TMath.cxx

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// @(#)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 <cmath>
#include <cstring>
#include <algorithm>
#include <iostream>
#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__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);
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns `sqrt(x*x + y*y)`
 
Long_t TMath::Hypot(Long_t x, Long_t y)
{
   return (Long_t) (hypot((Double_t)x, (Double_t)y) + 0.5);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns `sqrt(x*x + y*y)`
 
Double_t TMath::Hypot(Double_t x, Double_t y)
{
   return hypot(x, y);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the area hyperbolic sine of `x`.
 
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
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the nonnegative area hyperbolic cosine of `x`.
 
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
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the area hyperbolic tangent of `x`.
 
Double_t TMath::ATanH(Double_t x)
{
#if defined(WIN32)
   return log((1+x)/(1-x))/2;
#else
   return atanh(x);
#endif
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the binary (base-2) logarithm of `x`.
 
Double_t TMath::Log2(Double_t x)
{
   return log(x)/log(2.0);
}
 
////////////////////////////////////////////////////////////////////////////////
/// The DiLogarithm function
/// Code translated by R.Brun from CERNLIB DILOG function C332
 
Double_t TMath::DiLog(Double_t x)
{
   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, 0.40975987533077106,
     -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;
   t=h=y=s=a=alfa=b1=b2=b0=0.;
 
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computation of the error function erf(x).
/// Erf(x) = (2/sqrt(pi)) Integral(exp(-t^2))dt between 0 and x
 
Double_t TMath::Erf(Double_t x)
{
   return ::ROOT::Math::erf(x);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computes the complementary error function erfc(x).
/// Erfc(x) = (2/sqrt(pi)) Integral(exp(-t^2))dt between x and infinity
 
Double_t TMath::Erfc(Double_t x)
{
   return ::ROOT::Math::erfc(x);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the inverse error function.
/// x must be  <-1<x<1
 
Double_t TMath::ErfInverse(Double_t x)
{
   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
}
 
////////////////////////////////////////////////////////////////////////////////
/// 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
 
Double_t TMath::ErfcInverse(Double_t x)
{
 
   // erfc-1(x) = - 1/sqrt(2) * normal_quantile( 0.5 * x)
   return - 0.70710678118654752440 * TMath::NormQuantile( 0.5 * x);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computes factorial(n).
 
Double_t TMath::Factorial(Int_t n)
{
   if (n <= 0) return 1.;
   Double_t x = 1;
   Int_t b = 0;
   do {
      b++;
      x *= b;
   } while (b != n);
   return 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.
 
Double_t TMath::Freq(Double_t x)
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computation of gamma(z) for all z.
///
/// C.Lanczos, SIAM Journal of Numerical Analysis B1 (1964), 86.
 
Double_t TMath::Gamma(Double_t z)
{
   return ::ROOT::Math::tgamma(z);
}
 
////////////////////////////////////////////////////////////////////////////////
/// 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 .
///
///  \f[
///  P(a, x) = \frac{1}{\Gamma(a)} \int_{0}^{x} t^{a-1} e^{-t} dt
///  \f]
///
/// \author NvE 14-nov-1998 UU-SAP Utrecht
 
Double_t TMath::Gamma(Double_t a,Double_t x)
{
   return ::ROOT::Math::inc_gamma(a, x);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computation of the incomplete gamma function P(a,x)
/// via its continued fraction representation.
///
/// \author NvE 14-nov-1998 UU-SAP Utrecht
 
Double_t TMath::GamCf(Double_t a,Double_t x)
{
   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) std::cout << "*GamCf(a,x)* a too large or itmax too small" << std::endl;
   }
   Double_t v = Exp(-x+a*Log(x)-gln)*h;
   return (1-v);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computation of the incomplete gamma function P(a,x)
/// via its series representation.
///
/// \author NvE 14-nov-1998 UU-SAP Utrecht
 
Double_t TMath::GamSer(Double_t a,Double_t x)
{
   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) std::cout << "*GamSer(a,x)* a too large or itmax too small" << std::endl;
   }
   Double_t v = sum*Exp(-x+a*Log(x)-gln);
   return v;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Calculates a Breit Wigner function with mean and gamma.
 
Double_t TMath::BreitWigner(Double_t x, Double_t mean, Double_t gamma)
{
   Double_t bw = gamma/((x-mean)*(x-mean) + gamma*gamma/4);
   return bw/(2*Pi());
}
 
////////////////////////////////////////////////////////////////////////////////
/// Calculates a Relativistic Breit Wigner function with median and gamma.
// \f$ BW(E) = \frac{2\sqrt{2}}{\pi}\frac{M^{2}\gamma\sqrt{M^{2} + \gamma^{2}}}{\left(\sqrt{M^{2}+M\sqrt{M^{2} + \gamma^{2}}}\right)\left(\left(E^{2} - M^{2}\right)^{2} + M^{2}\gamma^{2}\right)} \f$
 
Double_t TMath::BreitWignerRelativistic(Double_t x, Double_t median, Double_t gamma)
{
  Double_t mm = median*median;
  Double_t gg = gamma*gamma;
  Double_t mg = median*gamma;
  Double_t xxMinusmm = x*x - mm;
 
  Double_t y = sqrt(mm * (mm + gg));
  Double_t k = (0.90031631615710606*mg*y)/(sqrt(mm+y)); //2*sqrt(2)/pi = 0.90031631615710606
 
  Double_t bw = k/(xxMinusmm*xxMinusmm + mg*mg);
  return bw;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Calculates a gaussian function with mean and sigma.
/// If norm=kTRUE (default is kFALSE) the result is divided
/// by sqrt(2*Pi)*sigma.
 
Double_t TMath::Gaus(Double_t x, Double_t mean, Double_t sigma, Bool_t norm)
{
   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
}
 
////////////////////////////////////////////////////////////////////////////////
/// The LANDAU function.
///
/// mu is a location parameter and correspond approximately 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 proper 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
 
Double_t TMath::Landau(Double_t x, Double_t mu, Double_t sigma, Bool_t norm)
{
   if (sigma <= 0) return 0;
   Double_t den = ::ROOT::Math::landau_pdf( (x-mu)/sigma );
   if (!norm) return den;
   return den/sigma;
}
 
////////////////////////////////////////////////////////////////////////////////
/// 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.
///
/// \author NvE 14-nov-1998 UU-SAP Utrecht
 
Double_t TMath::LnGamma(Double_t z)
{
   return ::ROOT::Math::lgamma(z);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Normalize a vector v in place.
/// Returns the norm of the original vector.
 
Float_t TMath::Normalize(Float_t v[3])
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// 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.
 
Double_t TMath::Normalize(Double_t v[3])
{
   // 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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computes 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 the correct Poisson distribution.
/// BUT for non-integer x values, it IS NOT equal to the Poisson distribution !
///
/// Begin_Macro
/// {
///   TCanvas *c1 = new TCanvas("c1", "c1", 700, 500);
///   TF1 *poisson = new TF1("poisson", "TMath::Poisson(x, 5)", 0, 15);
///   poisson->Draw("L");
/// }
/// End_Macro
 
Double_t TMath::Poisson(Double_t x, Double_t par)
{
   if (par < 0)
      return TMath::QuietNaN();
   if (x < 0)
      return 0;
   else if (x == 0.0 )
      return Exp(-par);
   else
   {
      return Exp( x * log(par) - LnGamma(x + 1.) - par);
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computes the Discrete Poisson distribution function for (x,par).
/// This is a discrete and a non-smooth function.
/// This function is equivalent to ROOT::Math::poisson_pdf
///
/// Begin_Macro
/// {
///   TCanvas *c1 = new TCanvas("c1", "c1", 700, 500);
///   TF1 *poissoni = new TF1("poissoni", "TMath::PoissonI(x, 5)", 0, 15);
///   poissoni->SetNpx(1000);
///   poissoni->Draw("L");
/// }
/// End_Macro
 
Double_t TMath::PoissonI(Double_t x, Double_t par)
{
   Int_t ix = Int_t(x);
   return Poisson(ix,par);
}
 
////////////////////////////////////////////////////////////////////////////////
/// 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.
///
/// \author NvE 14-nov-1998 UU-SAP Utrecht
 
Double_t TMath::Prob(Double_t chi2,Int_t ndf)
{
   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);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Calculates the Kolmogorov distribution function,
///
/// \f[
/// P(z) = 2 \sum_{j=1}^{\infty} (-1)^{j-1} e^{-2 j^2 z^2}
/// \f]
///
/// 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, "statistical Methods in Experimental
/// Physics', pp 269-270).
///
/// This function returns the confidence level for the null hypothesis, where:
///  - \f$ z = dn \sqrt{n} \f$, and
///     - \f$ dn \f$  is the maximum deviation between a hypothetical distribution
///           function and an experimental distribution with
///     - \f$ n \f$  events
///
/// NOTE: To compare two experimental distributions with m and n events,
///       use \f$ z = \sqrt{m n/(m+n)) dn} \f$
///
/// Accuracy: The function is far too accurate for any imaginable application.
///           Probabilities less than \f$ 10^{-15} \f$ 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 TMath::KolmogorovProb(Double_t z)
{
   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;
   }
 
////////////////////////////////////////////////////////////////////////////////
///  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:
///
/// ~~~ {.cpp}
///      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:
///
/// ~~~ {.cpp}
///         } 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;}
///         }
/// ~~~
///
/// ### Note:
///  A good description of the Kolmogorov test can be seen at:
///    http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm
 
Double_t TMath::KolmogorovTest(Int_t na, const Double_t *a, Int_t nb, const Double_t *b, Option_t *option)
{
//  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(ia < na && a[ia] == x) {
            rdiff -= sa;
            ia++;
         }
         while(ib < nb && b[ib] == x) {
            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;
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Computation of Voigt function (normalised).
/// Voigt is a convolution of the two functions:
/// \f[
/// gauss(xx) = \frac{1}{(\sqrt{2\pi} sigma)} e^{\frac{xx^{2}}{(2 sigma{^2})}}
/// \f]
/// and
/// \f[
/// lorentz(xx) = \frac{ \frac{1}{\pi} \frac{lg}{2} }{ (xx^{2} + \frac{lg^{2}}{4}) }
/// \f]
/// \.
///
/// The Voigt function is known to be the real part of Faddeeva function also
/// called complex error function [2].
///
/// The algorithm 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 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)
 
Double_t TMath::Voigt(Double_t xx, Double_t sigma, Double_t lg, Int_t r)
{
   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.
}
 
////////////////////////////////////////////////////////////////////////////////
/// 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 TMath::RootsCubic(const Double_t coef[4],Double_t &a, Double_t &b, Double_t &c)
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
///Computes sample quantiles, corresponding to the given probabilities
///
///  \param[in] x           the data sample
///  \param[in] n           its size
///  \param[out] quantiles  computed quantiles are returned in there
///  \param[in] prob        probabilities where to compute quantiles
///  \param[in] nprob       size of prob array
///  \param[in] isSorted    is the input array x sorted ?
///  \param[in] index       parameter index
///  \param[in] type        method to compute (from 1 to 9).
///
/// #### NOTE:
///  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
///
/// ### 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
///  - Piecewise 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}
 
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)
{
 
   if (type<1 || type>9){
      printf("illegal value of type\n");
      return;
   }
   Int_t *ind = nullptr;
   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
 
         // be careful with machine precision
         double eps = 4 * TMath::Limits<Double_t>::Epsilon();
         j = TMath::FloorNint(nppm + eps);
 
         // 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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// 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.
/// \param[in] Narr   number of array elements
/// \param[in] *arr1  is unchanged;
/// \param[in] *arr2  is the array of indices corresponding to the descending value
///            of arr1 with arr2[0] corresponding to the largest arr1 value and
///            arr2[Narr] the smallest.
///
/// \author Adrian Bevan (bevan@slac.stanford.edu)
 
void TMath::BubbleHigh(Int_t Narr, Double_t *arr1, Int_t *arr2)
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Opposite ordering of the array arr2[] to that of BubbleHigh.
///
/// \author Adrian Bevan (bevan@slac.stanford.edu)
 
void TMath::BubbleLow(Int_t Narr, Double_t *arr1, Int_t *arr2)
{
   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;
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Calculates hash index from any char string.
/// Based on pre-calculated table of 256 specially selected numbers.
/// These numbers are selected in such a way, that for string
/// length == 4 (integer number) the hash is unambiguous, 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
 
ULong_t TMath::Hash(const void *txt, Int_t ntxt)
{
   return TString::Hash(txt,ntxt);
}
 
 
////////////////////////////////////////////////////////////////////////////////
 
ULong_t TMath::Hash(const char *txt)
{
   return Hash(txt, Int_t(strlen(txt)));
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computes the modified Bessel function I_0(x) for any real x.
///
/// \author NvE 12-mar-2000 UU-SAP Utrecht
 
Double_t TMath::BesselI0(Double_t x)
{
   // 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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computes 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.
///
/// \author NvE 12-mar-2000 UU-SAP Utrecht
 
Double_t TMath::BesselK0(Double_t x)
{
   // 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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computes 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.
///
/// \author NvE 12-mar-2000 UU-SAP Utrecht
 
Double_t TMath::BesselI1(Double_t x)
{
   // 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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computes 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.
///
/// \author NvE 12-mar-2000 UU-SAP Utrecht
 
Double_t TMath::BesselK1(Double_t x)
{
   // 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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computes the Integer Order Modified Bessel function K_n(x)
/// for n=0,1,2,... and positive real x.
///
/// \author NvE 12-mar-2000 UU-SAP Utrecht
 
Double_t TMath::BesselK(Int_t n,Double_t x)
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computes the Integer Order Modified Bessel function I_n(x)
/// for n=0,1,2,... and any real x.
///
/// \author NvE 12-mar-2000 UU-SAP Utrecht
 
Double_t TMath::BesselI(Int_t n,Double_t x)
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the Bessel function J0(x) for any real x.
 
Double_t TMath::BesselJ0(Double_t 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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the Bessel function J1(x) for any real x.
 
Double_t TMath::BesselJ1(Double_t 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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the Bessel function Y0(x) for positive x.
 
Double_t TMath::BesselY0(Double_t 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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the Bessel function Y1(x) for positive x.
 
Double_t TMath::BesselY1(Double_t 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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Struve Functions of Order 0
///
/// Converted from CERNLIB M342 by Rene Brun.
 
Double_t TMath::StruveH0(Double_t x)
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Struve Functions of Order 1
///
/// Converted from CERNLIB M342 by Rene Brun.
 
Double_t TMath::StruveH1(Double_t x)
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Modified Struve Function of Order 0.
///
/// \author Kirill Filimonov.
 
Double_t TMath::StruveL0(Double_t x)
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Modified Struve Function of Order 1.
///
/// \author Kirill Filimonov.
 
Double_t TMath::StruveL1(Double_t x)
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Calculates Beta-function Gamma(p)*Gamma(q)/Gamma(p+q).
 
Double_t TMath::Beta(Double_t p, Double_t q)
{
   return ::ROOT::Math::beta(p, q);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Continued fraction evaluation by modified Lentz's method
/// used in calculation of incomplete Beta function.
 
Double_t TMath::BetaCf(Double_t x, Double_t a, Double_t b)
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computes the probability density function of the Beta distribution
/// (the cumulative 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.
 
Double_t TMath::BetaDist(Double_t x, Double_t p, Double_t q)
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computes the cumulative distribution function of the Beta distribution,
/// i.e. the lower tail integral of TMath::BetaDist
/// 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.
 
Double_t TMath::BetaDistI(Double_t x, Double_t p, Double_t q)
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Calculates the incomplete Beta-function.
 
Double_t TMath::BetaIncomplete(Double_t x, Double_t a, Double_t b)
{
   return ::ROOT::Math::inc_beta(x, a, b);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Calculates the binomial coefficient n over k.
/// \return the coefficient; or NaN if n < 0 or k < 0 or n < k; or 1 if k = 0 or n = k
 
Double_t TMath::Binomial(Int_t n, Int_t 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);
   auto k2 = static_cast<UInt_t>(n - k1);
   Double_t fact = k2 + 1;
   for (auto i = static_cast<UInt_t>(k1); i > 1; --i)
      fact *= (k2 + i) / static_cast<Double_t>(i);
   return fact;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Suppose an event occurs with probability _p_ per trial
/// Then the probability P of its occurring _k_ or more times
/// in _n_ trials is termed a cumulative binomial probability
/// the formula is:
/// ~~~ {cpp}
/// 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")
///
/// Note this function is not exactly implementing the cumulative or the complement of the cumulative of the
/// Binomial distribution. It is equivalent to ROOT::Math::binomial_cdf_c(k-1,p,n)
///
/// \author Anna Kreshuk
 
Double_t TMath::BinomialI(Double_t p, Int_t n, Int_t k)
{
   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));
}
 
////////////////////////////////////////////////////////////////////////////////
/// 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
///
/// Example:
///
/// ~~~ {cpp}
///    TF1* fc = new TF1("fc", "TMath::CauchyDist(x, [0], [1])", -5, 5);
///    fc->SetParameters(0, 1);
///    fc->Draw();
/// ~~~
///
/// \author Anna Kreshuk
 
Double_t TMath::CauchyDist(Double_t x, Double_t t, Double_t s)
{
   Double_t temp= (x-t)*(x-t)/(s*s);
   Double_t result = 1/(s*TMath::Pi()*(1+temp));
   return result;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Evaluate the quantiles of the chi-squared probability distribution function.
/// Algorithm AS 91   Appl. Statist. (1975) Vol.24, P.35
/// .
/// Incorporates the suggested changes in AS R85 (vol.40(1), pp.233-5, 1991)
///
/// \param[in] p     the probability value, at which the quantile is computed
/// \param[in] ndf   number of degrees of freedom
///
/// \author Anna Kreshuk
 
Double_t TMath::ChisquareQuantile(Double_t p, Double_t ndf)
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// 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 its number of degrees
/// of freedom.
///
/// \author Anna Kreshuk
 
Double_t TMath::FDist(Double_t F, Double_t N, Double_t M)
{
   return ::ROOT::Math::fdistribution_pdf(F,N,M);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Calculates the cumulative distribution function of F-distribution
/// (see ROOT::Math::fdistribution_cdf).
/// 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".
///
/// \author Anna Kreshuk
 
Double_t TMath::FDistI(Double_t F, Double_t N, Double_t M)
{
   Double_t fi = ::ROOT::Math::fdistribution_cdf(F,N,M);
   return fi;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computes the density function of Gamma distribution at point x.
///
/// \param[in] x       evaluation point
/// \param[in] gamma   shape parameter
/// \param[in] mu      location parameter
/// \param[in] 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_Macro
/// {
///   TCanvas *c1 = new TCanvas("c1", "c1", 700, 500);
///
///   c1->SetLogy();
///   c1->SetGridx();
///   c1->SetGridy();
///
///   TF1 *gdist = new TF1("gdist", "TMath::GammaDist(x, [0], [1], [2])", 0, 10);
///
///   gdist->SetParameters(0.5, 0., 1.);
///   gdist->SetLineColor(2);
///   TF1 *gdist1 = gdist->DrawCopy("L");
///   gdist->SetParameters(1.0, 0., 1.);
///   gdist->SetLineColor(3);
///   TF1 *gdist2 = gdist->DrawCopy("LSAME");
///   gdist->SetParameters(2.0, 0., 1.);
///   gdist->SetLineColor(4);
///   TF1 *gdist3 = gdist->DrawCopy("LSAME");
///   gdist->SetParameters(5.0, 0., 1.);
///   gdist->SetLineColor(6);
///   TF1 *gdist4 = gdist->DrawCopy("LSAME");
///
///   auto legend = new TLegend(0.15, 0.15, 0.5, 0.35);
///   legend->AddEntry(gdist1, "gamma = 0.5, mu = 0, beta = 1", "L");
///   legend->AddEntry(gdist2, "gamma = 1.0, mu = 0, beta = 1", "L");
///   legend->AddEntry(gdist3, "gamma = 2.0, mu = 0, beta = 1", "L");
///   legend->AddEntry(gdist4, "gamma = 5.0, mu = 0, beta = 1", "L");
///   legend->Draw();
/// }
/// End_Macro
 
Double_t TMath::GammaDist(Double_t x, Double_t gamma, Double_t mu, Double_t beta)
{
   if ((x<mu) || (gamma<=0) || (beta <=0)) {
      Error("TMath::GammaDist", "illegal parameter values x = %f , gamma = %f beta = %f",x,gamma,beta);
      return 0;
   }
   return ::ROOT::Math::gamma_pdf(x, gamma, beta, mu);
}
 
////////////////////////////////////////////////////////////////////////////////
/// 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 TMath::LaplaceDist(Double_t x, Double_t alpha, Double_t beta)
{
   Double_t temp;
   temp  = TMath::Exp(-TMath::Abs((x-alpha)/beta));
   temp /= (2.*beta);
   return temp;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computes the cumulative distribution function (lower tail integral)
/// 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 TMath::LaplaceDistI(Double_t x, Double_t alpha, Double_t beta)
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computes the density of LogNormal distribution at point x.
/// Variable X has lognormal distribution if Y=Ln(X) has normal distribution
///
/// \param[in] x      is the evaluation point
/// \param[in] sigma  is the shape parameter
/// \param[in] theta  is the location parameter
/// \param[in] 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_Macro
/// {
///   TCanvas *c1 = new TCanvas("c1", "c1", 700, 500);
///
///   c1->SetLogy();
///   c1->SetGridx();
///   c1->SetGridy();
///
///   TF1 *logn = new TF1("logn", "TMath::LogNormal(x, [0], [1], [2])", 0, 5);
///   logn->SetMinimum(1e-3);
///
///   logn->SetParameters(0.5, 0., 1.);
///   logn->SetLineColor(2);
///   TF1 *logn1 = logn->DrawCopy("L");
///   logn->SetParameters(1.0, 0., 1.);
///   logn->SetLineColor(3);
///   TF1 *logn2 = logn->DrawCopy("LSAME");
///   logn->SetParameters(2.0, 0., 1.);
///   logn->SetLineColor(4);
///   TF1 *logn3 = logn->DrawCopy("LSAME");
///   logn->SetParameters(5.0, 0., 1.);
///   logn->SetLineColor(6);
///   TF1 *logn4 = logn->DrawCopy("LSAME");
///
///   auto legend = new TLegend(0.15, 0.15, 0.5, 0.35);
///   legend->AddEntry(logn1, "sigma = 0.5, theta = 0, m = 1", "L");
///   legend->AddEntry(logn2, "sigma = 1.0, theta = 0, m = 1", "L");
///   legend->AddEntry(logn3, "sigma = 2.0, theta = 0, m = 1", "L");
///   legend->AddEntry(logn4, "sigma = 5.0, theta = 0, m = 1", "L");
///   legend->Draw();
/// }
/// End_Macro
 
Double_t TMath::LogNormal(Double_t x, Double_t sigma, Double_t theta, Double_t m)
{
   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);
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computes quantiles for standard normal distribution N(0, 1)
/// at probability p
///
/// ALGORITHM AS241  APPL. STATIST. (1988) VOL. 37, NO. 3, 477-484.
 
Double_t TMath::NormQuantile(Double_t p)
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// 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.
 
Bool_t TMath::Permute(Int_t n, Int_t *a)
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Computes density function for Student's t- distribution
/// \f[ p_{n}(x) = \frac{\Gamma(\frac{n+1}{2})}{\sqrt{n \pi}\Gamma(\frac{n}{2})} \left( 1+\frac{x^2}{n}\right)^{-(n+1)/2}  \f]
/// for \f$n \geq 0\f$, at point x for n (ndf) degrees of freedom. 
/// This is equivalent to ROOT::Math::tdistribution_pdf(x,ndf)
/// The probability function (integral of density) is computed in StudentI.
///
/// 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.
///
/// \author Anna Kreshuk
 
Double_t TMath::Student(Double_t T, Double_t ndf)
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// 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.
/// This is equivalent to ROOT::Math::tdistribution_cdf(T,ndf)
///
/// \author Anna Kreshuk
 
Double_t TMath::StudentI(Double_t T, Double_t ndf)
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// 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 TMath::StudentQuantile(Double_t p, Double_t ndf, Bool_t lower_tail)
{
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the value of the Vavilov probability density function
///
/// \param[in] x      the point were the density function is evaluated
/// \param[in] kappa  value of kappa (distribution parameter)
/// \param[in] beta2  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 results 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"
///
/// For a more accurate implementation see the documentation in the Vavilov class and
/// ROOT::Math::vavilov_accurate_pdf
///
/// Begin_Macro
/// {
///   TCanvas *c1 = new TCanvas("c1", "c1", 700, 500);
///
///   c1->SetGridx();
///   c1->SetGridy();
///
///   TF1 *vavilov = new TF1("vavilov", "TMath::Vavilov(x, [0], [1])", -3, 11);
///
///   vavilov->SetParameters(0.5, 0.);
///   vavilov->SetLineColor(2);
///   TF1 *vavilov1 = vavilov->DrawCopy("L");
///   vavilov->SetParameters(0.3, 0.);
///   vavilov->SetLineColor(3);
///   TF1 *vavilov2 = vavilov->DrawCopy("LSAME");
///   vavilov->SetParameters(0.2, 0.);
///   vavilov->SetLineColor(4);
///   TF1 *vavilov3 = vavilov->DrawCopy("LSAME");
///   vavilov->SetParameters(0.1, 0.);
///   vavilov->SetLineColor(6);
///   TF1 *vavilov4 = vavilov->DrawCopy("LSAME");
///
///   auto legend = new TLegend(0.5, 0.65, 0.85, 0.85);
///   legend->AddEntry(vavilov1, "kappa = 0.5, beta2 = 0", "L");
///   legend->AddEntry(vavilov2, "kappa = 0.3, beta2 = 0", "L");
///   legend->AddEntry(vavilov3, "kappa = 0.2, beta2 = 0", "L");
///   legend->AddEntry(vavilov4, "kappa = 0.1, beta2 = 0", "L");
///   legend->Draw();
/// }
/// End_Macro
 
Double_t TMath::Vavilov(Double_t x, Double_t kappa, Double_t beta2)
{
   Double_t *ac = new Double_t[14];
   Double_t *hc = new Double_t[9];
 
   Int_t itype;
   Int_t npt;
   TMath::VavilovSet(kappa, beta2, false, nullptr, ac, hc, itype, npt);
   Double_t v =  TMath::VavilovDenEval(x, ac, hc, itype);
   delete [] ac;
   delete [] hc;
   return v;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the value of the Vavilov cumulative distribution function
/// (lower tail integral of the probability distribution function)
///
/// \param[in] x      the point were the density function is evaluated
/// \param[in] kappa  value of kappa (distribution parameter)
/// \param[in] beta2  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 results 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"
///
/// For a more accurate implementation see the documentation of the Vavilov class and the cumulative
/// ROOT::Math::vavilov_accurate_cdf
 
Double_t TMath::VavilovI(Double_t x, Double_t kappa, Double_t beta2)
{
   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, true, 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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Returns the cumulative (lower tail integral) of the Landau distribution function at point x.
/// (see ROOT::Math::landau_cdf)
/// 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
 
Double_t TMath::LandauI(Double_t x)
{
   return ::ROOT::Math::landau_cdf(x);
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Internal function, called by Vavilov and VavilovI
 
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)
{
 
   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;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Internal function, called by Vavilov and VavilovSet
 
Double_t TMath::VavilovDenEval(Double_t rlam, Double_t *AC, Double_t *HC, Int_t itype)
{
   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;
}
 
/// \brief Computes a 1D k-nearest neighbor density estimate for a set of query points.
///
/// For each query point, this function estimates the local density based on
/// the distance to the k-th nearest neighbor. A minimum distance threshold
/// `dmin` is used to avoid division by zero and numerical instability when a
/// query point coincides with an observation or neighbors are extremely close.
///
/// \param observations Data points used for density estimation.
/// \param queries Points at which to estimate the density.
/// \param result Output for the density estimates. Must have the same size as `queries`.
/// \param k Number of nearest neighbors to consider. If k >= number of observations,
///          it is automatically reduced to n-1.
/// \param dmin Minimum allowed neighbor distance to prevent division by zero. Default is zero.
///
/// The density estimate for each query point x is:
///
/// \f[
/// \hat{p}(x) = \frac{n}{2(n+1)} \frac{k_{\text{actual}}}{d_\‍(k_{\text{actual}}\‍)(x)},
/// \f]
///
/// where \‍(d_\‍(k_{\text{actual}}\‍)(x)\‍) is the distance and \‍(k_{\text{actual}}\‍) may be different from \‍(k\‍)
/// when it needs to be greater than \‍(k\‍) to keep the \‍(d > d_{\text{min}} constraint.
/// If no neighbor is outside `dmin`, the minimum allowed distance will be used as the distance itself.
///
/// \note This function implements the math of the former `TH1K` class. There are some differences:
///
///   1. The `TH1K` class additionally multiplied all density estimates with the bin width.
///   2. If the `k` parameter was not specified by the user, the `TH1K` class used the bin width as `dmin`.
///   3. If the `k` parameter was explicitly specified, the `dmin` parameter was set to 1.e-6.
void TMath::KNNDensity(std::span<const double> observations, std::span<const double> queries, std::span<double> result,
                       int k, double dmin)
{
   int n = observations.size();
   if (k < 0) {
      Error("KNNDensity", "k must be positive");
      return;
   }
   if (k >= static_cast<int>(n))
      k = static_cast<int>(n - 1);
 
   if (result.size() != queries.size()) {
      Error("KNNDensity", "Result span size must match query span size");
      return;
   }
 
   // Make a sorted copy of the observations for binary search
   std::vector<double> sorted_obs(observations.begin(), observations.end());
   std::sort(sorted_obs.begin(), sorted_obs.end());
 
   // constant factor in the output
   const double factor = n * 0.5 / (n + 1.0);
 
   for (std::size_t i = 0; i < queries.size(); ++i) {
      double x = queries[i];
 
      // Find index in sorted_obs
      int jr = std::distance(sorted_obs.begin(), std::lower_bound(sorted_obs.begin(), sorted_obs.end(), x));
      int jl = jr - 1;
 
      double ff = 0.0;
      int k_actual = 0;
 
      for (; k_actual + 1 <= k || ff <= dmin; ++k_actual) {
 
         double fl = (jl >= 0) ? std::abs(sorted_obs[jl] - x) : std::numeric_limits<double>::max();
         double fr = (jr < n) ? std::abs(sorted_obs[jr] - x) : std::numeric_limits<double>::max();
 
         if (jl < 0 && jr >= n)
            break;
 
         ff = std::min(fl, fr);
         fl < fr ? --jl : ++jr;
      }
 
      if (ff == 0.0)
         ff = dmin; // avoid division by zero
 
      // Modified kNN density formula
      result[i] = factor == 0.0 ? 0.0 : factor * k_actual / ff;
   }
}