(TMath::Abs(derfi/erfi) < kEps) {if (x < 0) return -erfi; else return erfi;} } } return 0; //did not converge } Double_t TMath::ErfcInverse(Double_t x) { // returns the inverse of the complementary error function // x must be 0<x<2 // implement using the quantile of the normal distribution // instead of ErfInverse for better numerical precision for large x // erfc-1(x) = - 1/sqrt(2) * normal_quantile( 0.5 * x) return - 0.70710678118654752440 * TMath::NormQuantile( 0.5 * x); } //______________________________________________________________________________ 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. // // C.Lanczos, SIAM Journal of Numerical Analysis B1 (1964), 86. // return ::ROOT::Math::tgamma(z); } //______________________________________________________________________________ Double_t TMath::Gamma(Double_t a,Double_t x) { // Computation of the normalized lower incomplete gamma function P(a,x) as defined in the // Handbook of Mathematical Functions by Abramowitz and Stegun, formula 6.5.1 on page 260 . // Its normalization is such that TMath::Gamma(a,+infinity) = 1 . // // Begin_Latex // P(a, x) = #frac{1}{#Gamma(a) } #int_{0}^{x} t^{a-1} e^{-t} dt // End_Latex // // //--- Nve 14-nov-1998 UU-SAP Utrecht return ::ROOT::Math::inc_gamma(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-14; // 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-14; // 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, Bool_t norm) { // Calculate a gaussian function with mean and sigma. // If norm=kTRUE (default is kFALSE) the result is divided // by sqrt(2*Pi)*sigma. if (sigma == 0) return 1.e30; Double_t arg = (x-mean)/sigma; // for |arg| > 39 result is zero in double precision if (arg < -39.0 || arg > 39.0) return 0.0; Double_t res = TMath::Exp(-0.5*arg*arg); if (!norm) return res; return res/(2.50662827463100024*sigma); //sqrt(2*Pi)=2.50662827463100024 } //______________________________________________________________________________ Double_t TMath::Landau(Double_t x, Double_t mu, Double_t sigma, Bool_t norm) { // The LANDAU function. // mu is a location parameter and correspond approximatly to the most probable value // and sigma is a scale parameter (not the sigma of the full distribution which is not defined) // Note that for mu=0 and sigma=1 (default values) the exact location of the maximum of the distribution // (most proble value) is at x = -0.22278 // This function has been adapted from the CERNLIB routine G110 denlan. // If norm=kTRUE (default is kFALSE) the result is divided by sigma if (sigma <= 0) return 0; Double_t den = ::ROOT::Math::landau_pdf( (x-mu)/sigma ); if (!norm) return den; return den/sigma; } //______________________________________________________________________________ Double_t TMath::LnGamma(Double_t z) { // Computation of ln[gamma(z)] for all z. // // 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 return ::ROOT::Math::lgamma(z); } //______________________________________________________________________________ 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. // This implementation (thanks Kevin Lynch <krlynch@bu.edu>) is protected // against possible overflows. // Find the largest element, and divide that one out. Double_t av0 = Abs(v[0]), av1 = Abs(v[1]), av2 = Abs(v[2]); Double_t amax, foo, bar; // 0 >= {1, 2} if( av0 >= av1 && av0 >= av2 ) { amax = av0; foo = av1; bar = av2; } // 1 >= {0, 2} else if (av1 >= av0 && av1 >= av2) { amax = av1; foo = av0; bar = av2; } // 2 >= {0, 1} else { amax = av2; foo = av0; bar = av1; } if (amax == 0.0) return 0.; Double_t foofrac = foo/amax, barfrac = bar/amax; Double_t d = amax * Sqrt(1.+foofrac*foofrac+barfrac*barfrac); v[0] /= d; v[1] /= d; v[2] /= d; return d; } //______________________________________________________________________________ Double_t TMath::Poisson(Double_t x, Double_t par) { // compute the Poisson distribution function for (x,par) // The Poisson PDF is implemented by means of Euler's Gamma-function // (for the factorial), so for any x integer argument it is correct. // BUT for non-integer x values, it IS NOT equal to the Poisson distribution. // see TMath::PoissonI to get a non-smooth function. // Note that for large values of par, it is better to call // TMath::Gaus(x,par,sqrt(par),kTRUE) //Begin_Html /* <img src="gif/Poisson.gif"> */ //End_Html if (x<0) return 0; else if (x == 0.0) return 1./Exp(par); else { Double_t lnpoisson = x*log(par)-par-LnGamma(x+1.); return Exp(lnpoisson); } // An alternative strategy is to transition to a Gaussian approximation for // large values of par ... // else { // return Gaus(x,par,Sqrt(par),kTRUE); // } } //______________________________________________________________________________ Double_t TMath::PoissonI(Double_t x, Double_t par) { // compute the Poisson distribution function for (x,par) // This is a non-smooth function. // This function is equivalent to ROOT::Math::poisson_pdf //Begin_Html /* <img src="gif/PoissonI.gif"> */ //End_Html Int_t ix = Int_t(x); return Poisson(ix,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; } return ::ROOT::Math::chisquared_cdf_c(chi2,ndf); } //______________________________________________________________________________ Double_t TMath::KolmogorovProb(Double_t z) { // Calculates the Kolmogorov distribution function, //Begin_Html /* <img src="gif/kolmogorov.gif"> */ //End_Html // 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. Double_t fj[4] = {-2,-8,-18,-32}, r[4]; const Double_t w = 2.50662827; // c1 - -pi**2/8, c2 = 9*c1, c3 = 25*c1 const Double_t c1 = -1.2337005501361697; const Double_t c2 = -11.103304951225528; const Double_t c3 = -30.842513753404244; Double_t u = TMath::Abs(z); Double_t p; if (u < 0.2) { p = 1; } else if (u < 0.755) { Double_t v = 1./(u*u); p = 1 - w*(TMath::Exp(c1*v) + TMath::Exp(c2*v) + TMath::Exp(c3*v))/u; } else if (u < 6.8116) { r[1] = 0; r[2] = 0; r[3] = 0; Double_t v = u*u; Int_t maxj = TMath::Max(1,TMath::Nint(3./u)); for (Int_t j=0; j<maxj;j++) { r[j] = TMath::Exp(fj[j]*v); } p = 2*(r[0] - r[1] +r[2] - r[3]); } else { p = 0; } return p; } //______________________________________________________________________________ Double_t TMath::KolmogorovTest(Int_t na, const Double_t *a, Int_t nb, const Double_t *b, Option_t *option) { // Statistical test whether two one-dimensional sets of points are compatible // with coming from the same parent distribution, using the Kolmogorov test. // That is, it is used to compare two experimental distributions of unbinned data. // // Input: // a,b: One-dimensional arrays of length na, nb, respectively. // The elements of a and b must be given in ascending order. // option is a character string to specify options // "D" Put out a line of "Debug" printout // "M" Return the Maximum Kolmogorov distance instead of prob // // Output: // The returned value prob is a calculated confidence level which gives a // statistical test for compatibility of a and b. // Values of prob close to zero are taken as indicating a small probability // of compatibility. For two point sets drawn randomly from the same parent // distribution, the value of prob should be uniformly distributed between // zero and one. // in case of error the function return -1 // If the 2 sets have a different number of points, the minimum of // the two sets is used. // // Method: // The Kolmogorov test is used. The test statistic is the maximum deviation // between the two integrated distribution functions, multiplied by the // normalizing factor (rdmax*sqrt(na*nb/(na+nb)). // // Code adapted by Rene Brun from CERNLIB routine TKOLMO (Fred James) // (W.T. Eadie, D. Drijard, F.E. James, M. Roos and B. Sadoulet, // Statistical Methods in Experimental Physics, (North-Holland, // Amsterdam 1971) 269-271) // // Method Improvement by Jason A Detwiler (JADetwiler@lbl.gov) // ----------------------------------------------------------- // The nuts-and-bolts of the TMath::KolmogorovTest() algorithm is a for-loop // over the two sorted arrays a and b representing empirical distribution // functions. The for-loop handles 3 cases: when the next points to be // evaluated satisfy a>b, a<b, or a=b: // // for (Int_t i=0;i<na+nb;i++) { // if (a[ia-1] < b[ib-1]) { // rdiff -= sa; // ia++; // if (ia > na) {ok = kTRUE; break;} // } else if (a[ia-1] > b[ib-1]) { // rdiff += sb; // ib++; // if (ib > nb) {ok = kTRUE; break;} // } else { // rdiff += sb - sa; // ia++; // ib++; // if (ia > na) {ok = kTRUE; break;} // if (ib > nb) {ok = kTRUE; break;} // } // rdmax = TMath::Max(rdmax,TMath::Abs(rdiff)); // } // // For the last case, a=b, the algorithm advances each array by one index in an // attempt to move through the equality. However, this is incorrect when one or // the other of a or b (or both) have a repeated value, call it x. For the KS // statistic to be computed properly, rdiff needs to be calculated after all of // the a and b at x have been tallied (this is due to the definition of the // empirical distribution function; another way to convince yourself that the // old CERNLIB method is wrong is that it implies that the function defined as the // difference between a and b is multi-valued at x -- besides being ugly, this // would invalidate Kolmogorov's theorem). // // The solution is to just add while-loops into the equality-case handling to // perform the tally: // // } else { // double x = a[ia-1]; // while(a[ia-1] == x && ia <= na) { // rdiff -= sa; // ia++; // } // while(b[ib-1] == x && ib <= nb) { // rdiff += sb; // ib++; // } // if (ia > na) {ok = kTRUE; break;} // if (ib > nb) {ok = kTRUE; break;} // } // // // NOTE1 // A good description of the Kolmogorov test can be seen at: // http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm // LM: Nov 2010: clean up and returns now a zero distance when vectors are the same TString opt = option; opt.ToUpper(); Double_t prob = -1; // Require at least two points in each graph if (!a || !b || na <= 2 || nb <= 2) { Error("KolmogorovTest","Sets must have more than 2 points"); return prob; } // Constants needed Double_t rna = na; Double_t rnb = nb; Double_t sa = 1./rna; Double_t sb = 1./rnb; Double_t rdiff = 0; Double_t rdmax = 0; Int_t ia = 0; Int_t ib = 0; // Main loop over point sets to find max distance // rdiff is the running difference, and rdmax the max. Bool_t ok = kFALSE; for (Int_t i=0;i<na+nb;i++) { if (a[ia] < b[ib]) { rdiff -= sa; ia++; if (ia >= na) {ok = kTRUE; break;} } else if (a[ia] > b[ib]) { rdiff += sb; ib++; if (ib >= nb) {ok = kTRUE; break;} } else { // special cases for the ties double x = a[ia]; while(a[ia] == x && ia < na) { rdiff -= sa; ia++; } while(b[ib] == x && ib < nb) { rdiff += sb; ib++; } if (ia >= na) {ok = kTRUE; break;} if (ib >= nb) {ok = kTRUE; break;} } rdmax = TMath::Max(rdmax,TMath::Abs(rdiff)); } // Should never terminate this loop with ok = kFALSE! R__ASSERT(ok); if (ok) { rdmax = TMath::Max(rdmax,TMath::Abs(rdiff)); Double_t z = rdmax * TMath::Sqrt(rna*rnb/(rna+rnb)); prob = TMath::KolmogorovProb(z); } // debug printout if (opt.Contains("D")) { printf(" Kolmogorov Probability = %g, Max Dist = %g\n",prob,rdmax); } if(opt.Contains("M")) return rdmax; else return prob; } //______________________________________________________________________________ Double_t TMath::Voigt(Double_t xx, Double_t sigma, Double_t lg, Int_t r) { // Computation of Voigt function (normalised). // Voigt is a convolution of // gauss(xx) = 1/(sqrt(2*pi)*sigma) * exp(xx*xx/(2*sigma*sigma) // and // lorentz(xx) = (1/pi) * (lg/2) / (xx*xx + lg*lg/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 / (xx*xx + lg*lg /4); //pure Lorentz } if (lg == 0) { //pure gauss return 0.39894228 / sigma * TMath::Exp(-xx*xx / (2*sigma*sigma)); } Double_t x, y, k; x = xx / 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. } //______________________________________________________________________________ Bool_t TMath::RootsCubic(const Double_t coef[4],Double_t &a, Double_t &b, Double_t &c) { // Calculates roots of polynomial of 3rd order a*x^3 + b*x^2 + c*x + d, where // a == coef[3], b == coef[2], c == coef[1], d == coef[0] //coef[3] must be different from 0 // If the boolean returned by the method is false: // ==> there are 3 real roots a,b,c // If the boolean returned by the method is true: // ==> there is one real root a and 2 complex conjugates roots (b+i*c,b-i*c) // Author: Francois-Xavier Gentit Bool_t complex = kFALSE; Double_t r,s,t,p,q,d,ps3,ps33,qs2,u,v,tmp,lnu,lnv,su,sv,y1,y2,y3; a = 0; b = 0; c = 0; if (coef[3] == 0) return complex; r = coef[2]/coef[3]; s = coef[1]/coef[3]; t = coef[0]/coef[3]; p = s - (r*r)/3; ps3 = p/3; q = (2*r*r*r)/27.0 - (r*s)/3 + t; qs2 = q/2; ps33 = ps3*ps3*ps3; d = ps33 + qs2*qs2; if (d>=0) { complex = kTRUE; d = TMath::Sqrt(d); u = -qs2 + d; v = -qs2 - d; tmp = 1./3.; lnu = TMath::Log(TMath::Abs(u)); lnv = TMath::Log(TMath::Abs(v)); su = TMath::Sign(1.,u); sv = TMath::Sign(1.,v); u = su*TMath::Exp(tmp*lnu); v = sv*TMath::Exp(tmp*lnv); y1 = u + v; y2 = -y1/2; y3 = ((u-v)*TMath::Sqrt(3.))/2; tmp = r/3; a = y1 - tmp; b = y2 - tmp; c = y3; } else { Double_t phi,cphi,phis3,c1,c2,c3,pis3; ps3 = -ps3; ps33 = -ps33; cphi = -qs2/TMath::Sqrt(ps33); phi = TMath::ACos(cphi); phis3 = phi/3; pis3 = TMath::Pi()/3; c1 = TMath::Cos(phis3); c2 = TMath::Cos(pis3 + phis3); c3 = TMath::Cos(pis3 - phis3); tmp = TMath::Sqrt(ps3); y1 = 2*tmp*c1; y2 = -2*tmp*c2; y3 = -2*tmp*c3; tmp = r/3; a = y1 - tmp; b = y2 - tmp; c = y3 - tmp; } return complex; } //______________________________________________________________________________ void TMath::Quantiles(Int_t n, Int_t nprob, Double_t *x, Double_t *quantiles, Double_t *prob, Bool_t isSorted, Int_t *index, Int_t type) { //Computes sample quantiles, corresponding to the given probabilities //Parameters: // x -the data sample // n - its size // quantiles - computed quantiles are returned in there // prob - probabilities where to compute quantiles // nprob - size of prob array // isSorted - is the input array x sorted? // NOTE, that when the input is not sorted, an array of integers of size n needs // to be allocated. It can be passed by the user in parameter index, // or, if not passed, it will be allocated inside the function // // type - method to compute (from 1 to 9). Following types are provided: // Discontinuous: // type=1 - inverse of the empirical distribution function // type=2 - like type 1, but with averaging at discontinuities // type=3 - SAS definition: nearest even order statistic // Piecwise linear continuous: // In this case, sample quantiles can be obtained by linear interpolation // between the k-th order statistic and p(k). // type=4 - linear interpolation of empirical cdf, p(k)=k/n; // type=5 - a very popular definition, p(k) = (k-0.5)/n; // type=6 - used by Minitab and SPSS, p(k) = k/(n+1); // type=7 - used by S-Plus and R, p(k) = (k-1)/(n-1); // type=8 - resulting sample quantiles are approximately median unbiased // regardless of the distribution of x. p(k) = (k-1/3)/(n+1/3); // type=9 - resulting sample quantiles are approximately unbiased, when // the sample comes from Normal distribution. p(k)=(k-3/8)/(n+1/4); // // default type = 7 // // References: // 1) Hyndman, R.J and Fan, Y, (1996) "Sample quantiles in statistical packages" // American Statistician, 50, 361-365 // 2) R Project documentation for the function quantile of package {stats} if (type<1 || type>9){ printf("illegal value of type\n"); return; } Int_t *ind = 0; Bool_t isAllocated = kFALSE; if (!isSorted){ if (index) ind = index; else { ind = new Int_t[n]; isAllocated = kTRUE; } } // re-implemented by L.M (9/11/2011) to fix bug https://savannah.cern.ch/bugs/?87251 // following now exactly R implementation (available in library/stats/R/quantile.R ) // which follows precisely Hyndman-Fan paper // (older implementation had a bug for type =3) for (Int_t i=0; i<nprob; i++){ Double_t nppm = 0; Double_t gamma = 0; Int_t j = 0; //Discontinuous functions // type = 1,2, or 3 if (type < 4 ){ if (type == 3) nppm = n*prob[i]-0.5; // use m = -0.5 else nppm = n*prob[i]; // use m = 0 j = TMath::FloorNint(nppm); // LM : fix for numerical problems if nppm is actually equal to j, but results different for numerical error // g in the paper is nppm -j if (type == 1) gamma = ( (nppm -j) > j*TMath::Limits<Double_t>::Epsilon() ) ? 1 : 0; else if (type == 2) gamma = ( (nppm -j) > j*TMath::Limits<Double_t>::Epsilon() ) ? 1 : 0.5; else if (type == 3) // gamma = 0 for g=0 and j even gamma = ( TMath::Abs(nppm -j) <= j*TMath::Limits<Double_t>::Epsilon() && TMath::Even(j) ) ? 0 : 1; } else { // for continuous quantiles type 4 to 9) // define alpha and beta double a = 0; double b = 0; if (type == 4) { a = 0; b = 1; } else if (type == 5) { a = 0.5; b = 0.5; } else if (type == 6) { a = 0.; b = 0.; } else if (type == 7) { a = 1.; b = 1.; } else if (type == 8) { a = 1./3.; b = a; } else if (type == 9) { a = 3./8.; b = a; } // be careful with machine precision double eps = 4 * TMath::Limits<Double_t>::Epsilon(); nppm = a + prob[i] * ( n + 1 -a -b); // n * p + m j = TMath::FloorNint(nppm + eps); gamma = nppm - j; if (gamma < eps) gamma = 0; } // since index j starts from 1 first is j-1 and second is j // add protection to keep indices in range [0,n-1] int first = (j > 0 && j <=n) ? j-1 : ( j <=0 ) ? 0 : n-1; int second = (j > 0 && j < n) ? j : ( j <=0 ) ? 0 : n-1; Double_t xj, xjj; if (isSorted){ xj = x[first]; xjj = x[second]; } else { xj = TMath::KOrdStat(n, x, first, ind); xjj = TMath::KOrdStat(n, x, second, ind); } quantiles[i] = (1-gamma)*xj + gamma*xjj; // printf("x[j] = %12f x[j+1] = %12f \n",xj, xjj); } if (isAllocated) delete [] ind; } //______________________________________________________________________________ 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) 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) 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; } //______________________________________________________________________________ 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 // This function is kept for back compatibility. The code previously in this function // has been moved to the static function TString::Hash return TString::Hash(txt,ntxt); } //______________________________________________________________________________ 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-6; 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 = %g\n",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 = %g\n",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 (x == 0) return 0; 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. // By 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 (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. // By 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; } //______________________________________________________________________________ Double_t TMath::Beta(Double_t p, Double_t q) { // Calculates Beta-function Gamma(p)*Gamma(q)/Gamma(p+q). return ::ROOT::Math::beta(p, q); } //______________________________________________________________________________ Double_t TMath::BetaCf(Double_t x, Double_t a, Double_t b) { // Continued fraction evaluation by modified Lentz's method // used in calculation of incomplete Beta function. Int_t itmax = 500; Double_t eps = 3.e-14; Double_t fpmin = 1.e-30; Int_t m, m2; Double_t aa, c, d, del, qab, qam, qap; Double_t h; qab = a+b; qap = a+1.0; qam = a-1.0; c = 1.0; d = 1.0 - qab*x/qap; if (TMath::Abs(d)<fpmin) d=fpmin; d=1.0/d; h=d; for (m=1; m<=itmax; m++) { m2=m*2; aa = m*(b-m)*x/((qam+ m2)*(a+m2)); d = 1.0 +aa*d; if(TMath::Abs(d)<fpmin) d = fpmin; c = 1 +aa/c; if (TMath::Abs(c)<fpmin) c = fpmin; d=1.0/d; h*=d*c; aa = -(a+m)*(qab +m)*x/((a+m2)*(qap+m2)); d=1.0+aa*d; if(TMath::Abs(d)<fpmin) d = fpmin; c = 1.0 +aa/c; if (TMath::Abs(c)<fpmin) c = fpmin; d=1.0/d; del = d*c; h*=del; if (TMath::Abs(del-1)<=eps) break; } if (m>itmax) { Info("TMath::BetaCf", "a or b too big, or itmax too small, a=%g, b=%g, x=%g, h=%g, itmax=%d", a,b,x,h,itmax); } return h; } //______________________________________________________________________________ Double_t TMath::BetaDist(Double_t x, Double_t p, Double_t q) { // Computes the probability density function of the Beta distribution // (the distribution function is computed in BetaDistI). // The first argument is the point, where the function will be // computed, second and third are the function parameters. // Since the Beta distribution is bounded on both sides, it's often // used to represent processes with natural lower and upper limits. if ((x<0) || (x>1) || (p<=0) || (q<=0)){ Error("TMath::BetaDist", "parameter value outside allowed range"); return 0; } Double_t beta = TMath::Beta(p, q); Double_t r = TMath::Power(x, p-1)*TMath::Power(1-x, q-1)/beta; return r; } //______________________________________________________________________________ Double_t TMath::BetaDistI(Double_t x, Double_t p, Double_t q) { // Computes the distribution function of the Beta distribution. // The first argument is the point, where the function will be // computed, second and third are the function parameters. // Since the Beta distribution is bounded on both sides, it's often // used to represent processes with natural lower and upper limits. if ((x<0) || (x>1) || (p<=0) || (q<=0)){ Error("TMath::BetaDistI", "parameter value outside allowed range"); return 0; } Double_t betai = TMath::BetaIncomplete(x, p, q); return betai; } //______________________________________________________________________________ Double_t TMath::BetaIncomplete(Double_t x, Double_t a, Double_t b) { // Calculates the incomplete Beta-function. return ::ROOT::Math::inc_beta(x, a, b); } //______________________________________________________________________________ Double_t TMath::Binomial(Int_t n,Int_t k) { // Calculate the binomial coefficient n over k. if (n<0 || k<0 || n<k) return TMath::SignalingNaN(); if (k==0 || n==k) return 1; Int_t k1=TMath::Min(k,n-k); Int_t k2=n-k1; Double_t fact=k2+1; for (Double_t i=k1;i>1.;--i) fact *= (k2+i)/i; return fact; } //______________________________________________________________________________ Double_t TMath::BinomialI(Double_t p, Int_t n, Int_t k) { // Suppose an event occurs with probability _p_ per trial // Then the probability P of its occuring _k_ or more times // in _n_ trials is termed a cumulative binomial probability // the formula is P = sum_from_j=k_to_n(TMath::Binomial(n, j)* // *TMath::Power(p, j)*TMath::Power(1-p, n-j) // For _n_ larger than 12 BetaIncomplete is a much better way // to evaluate the sum than would be the straightforward sum calculation // for _n_ smaller than 12 either method is acceptable // ("Numerical Recipes") // --implementation by Anna Kreshuk if(k <= 0) return 1.0; if(k > n) return 0.0; if(k == n) return TMath::Power(p, n); return BetaIncomplete(p, Double_t(k), Double_t(n-k+1)); } //______________________________________________________________________________ Double_t TMath::CauchyDist(Double_t x, Double_t t, Double_t s) { // Computes the density of Cauchy distribution at point x // by default, standard Cauchy distribution is used (t=0, s=1) // t is the location parameter // s is the scale parameter // The Cauchy distribution, also called Lorentzian distribution, // is a continuous distribution describing resonance behavior // The mean and standard deviation of the Cauchy distribution are undefined. // The practical meaning of this is that collecting 1,000 data points gives // no more accurate an estimate of the mean and standard deviation than // does a single point. // The formula was taken from "Engineering Statistics Handbook" on site // http://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm // Implementation by Anna Kreshuk. // Example: // TF1* fc = new TF1("fc", "TMath::CauchyDist(x, [0], [1])", -5, 5); // fc->SetParameters(0, 1); // fc->Draw(); Double_t temp= (x-t)*(x-t)/(s*s); Double_t result = 1/(s*TMath::Pi()*(1+temp)); return result; } //______________________________________________________________________________ Double_t TMath::ChisquareQuantile(Double_t p, Double_t ndf) { // Evaluate the quantiles of the chi-squared probability distribution function. // Algorithm AS 91 Appl. Statist. (1975) Vol.24, P.35 // implemented by Anna Kreshuk. // Incorporates the suggested changes in AS R85 (vol.40(1), pp.233-5, 1991) // Parameters: // p - the probability value, at which the quantile is computed // ndf - number of degrees of freedom Double_t c[]={0, 0.01, 0.222222, 0.32, 0.4, 1.24, 2.2, 4.67, 6.66, 6.73, 13.32, 60.0, 70.0, 84.0, 105.0, 120.0, 127.0, 140.0, 175.0, 210.0, 252.0, 264.0, 294.0, 346.0, 420.0, 462.0, 606.0, 672.0, 707.0, 735.0, 889.0, 932.0, 966.0, 1141.0, 1182.0, 1278.0, 1740.0, 2520.0, 5040.0}; Double_t e = 5e-7; Double_t aa = 0.6931471806; Int_t maxit = 20; Double_t ch, p1, p2, q, t, a, b, x; Double_t s1, s2, s3, s4, s5, s6; if (ndf <= 0) return 0; Double_t g = TMath::LnGamma(0.5*ndf); Double_t xx = 0.5 * ndf; Double_t cp = xx - 1; if (ndf >= TMath::Log(p)*(-c[5])){ //starting approximation for ndf less than or equal to 0.32 if (ndf > c[3]) { x = TMath::NormQuantile(p); //starting approximation using Wilson and Hilferty estimate p1=c[2]/ndf; ch = ndf*TMath::Power((x*TMath::Sqrt(p1) + 1 - p1), 3); if (ch > c[6]*ndf + 6) ch = -2 * (TMath::Log(1-p) - cp * TMath::Log(0.5 * ch) + g); } else { ch = c[4]; a = TMath::Log(1-p); do{ q = ch; p1 = 1 + ch * (c[7]+ch); p2 = ch * (c[9] + ch * (c[8] + ch)); t = -0.5 + (c[7] + 2 * ch) / p1 - (c[9] + ch * (c[10] + 3 * ch)) / p2; ch = ch - (1 - TMath::Exp(a + g + 0.5 * ch + cp * aa) *p2 / p1) / t; }while (TMath::Abs(q/ch - 1) > c[1]); } } else { ch = TMath::Power((p * xx * TMath::Exp(g + xx * aa)),(1./xx)); if (ch < e) return ch; } //call to algorithm AS 239 and calculation of seven term Taylor series for (Int_t i=0; i<maxit; i++){ q = ch; p1 = 0.5 * ch; p2 = p - TMath::Gamma(xx, p1); t = p2 * TMath::Exp(xx * aa + g + p1 - cp * TMath::Log(ch)); b = t / ch; a = 0.5 * t - b * cp; s1 = (c[19] + a * (c[17] + a * (c[14] + a * (c[13] + a * (c[12] +c[11] * a))))) / c[24]; s2 = (c[24] + a * (c[29] + a * (c[32] + a * (c[33] + c[35] * a)))) / c[37]; s3 = (c[19] + a * (c[25] + a * (c[28] + c[31] * a))) / c[37]; s4 = (c[20] + a * (c[27] + c[34] * a) + cp * (c[22] + a * (c[30] + c[36] * a))) / c[38]; s5 = (c[13] + c[21] * a + cp * (c[18] + c[26] * a)) / c[37]; s6 = (c[15] + cp * (c[23] + c[16] * cp)) / c[38]; ch = ch + t * (1 + 0.5 * t * s1 - b * cp * (s1 - b * (s2 - b * (s3 - b * (s4 - b * (s5 - b * s6)))))); if (TMath::Abs(q / ch - 1) > e) break; } return ch; } //______________________________________________________________________________ Double_t TMath::FDist(Double_t F, Double_t N, Double_t M) { // Computes the density function of F-distribution // (probability function, integral of density, is computed in FDistI). // // Parameters N and M stand for degrees of freedom of chi-squares // mentioned above parameter F is the actual variable x of the // density function p(x) and the point at which the density function // is calculated. // // About F distribution: // F-distribution arises in testing whether two random samples // have the same variance. It is the ratio of two chi-square // distributions, with N and M degrees of freedom respectively, // where each chi-square is first divided by it's number of degrees // of freedom. // Implementation by Anna Kreshuk. return ::ROOT::Math::fdistribution_pdf(F,N,M); } //______________________________________________________________________________ Double_t TMath::FDistI(Double_t F, Double_t N, Double_t M) { // Calculates the cumulative distribution function of F-distribution, // this function occurs in the statistical test of whether two observed // samples have the same variance. For this test a certain statistic F, // the ratio of observed dispersion of the first sample to that of the // second sample, is calculated. N and M stand for numbers of degrees // of freedom in the samples 1-FDistI() is the significance level at // which the hypothesis "1 has smaller variance than 2" can be rejected. // A small numerical value of 1 - FDistI() implies a very significant // rejection, in turn implying high confidence in the hypothesis // "1 has variance greater than 2". // Implementation by Anna Kreshuk. Double_t fi = ::ROOT::Math::fdistribution_cdf(F,N,M); return fi; } //______________________________________________________________________________ Double_t TMath::GammaDist(Double_t x, Double_t gamma, Double_t mu, Double_t beta) { // Computes the density function of Gamma distribution at point x. // gamma - shape parameter // mu - location parameter // beta - scale parameter // // The definition can be found in "Engineering Statistics Handbook" on site // http://www.itl.nist.gov/div898/handbook/eda/section3/eda366b.htm // use now implementation in ROOT::Math::gamma_pdf //Begin_Html /* <img src="gif/gammadist.gif"> */ //End_Html if ((x<mu) || (gamma<=0) || (beta <=0)) { Error("TMath::GammaDist", "illegal parameter values"); return 0; } return ::ROOT::Math::gamma_pdf(x, gamma, beta, mu); } //______________________________________________________________________________ Double_t TMath::LaplaceDist(Double_t x, Double_t alpha, Double_t beta) { // Computes the probability density function of Laplace distribution // at point x, with location parameter alpha and shape parameter beta. // By default, alpha=0, beta=1 // This distribution is known under different names, most common is // double exponential distribution, but it also appears as // the two-tailed exponential or the bilateral exponential distribution Double_t temp; temp = TMath::Exp(-TMath::Abs((x-alpha)/beta)); temp /= (2.*beta); return temp; } //______________________________________________________________________________ Double_t TMath::LaplaceDistI(Double_t x, Double_t alpha, Double_t beta) { // Computes the distribution function of Laplace distribution // at point x, with location parameter alpha and shape parameter beta. // By default, alpha=0, beta=1 // This distribution is known under different names, most common is // double exponential distribution, but it also appears as // the two-tailed exponential or the bilateral exponential distribution Double_t temp; if (x<=alpha){ temp = 0.5*TMath::Exp(-TMath::Abs((x-alpha)/beta)); } else { temp = 1-0.5*TMath::Exp(-TMath::Abs((x-alpha)/beta)); } return temp; } //______________________________________________________________________________ Double_t TMath::LogNormal(Double_t x, Double_t sigma, Double_t theta, Double_t m) { // Computes the density of LogNormal distribution at point x. // Variable X has lognormal distribution if Y=Ln(X) has normal distribution // sigma is the shape parameter // theta is the location parameter // m is the scale parameter // The formula was taken from "Engineering Statistics Handbook" on site // http://www.itl.nist.gov/div898/handbook/eda/section3/eda3669.htm // Implementation using ROOT::Math::lognormal_pdf //Begin_Html /* <img src="gif/lognormal.gif"> */ //End_Html if ((x<theta) || (sigma<=0) || (m<=0)) { Error("TMath::Lognormal", "illegal parameter values"); return 0; } // lognormal_pdf uses same definition of http://en.wikipedia.org/wiki/Log-normal_distribution // where mu = log(m) return ::ROOT::Math::lognormal_pdf(x, TMath::Log(m), sigma, theta); } //______________________________________________________________________________ Double_t TMath::NormQuantile(Double_t p) { // Computes quantiles for standard normal distribution N(0, 1) // at probability p // ALGORITHM AS241 APPL. STATIST. (1988) VOL. 37, NO. 3, 477-484. if ((p<=0)||(p>=1)) { Error("TMath::NormQuantile", "probability outside (0, 1)"); return 0; } Double_t a0 = 3.3871328727963666080e0; Double_t a1 = 1.3314166789178437745e+2; Double_t a2 = 1.9715909503065514427e+3; Double_t a3 = 1.3731693765509461125e+4; Double_t a4 = 4.5921953931549871457e+4; Double_t a5 = 6.7265770927008700853e+4; Double_t a6 = 3.3430575583588128105e+4; Double_t a7 = 2.5090809287301226727e+3; Double_t b1 = 4.2313330701600911252e+1; Double_t b2 = 6.8718700749205790830e+2; Double_t b3 = 5.3941960214247511077e+3; Double_t b4 = 2.1213794301586595867e+4; Double_t b5 = 3.9307895800092710610e+4; Double_t b6 = 2.8729085735721942674e+4; Double_t b7 = 5.2264952788528545610e+3; Double_t c0 = 1.42343711074968357734e0; Double_t c1 = 4.63033784615654529590e0; Double_t c2 = 5.76949722146069140550e0; Double_t c3 = 3.64784832476320460504e0; Double_t c4 = 1.27045825245236838258e0; Double_t c5 = 2.41780725177450611770e-1; Double_t c6 = 2.27238449892691845833e-2; Double_t c7 = 7.74545014278341407640e-4; Double_t d1 = 2.05319162663775882187e0; Double_t d2 = 1.67638483018380384940e0; Double_t d3 = 6.89767334985100004550e-1; Double_t d4 = 1.48103976427480074590e-1; Double_t d5 = 1.51986665636164571966e-2; Double_t d6 = 5.47593808499534494600e-4; Double_t d7 = 1.05075007164441684324e-9; Double_t e0 = 6.65790464350110377720e0; Double_t e1 = 5.46378491116411436990e0; Double_t e2 = 1.78482653991729133580e0; Double_t e3 = 2.96560571828504891230e-1; Double_t e4 = 2.65321895265761230930e-2; Double_t e5 = 1.24266094738807843860e-3; Double_t e6 = 2.71155556874348757815e-5; Double_t e7 = 2.01033439929228813265e-7; Double_t f1 = 5.99832206555887937690e-1; Double_t f2 = 1.36929880922735805310e-1; Double_t f3 = 1.48753612908506148525e-2; Double_t f4 = 7.86869131145613259100e-4; Double_t f5 = 1.84631831751005468180e-5; Double_t f6 = 1.42151175831644588870e-7; Double_t f7 = 2.04426310338993978564e-15; Double_t split1 = 0.425; Double_t split2=5.; Double_t konst1=0.180625; Double_t konst2=1.6; Double_t q, r, quantile; q=p-0.5; if (TMath::Abs(q)<split1) { r=konst1-q*q; quantile = q* (((((((a7 * r + a6) * r + a5) * r + a4) * r + a3) * r + a2) * r + a1) * r + a0) / (((((((b7 * r + b6) * r + b5) * r + b4) * r + b3) * r + b2) * r + b1) * r + 1.); } else { if(q<0) r=p; else r=1-p; //error case if (r<=0) quantile=0; else { r=TMath::Sqrt(-TMath::Log(r)); if (r<=split2) { r=r-konst2; quantile=(((((((c7 * r + c6) * r + c5) * r + c4) * r + c3) * r + c2) * r + c1) * r + c0) / (((((((d7 * r + d6) * r + d5) * r + d4) * r + d3) * r + d2) * r + d1) * r + 1); } else{ r=r-split2; quantile=(((((((e7 * r + e6) * r + e5) * r + e4) * r + e3) * r + e2) * r + e1) * r + e0) / (((((((f7 * r + f6) * r + f5) * r + f4) * r + f3) * r + f2) * r + f1) * r + 1); } if (q<0) quantile=-quantile; } } return quantile; } //______________________________________________________________________________ Bool_t TMath::Permute(Int_t n, Int_t *a) { // Simple recursive algorithm to find the permutations of // n natural numbers, not necessarily all distinct // adapted from CERNLIB routine PERMU. // The input array has to be initialised with a non descending // sequence. The method returns kFALSE when // all combinations are exhausted. Int_t i,itmp; Int_t i1=-1; // find rightmost upward transition for(i=n-2; i>-1; i--) { if(a[i]<a[i+1]) { i1=i; break; } } // no more upward transitions, end of the story if(i1==-1) return kFALSE; else { // find lower right element higher than the lower // element of the upward transition for(i=n-1;i>i1;i--) { if(a[i] > a[i1]) { // swap the two itmp=a[i1]; a[i1]=a[i]; a[i]=itmp; break; } } // order the rest, in fact just invert, as there // are only downward transitions from here on for(i=0;i<(n-i1-1)/2;i++) { itmp=a[i1+i+1]; a[i1+i+1]=a[n-i-1]; a[n-i-1]=itmp; } } return kTRUE; } //______________________________________________________________________________ Double_t TMath::Student(Double_t T, Double_t ndf) { // Computes density function for Student's t- distribution // (the probability function (integral of density) is computed in StudentI). // // First parameter stands for x - the actual variable of the // density function p(x) and the point at which the density is calculated. // Second parameter stands for number of degrees of freedom. // // About Student distribution: // Student's t-distribution is used for many significance tests, for example, // for the Student's t-tests for the statistical significance of difference // between two sample means and for confidence intervals for the difference // between two population means. // // Example: suppose we have a random sample of size n drawn from normal // distribution with mean Mu and st.deviation Sigma. Then the variable // // t = (sample_mean - Mu)/(sample_deviation / sqrt(n)) // // has Student's t-distribution with n-1 degrees of freedom. // // NOTE that this function's second argument is number of degrees of freedom, // not the sample size. // // As the number of degrees of freedom grows, t-distribution approaches // Normal(0,1) distribution. // Implementation by Anna Kreshuk. if (ndf < 1) { return 0; } Double_t r = ndf; Double_t rh = 0.5*r; Double_t rh1 = rh + 0.5; Double_t denom = TMath::Sqrt(r*TMath::Pi())*TMath::Gamma(rh)*TMath::Power(1+T*T/r, rh1); return TMath::Gamma(rh1)/denom; } //______________________________________________________________________________ Double_t TMath::StudentI(Double_t T, Double_t ndf) { // Calculates the cumulative distribution function of Student's // t-distribution second parameter stands for number of degrees of freedom, // not for the number of samples // if x has Student's t-distribution, the function returns the probability of // x being less than T. // Implementation by Anna Kreshuk. Double_t r = ndf; Double_t si = (T>0) ? (1 - 0.5*BetaIncomplete((r/(r + T*T)), r*0.5, 0.5)) : 0.5*BetaIncomplete((r/(r + T*T)), r*0.5, 0.5); return si; } //______________________________________________________________________________ Double_t TMath::StudentQuantile(Double_t p, Double_t ndf, Bool_t lower_tail) { // Computes quantiles of the Student's t-distribution // 1st argument is the probability, at which the quantile is computed // 2nd argument - the number of degrees of freedom of the // Student distribution // When the 3rd argument lower_tail is kTRUE (default)- // the algorithm returns such x0, that // P(x < x0)=p // upper tail (lower_tail is kFALSE)- the algorithm returns such x0, that // P(x > x0)=p // the algorithm was taken from // G.W.Hill, "Algorithm 396, Student's t-quantiles" // "Communications of the ACM", 13(10), October 1970 Double_t quantile; Double_t temp; Bool_t neg; Double_t q; if (ndf<1 || p>=1 || p<=0) { Error("TMath::StudentQuantile", "illegal parameter values"); return 0; } if ((lower_tail && p>0.5)||(!lower_tail && p<0.5)){ neg=kFALSE; q=2*(lower_tail ? (1-p) : p); } else { neg=kTRUE; q=2*(lower_tail? p : (1-p)); } if ((ndf-1)<1e-8) { temp=TMath::PiOver2()*q; quantile = TMath::Cos(temp)/TMath::Sin(temp); } else { if ((ndf-2)<1e-8) { quantile = TMath::Sqrt(2./(q*(2-q))-2); } else { Double_t a=1./(ndf-0.5); Double_t b=48./(a*a); Double_t c=((20700*a/b -98)*a-16)*a+96.36; Double_t d=((94.5/(b+c)-3.)/b+1)*TMath::Sqrt(a*TMath::PiOver2())*ndf; Double_t x=q*d; Double_t y=TMath::Power(x, (2./ndf)); if (y>0.05+a){ //asymptotic inverse expansion about normal x=TMath::NormQuantile(q*0.5); y=x*x; if (ndf<5) c+=0.3*(ndf-4.5)*(x+0.6); c+=(((0.05*d*x-5.)*x-7.)*x-2.)*x +b; y=(((((0.4*y+6.3)*y+36.)*y+94.5)/c - y-3.)/b+1)*x; y=a*y*y; if(y>0.002) y = TMath::Exp(y)-1; else y += 0.5*y*y; } else { y=((1./(((ndf+6.)/(ndf*y)-0.089*d-0.822)*(ndf+2.)*3)+0.5/(ndf+4.))*y-1.)* (ndf+1.)/(ndf+2.)+1/y; } quantile = TMath::Sqrt(ndf*y); } } if(neg) quantile=-quantile; return quantile; } //______________________________________________________________________________ Double_t TMath::Vavilov(Double_t x, Double_t kappa, Double_t beta2) { //Returns the value of the Vavilov density function //Parameters: 1st - the point were the density function is evaluated // 2nd - value of kappa (distribution parameter) // 3rd - value of beta2 (distribution parameter) //The algorithm was taken from the CernLib function vavden(G115) //Reference: A.Rotondi and P.Montagna, Fast Calculation of Vavilov distribution //Nucl.Instr. and Meth. B47(1990), 215-224 //Accuracy: quote from the reference above: //"The resuls of our code have been compared with the values of the Vavilov //density function computed numerically in an accurate way: our approximation //shows a difference of less than 3% around the peak of the density function, slowly //increasing going towards the extreme tails to the right and to the left" //Begin_Html /* <img src="gif/Vavilov.gif"> */ //End_Html Double_t *ac = new Double_t[14]; Double_t *hc = new Double_t[9]; Int_t itype; Int_t npt; TMath::VavilovSet(kappa, beta2, 0, 0, ac, hc, itype, npt); Double_t v = TMath::VavilovDenEval(x, ac, hc, itype); delete [] ac; delete [] hc; return v; } //______________________________________________________________________________ Double_t TMath::VavilovI(Double_t x, Double_t kappa, Double_t beta2) { //Returns the value of the Vavilov distribution function //Parameters: 1st - the point were the density function is evaluated // 2nd - value of kappa (distribution parameter) // 3rd - value of beta2 (distribution parameter) //The algorithm was taken from the CernLib function vavden(G115) //Reference: A.Rotondi and P.Montagna, Fast Calculation of Vavilov distribution //Nucl.Instr. and Meth. B47(1990), 215-224 //Accuracy: quote from the reference above: //"The resuls of our code have been compared with the values of the Vavilov //density function computed numerically in an accurate way: our approximation //shows a difference of less than 3% around the peak of the density function, slowly //increasing going towards the extreme tails to the right and to the left" Double_t *ac = new Double_t[14]; Double_t *hc = new Double_t[9]; Double_t *wcm = new Double_t[201]; Int_t itype; Int_t npt; Int_t k; Double_t xx, v; TMath::VavilovSet(kappa, beta2, 1, wcm, ac, hc, itype, npt); if (x < ac[0]) v = 0; else if (x >=ac[8]) v = 1; else { xx = x - ac[0]; k = Int_t(xx*ac[10]); v = TMath::Min(wcm[k] + (xx - k*ac[9])*(wcm[k+1]-wcm[k])*ac[10], 1.); } delete [] ac; delete [] hc; delete [] wcm; return v; } //______________________________________________________________________________ Double_t TMath::LandauI(Double_t x) { //Returns the value of the Landau distribution function at point x. //The algorithm was taken from the Cernlib function dislan(G110) //Reference: K.S.Kolbig and B.Schorr, "A program package for the Landau //distribution", Computer Phys.Comm., 31(1984), 97-111 return ::ROOT::Math::landau_cdf(x); } //______________________________________________________________________________ void TMath::VavilovSet(Double_t rkappa, Double_t beta2, Bool_t mode, Double_t *WCM, Double_t *AC, Double_t *HC, Int_t &itype, Int_t &npt) { //Internal function, called by Vavilov and VavilovI Double_t BKMNX1 = 0.02, BKMNY1 = 0.05, BKMNX2 = 0.12, BKMNY2 = 0.05, BKMNX3 = 0.22, BKMNY3 = 0.05, BKMXX1 = 0.1 , BKMXY1 = 1, BKMXX2 = 0.2 , BKMXY2 = 1 , BKMXX3 = 0.3 , BKMXY3 = 1; Double_t FBKX1 = 2/(BKMXX1-BKMNX1), FBKX2 = 2/(BKMXX2-BKMNX2), FBKX3 = 2/(BKMXX3-BKMNX3), FBKY1 = 2/(BKMXY1-BKMNY1), FBKY2 = 2/(BKMXY2-BKMNY2), FBKY3 = 2/(BKMXY3-BKMNY3); Double_t FNINV[] = {0, 1, 0.5, 0.33333333, 0.25, 0.2}; Double_t EDGEC[]= {0, 0, 0.16666667e+0, 0.41666667e-1, 0.83333333e-2, 0.13888889e-1, 0.69444444e-2, 0.77160493e-3}; Double_t U1[] = {0, 0.25850868e+0, 0.32477982e-1, -0.59020496e-2, 0. , 0.24880692e-1, 0.47404356e-2, -0.74445130e-3, 0.73225731e-2, 0. , 0.11668284e-2, 0. , -0.15727318e-2,-0.11210142e-2}; Double_t U2[] = {0, 0.43142611e+0, 0.40797543e-1, -0.91490215e-2, 0. , 0.42127077e-1, 0.73167928e-2, -0.14026047e-2, 0.16195241e-1, 0.24714789e-2, 0.20751278e-2, 0. , -0.25141668e-2,-0.14064022e-2}; Double_t U3[] = {0, 0.25225955e+0, 0.64820468e-1, -0.23615759e-1, 0. , 0.23834176e-1, 0.21624675e-2, -0.26865597e-2, -0.54891384e-2, 0.39800522e-2, 0.48447456e-2, -0.89439554e-2, -0.62756944e-2,-0.24655436e-2}; Double_t U4[] = {0, 0.12593231e+1, -0.20374501e+0, 0.95055662e-1, -0.20771531e-1, -0.46865180e-1, -0.77222986e-2, 0.32241039e-2, 0.89882920e-2, -0.67167236e-2, -0.13049241e-1, 0.18786468e-1, 0.14484097e-1}; Double_t U5[] = {0, -0.24864376e-1, -0.10368495e-2, 0.14330117e-2, 0.20052730e-3, 0.18751903e-2, 0.12668869e-2, 0.48736023e-3, 0.34850854e-2, 0. , -0.36597173e-3, 0.19372124e-2, 0.70761825e-3, 0.46898375e-3}; Double_t U6[] = {0, 0.35855696e-1, -0.27542114e-1, 0.12631023e-1, -0.30188807e-2, -0.84479939e-3, 0. , 0.45675843e-3, -0.69836141e-2, 0.39876546e-2, -0.36055679e-2, 0. , 0.15298434e-2, 0.19247256e-2}; Double_t U7[] = {0, 0.10234691e+2, -0.35619655e+1, 0.69387764e+0, -0.14047599e+0, -0.19952390e+1, -0.45679694e+0, 0. , 0.50505298e+0}; Double_t U8[] = {0, 0.21487518e+2, -0.11825253e+2, 0.43133087e+1, -0.14500543e+1, -0.34343169e+1, -0.11063164e+1, -0.21000819e+0, 0.17891643e+1, -0.89601916e+0, 0.39120793e+0, 0.73410606e+0, 0. ,-0.32454506e+0}; Double_t V1[] = {0, 0.27827257e+0, -0.14227603e-2, 0.24848327e-2, 0. , 0.45091424e-1, 0.80559636e-2, -0.38974523e-2, 0. , -0.30634124e-2, 0.75633702e-3, 0.54730726e-2, 0.19792507e-2}; Double_t V2[] = {0, 0.41421789e+0, -0.30061649e-1, 0.52249697e-2, 0. , 0.12693873e+0, 0.22999801e-1, -0.86792801e-2, 0.31875584e-1, -0.61757928e-2, 0. , 0.19716857e-1, 0.32596742e-2}; Double_t V3[] = {0, 0.20191056e+0, -0.46831422e-1, 0.96777473e-2, -0.17995317e-2, 0.53921588e-1, 0.35068740e-2, -0.12621494e-1, -0.54996531e-2, -0.90029985e-2, 0.34958743e-2, 0.18513506e-1, 0.68332334e-2,-0.12940502e-2}; Double_t V4[] = {0, 0.13206081e+1, 0.10036618e+0, -0.22015201e-1, 0.61667091e-2, -0.14986093e+0, -0.12720568e-1, 0.24972042e-1, -0.97751962e-2, 0.26087455e-1, -0.11399062e-1, -0.48282515e-1, -0.98552378e-2}; Double_t V5[] = {0, 0.16435243e-1, 0.36051400e-1, 0.23036520e-2, -0.61666343e-3, -0.10775802e-1, 0.51476061e-2, 0.56856517e-2, -0.13438433e-1, 0. , 0. , -0.25421507e-2, 0.20169108e-2,-0.15144931e-2}; Double_t V6[] = {0, 0.33432405e-1, 0.60583916e-2, -0.23381379e-2, 0.83846081e-3, -0.13346861e-1, -0.17402116e-2, 0.21052496e-2, 0.15528195e-2, 0.21900670e-2, -0.13202847e-2, -0.45124157e-2, -0.15629454e-2, 0.22499176e-3}; Double_t V7[] = {0, 0.54529572e+1, -0.90906096e+0, 0.86122438e-1, 0. , -0.12218009e+1, -0.32324120e+0, -0.27373591e-1, 0.12173464e+0, 0. , 0. , 0.40917471e-1}; Double_t V8[] = {0, 0.93841352e+1, -0.16276904e+1, 0.16571423e+0, 0. , -0.18160479e+1, -0.50919193e+0, -0.51384654e-1, 0.21413992e+0, 0. , 0. , 0.66596366e-1}; Double_t W1[] = {0, 0.29712951e+0, 0.97572934e-2, 0. , -0.15291686e-2, 0.35707399e-1, 0.96221631e-2, -0.18402821e-2, -0.49821585e-2, 0.18831112e-2, 0.43541673e-2, 0.20301312e-2, -0.18723311e-2,-0.73403108e-3}; Double_t W2[] = {0, 0.40882635e+0, 0.14474912e-1, 0.25023704e-2, -0.37707379e-2, 0.18719727e+0, 0.56954987e-1, 0. , 0.23020158e-1, 0.50574313e-2, 0.94550140e-2, 0.19300232e-1}; Double_t W3[] = {0, 0.16861629e+0, 0. , 0.36317285e-2, -0.43657818e-2, 0.30144338e-1, 0.13891826e-1, -0.58030495e-2, -0.38717547e-2, 0.85359607e-2, 0.14507659e-1, 0.82387775e-2, -0.10116105e-1,-0.55135670e-2}; Double_t W4[] = {0, 0.13493891e+1, -0.26863185e-2, -0.35216040e-2, 0.24434909e-1, -0.83447911e-1, -0.48061360e-1, 0.76473951e-2, 0.24494430e-1, -0.16209200e-1, -0.37768479e-1, -0.47890063e-1, 0.17778596e-1, 0.13179324e-1}; Double_t W5[] = {0, 0.10264945e+0, 0.32738857e-1, 0. , 0.43608779e-2, -0.43097757e-1, -0.22647176e-2, 0.94531290e-2, -0.12442571e-1, -0.32283517e-2, -0.75640352e-2, -0.88293329e-2, 0.52537299e-2, 0.13340546e-2}; Double_t W6[] = {0, 0.29568177e-1, -0.16300060e-2, -0.21119745e-3, 0.23599053e-2, -0.48515387e-2, -0.40797531e-2, 0.40403265e-3, 0.18200105e-2, -0.14346306e-2, -0.39165276e-2, -0.37432073e-2, 0.19950380e-2, 0.12222675e-2}; Double_t W8[] = {0, 0.66184645e+1, -0.73866379e+0, 0.44693973e-1, 0. , -0.14540925e+1, -0.39529833e+0, -0.44293243e-1, 0.88741049e-1}; itype = 0; if (rkappa <0.01 || rkappa >12) { Error("Vavilov distribution", "illegal value of kappa"); return; } Double_t DRK[6]; Double_t DSIGM[6]; Double_t ALFA[8]; Int_t j; Double_t x, y, xx, yy, x2, x3, y2, y3, xy, p2, p3, q2, q3, pq; if (rkappa >= 0.29) { itype = 1; npt = 100; Double_t wk = 1./TMath::Sqrt(rkappa); AC[0] = (-0.032227*beta2-0.074275)*rkappa + (0.24533*beta2+0.070152)*wk + (-0.55610*beta2-3.1579); AC[8] = (-0.013483*beta2-0.048801)*rkappa + (-1.6921*beta2+8.3656)*wk + (-0.73275*beta2-3.5226); DRK[1] = wk*wk; DSIGM[1] = TMath::Sqrt(rkappa/(1-0.5*beta2)); for (j=1; j<=4; j++) { DRK[j+1] = DRK[1]*DRK[j]; DSIGM[j+1] = DSIGM[1]*DSIGM[j]; ALFA[j+1] = (FNINV[j]-beta2*FNINV[j+1])*DRK[j]; } HC[0]=TMath::Log(rkappa)+beta2+0.42278434; HC[1]=DSIGM[1]; HC[2]=ALFA[3]*DSIGM[3]; HC[3]=(3*ALFA[2]*ALFA[2] + ALFA[4])*DSIGM[4]-3; HC[4]=(10*ALFA[2]*ALFA[3]+ALFA[5])*DSIGM[5]-10*HC[2]; HC[5]=HC[2]*HC[2]; HC[6]=HC[2]*HC[3]; HC[7]=HC[2]*HC[5]; for (j=2; j<=7; j++) HC[j]*=EDGEC[j]; HC[8]=0.39894228*HC[1]; } else if (rkappa >=0.22) { itype = 2; npt = 150; x = 1+(rkappa-BKMXX3)*FBKX3; y = 1+(TMath::Sqrt(beta2)-BKMXY3)*FBKY3; xx = 2*x; yy = 2*y; x2 = xx*x-1; x3 = xx*x2-x; y2 = yy*y-1; y3 = yy*y2-y; xy = x*y; p2 = x2*y; p3 = x3*y; q2 = y2*x; q3 = y3*x; pq = x2*y2; AC[1] = W1[1] + W1[2]*x + W1[4]*x3 + W1[5]*y + W1[6]*y2 + W1[7]*y3 + W1[8]*xy + W1[9]*p2 + W1[10]*p3 + W1[11]*q2 + W1[12]*q3 + W1[13]*pq; AC[2] = W2[1] + W2[2]*x + W2[3]*x2 + W2[4]*x3 + W2[5]*y + W2[6]*y2 + W2[8]*xy + W2[9]*p2 + W2[10]*p3 + W2[11]*q2; AC[3] = W3[1] + W3[3]*x2 + W3[4]*x3 + W3[5]*y + W3[6]*y2 + W3[7]*y3 + W3[8]*xy + W3[9]*p2 + W3[10]*p3 + W3[11]*q2 + W3[12]*q3 + W3[13]*pq; AC[4] = W4[1] + W4[2]*x + W4[3]*x2 + W4[4]*x3 + W4[5]*y + W4[6]*y2 + W4[7]*y3 + W4[8]*xy + W4[9]*p2 + W4[10]*p3 + W4[11]*q2 + W4[12]*q3 + W4[13]*pq; AC[5] = W5[1] + W5[2]*x + W5[4]*x3 + W5[5]*y + W5[6]*y2 + W5[7]*y3 + W5[8]*xy + W5[9]*p2 + W5[10]*p3 + W5[11]*q2 + W5[12]*q3 + W5[13]*pq; AC[6] = W6[1] + W6[2]*x + W6[3]*x2 + W6[4]*x3 + W6[5]*y + W6[6]*y2 + W6[7]*y3 + W6[8]*xy + W6[9]*p2 + W6[10]*p3 + W6[11]*q2 + W6[12]*q3 + W6[13]*pq; AC[8] = W8[1] + W8[2]*x + W8[3]*x2 + W8[5]*y + W8[6]*y2 + W8[7]*y3 + W8[8]*xy; AC[0] = -3.05; } else if (rkappa >= 0.12) { itype = 3; npt = 200; x = 1 + (rkappa-BKMXX2)*FBKX2; y = 1 + (TMath::Sqrt(beta2)-BKMXY2)*FBKY2; xx = 2*x; yy = 2*y; x2 = xx*x-1; x3 = xx*x2-x; y2 = yy*y-1; y3 = yy*y2-y; xy = x*y; p2 = x2*y; p3 = x3*y; q2 = y2*x; q3 = y3*x; pq = x2*y2; AC[1] = V1[1] + V1[2]*x + V1[3]*x2 + V1[5]*y + V1[6]*y2 + V1[7]*y3 + V1[9]*p2 + V1[10]*p3 + V1[11]*q2 + V1[12]*q3; AC[2] = V2[1] + V2[2]*x + V2[3]*x2 + V2[5]*y + V2[6]*y2 + V2[7]*y3 + V2[8]*xy + V2[9]*p2 + V2[11]*q2 + V2[12]*q3; AC[3] = V3[1] + V3[2]*x + V3[3]*x2 + V3[4]*x3 + V3[5]*y + V3[6]*y2 + V3[7]*y3 + V3[8]*xy + V3[9]*p2 + V3[10]*p3 + V3[11]*q2 + V3[12]*q3 + V3[13]*pq; AC[4] = V4[1] + V4[2]*x + V4[3]*x2 + V4[4]*x3 + V4[5]*y + V4[6]*y2 + V4[7]*y3 + V4[8]*xy + V4[9]*p2 + V4[10]*p3 + V4[11]*q2 + V4[12]*q3; AC[5] = V5[1] + V5[2]*x + V5[3]*x2 + V5[4]*x3 + V5[5]*y + V5[6]*y2 + V5[7]*y3 + V5[8]*xy + V5[11]*q2 + V5[12]*q3 + V5[13]*pq; AC[6] = V6[1] + V6[2]*x + V6[3]*x2 + V6[4]*x3 + V6[5]*y + V6[6]*y2 + V6[7]*y3 + V6[8]*xy + V6[9]*p2 + V6[10]*p3 + V6[11]*q2 + V6[12]*q3 + V6[13]*pq; AC[7] = V7[1] + V7[2]*x + V7[3]*x2 + V7[5]*y + V7[6]*y2 + V7[7]*y3 + V7[8]*xy + V7[11]*q2; AC[8] = V8[1] + V8[2]*x + V8[3]*x2 + V8[5]*y + V8[6]*y2 + V8[7]*y3 + V8[8]*xy + V8[11]*q2; AC[0] = -3.04; } else { itype = 4; if (rkappa >=0.02) itype = 3; npt = 200; x = 1+(rkappa-BKMXX1)*FBKX1; y = 1+(TMath::Sqrt(beta2)-BKMXY1)*FBKY1; xx = 2*x; yy = 2*y; x2 = xx*x-1; x3 = xx*x2-x; y2 = yy*y-1; y3 = yy*y2-y; xy = x*y; p2 = x2*y; p3 = x3*y; q2 = y2*x; q3 = y3*x; pq = x2*y2; if (itype==3){ AC[1] = U1[1] + U1[2]*x + U1[3]*x2 + U1[5]*y + U1[6]*y2 + U1[7]*y3 + U1[8]*xy + U1[10]*p3 + U1[12]*q3 + U1[13]*pq; AC[2] = U2[1] + U2[2]*x + U2[3]*x2 + U2[5]*y + U2[6]*y2 + U2[7]*y3 + U2[8]*xy + U2[9]*p2 + U2[10]*p3 + U2[12]*q3 + U2[13]*pq; AC[3] = U3[1] + U3[2]*x + U3[3]*x2 + U3[5]*y + U3[6]*y2 + U3[7]*y3 + U3[8]*xy + U3[9]*p2 + U3[10]*p3 + U3[11]*q2 + U3[12]*q3 + U3[13]*pq; AC[4] = U4[1] + U4[2]*x + U4[3]*x2 + U4[4]*x3 + U4[5]*y + U4[6]*y2 + U4[7]*y3 + U4[8]*xy + U4[9]*p2 + U4[10]*p3 + U4[11]*q2 + U4[12]*q3; AC[5] = U5[1] + U5[2]*x + U5[3]*x2 + U5[4]*x3 + U5[5]*y + U5[6]*y2 + U5[7]*y3 + U5[8]*xy + U5[10]*p3 + U5[11]*q2 + U5[12]*q3 + U5[13]*pq; AC[6] = U6[1] + U6[2]*x + U6[3]*x2 + U6[4]*x3 + U6[5]*y + U6[7]*y3 + U6[8]*xy + U6[9]*p2 + U6[10]*p3 + U6[12]*q3 + U6[13]*pq; AC[7] = U7[1] + U7[2]*x + U7[3]*x2 + U7[4]*x3 + U7[5]*y + U7[6]*y2 + U7[8]*xy; } AC[8] = U8[1] + U8[2]*x + U8[3]*x2 + U8[4]*x3 + U8[5]*y + U8[6]*y2 + U8[7]*y3 + U8[8]*xy + U8[9]*p2 + U8[10]*p3 + U8[11]*q2 + U8[13]*pq; AC[0] = -3.03; } AC[9] = (AC[8] - AC[0])/npt; AC[10] = 1./AC[9]; if (itype == 3) { x = (AC[7]-AC[8])/(AC[7]*AC[8]); y = 1./TMath::Log(AC[8]/AC[7]); p2 = AC[7]*AC[7]; AC[11] = p2*(AC[1]*TMath::Exp(-AC[2]*(AC[7]+AC[5]*p2)- AC[3]*TMath::Exp(-AC[4]*(AC[7]+AC[6]*p2)))-0.045*y/AC[7])/(1+x*y*AC[7]); AC[12] = (0.045+x*AC[11])*y; } if (itype == 4) AC[13] = 0.995/LandauI(AC[8]); if (mode==0) return; // x = AC[0]; WCM[0] = 0; Double_t fl, fu; Int_t k; fl = TMath::VavilovDenEval(x, AC, HC, itype); for (k=1; k<=npt; k++) { x += AC[9]; fu = TMath::VavilovDenEval(x, AC, HC, itype); WCM[k] = WCM[k-1] + fl + fu; fl = fu; } x = 0.5*AC[9]; for (k=1; k<=npt; k++) WCM[k]*=x; } //______________________________________________________________________________ Double_t TMath::VavilovDenEval(Double_t rlam, Double_t *AC, Double_t *HC, Int_t itype) { //Internal function, called by Vavilov and VavilovSet Double_t v = 0; if (rlam < AC[0] || rlam > AC[8]) return 0; Int_t k; Double_t x, fn, s; Double_t h[10]; if (itype ==1 ) { fn = 1; x = (rlam + HC[0])*HC[1]; h[1] = x; h[2] = x*x -1; for (k=2; k<=8; k++) { fn++; h[k+1] = x*h[k]-fn*h[k-1]; } s = 1 + HC[7]*h[9]; for (k=2; k<=6; k++) s+=HC[k]*h[k+1]; v = HC[8]*TMath::Exp(-0.5*x*x)*TMath::Max(s, 0.); } else if (itype == 2) { x = rlam*rlam; v = AC[1]*TMath::Exp(-AC[2]*(rlam+AC[5]*x) - AC[3]*TMath::Exp(-AC[4]*(rlam+AC[6]*x))); } else if (itype == 3) { if (rlam < AC[7]) { x = rlam*rlam; v = AC[1]*TMath::Exp(-AC[2]*(rlam+AC[5]*x)-AC[3]*TMath::Exp(-AC[4]*(rlam+AC[6]*x))); } else { x = 1./rlam; v = (AC[11]*x + AC[12])*x; } } else if (itype == 4) { v = AC[13]*TMath::Landau(rlam); } return v; }
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