TMath.h

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// @(#)root/mathcore:$Id: TMath.h 36792 2010-11-19 16:53:42Z rdm $
// 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.             *
 *************************************************************************/

#ifndef ROOT_TMath
#define ROOT_TMath


//////////////////////////////////////////////////////////////////////////
//                                                                      //
// TMath                                                                //
//                                                                      //
// Encapsulate most frequently used Math functions.                     //
// NB. The basic functions Min, Max, Abs and Sign are defined           //
// in TMathBase.                                                        //
//                                                                      //
//////////////////////////////////////////////////////////////////////////

#ifndef ROOT_Rtypes
#include "Rtypes.h"
#endif
#ifndef ROOT_TMathBase
#include "TMathBase.h"
#endif

#include "TError.h"
#include <algorithm>

namespace TMath {

   /* ************************* */
   /* * Fundamental constants * */
   /* ************************* */

   inline Double_t Pi()       { return 3.14159265358979323846; }
   inline Double_t TwoPi()    { return 2.0 * Pi(); }
   inline Double_t PiOver2()  { return Pi() / 2.0; }
   inline Double_t PiOver4()  { return Pi() / 4.0; }
   inline Double_t InvPi()    { return 1.0 / Pi(); }
   inline Double_t RadToDeg() { return 180.0 / Pi(); }
   inline Double_t DegToRad() { return Pi() / 180.0; }
   inline Double_t Sqrt2()    { return 1.4142135623730950488016887242097; }

   // e (base of natural log)
   inline Double_t E()        { return 2.71828182845904523536; }

   // natural log of 10 (to convert log to ln)
   inline Double_t Ln10()     { return 2.30258509299404568402; }

   // base-10 log of e  (to convert ln to log)
   inline Double_t LogE()     { return 0.43429448190325182765; }

   // velocity of light
   inline Double_t C()        { return 2.99792458e8; }        // m s^-1
   inline Double_t Ccgs()     { return 100.0 * C(); }         // cm s^-1
   inline Double_t CUncertainty() { return 0.0; }             // exact

   // gravitational constant
   inline Double_t G()        { return 6.673e-11; }           // m^3 kg^-1 s^-2
   inline Double_t Gcgs()     { return G() / 1000.0; }        // cm^3 g^-1 s^-2
   inline Double_t GUncertainty() { return 0.010e-11; }

   // G over h-bar C
   inline Double_t GhbarC()   { return 6.707e-39; }           // (GeV/c^2)^-2
   inline Double_t GhbarCUncertainty() { return 0.010e-39; }

   // standard acceleration of gravity
   inline Double_t Gn()       { return 9.80665; }             // m s^-2
   inline Double_t GnUncertainty() { return 0.0; }            // exact

   // Planck's constant
   inline Double_t H()        { return 6.62606876e-34; }      // J s
   inline Double_t Hcgs()     { return 1.0e7 * H(); }         // erg s
   inline Double_t HUncertainty() { return 0.00000052e-34; }

   // h-bar (h over 2 pi)
   inline Double_t Hbar()     { return 1.054571596e-34; }     // J s
   inline Double_t Hbarcgs()  { return 1.0e7 * Hbar(); }      // erg s
   inline Double_t HbarUncertainty() { return 0.000000082e-34; }

   // hc (h * c)
   inline Double_t HC()       { return H() * C(); }           // J m
   inline Double_t HCcgs()    { return Hcgs() * Ccgs(); }     // erg cm

   // Boltzmann's constant
   inline Double_t K()        { return 1.3806503e-23; }       // J K^-1
   inline Double_t Kcgs()     { return 1.0e7 * K(); }         // erg K^-1
   inline Double_t KUncertainty() { return 0.0000024e-23; }

   // Stefan-Boltzmann constant
   inline Double_t Sigma()    { return 5.6704e-8; }           // W m^-2 K^-4
   inline Double_t SigmaUncertainty() { return 0.000040e-8; }

   // Avogadro constant (Avogadro's Number)
   inline Double_t Na()       { return 6.02214199e+23; }      // mol^-1
   inline Double_t NaUncertainty() { return 0.00000047e+23; }

   // universal gas constant (Na * K)
   // http://scienceworld.wolfram.com/physics/UniversalGasConstant.html
   inline Double_t R()        { return K() * Na(); }          // J K^-1 mol^-1
   inline Double_t RUncertainty() { return R()*((KUncertainty()/K()) + (NaUncertainty()/Na())); }

   // Molecular weight of dry air
   // 1976 US Standard Atmosphere,
   // also see http://atmos.nmsu.edu/jsdap/encyclopediawork.html
   inline Double_t MWair()    { return 28.9644; }             // kg kmol^-1 (or gm mol^-1)

   // Dry Air Gas Constant (R / MWair)
   // http://atmos.nmsu.edu/education_and_outreach/encyclopedia/gas_constant.htm
   inline Double_t Rgair()    { return (1000.0 * R()) / MWair(); }  // J kg^-1 K^-1

   // Euler-Mascheroni Constant
   inline Double_t EulerGamma() { return 0.577215664901532860606512090082402431042; }

   // Elementary charge
   inline Double_t Qe()       { return 1.602176462e-19; }     // C
   inline Double_t QeUncertainty() { return 0.000000063e-19; }

   /* ************************** */
   /* * Mathematical Functions * */
   /* ************************** */

   /* ***************************** */
   /* * Trigonometrical Functions * */
   /* ***************************** */
   inline Double_t Sin(Double_t);
   inline Double_t Cos(Double_t);
   inline Double_t Tan(Double_t);
   inline Double_t SinH(Double_t);
   inline Double_t CosH(Double_t);
   inline Double_t TanH(Double_t);
   inline Double_t ASin(Double_t);
   inline Double_t ACos(Double_t);
   inline Double_t ATan(Double_t);
   inline Double_t ATan2(Double_t, Double_t);
          Double_t ASinH(Double_t);
          Double_t ACosH(Double_t);
          Double_t ATanH(Double_t);
          Double_t Hypot(Double_t x, Double_t y);


   /* ************************ */
   /* * Elementary Functions * */
   /* ************************ */
   inline Double_t Sqrt(Double_t x);
   inline Double_t Ceil(Double_t x);
   inline Int_t    CeilNint(Double_t x);
   inline Double_t Floor(Double_t x);
   inline Int_t    FloorNint(Double_t x);
   inline Double_t Exp(Double_t x);
   inline Double_t Ldexp(Double_t x, Int_t exp);
          Double_t Factorial(Int_t i);
   inline Double_t Power(Double_t x, Double_t y);
   inline Double_t Log(Double_t x);
          Double_t Log2(Double_t x);
   inline Double_t Log10(Double_t x);
          Int_t    Nint(Float_t x);
          Int_t    Nint(Double_t x);
   inline Int_t    Finite(Double_t x);
   inline Int_t    IsNaN(Double_t x);

   // Some integer math
   Long_t   Hypot(Long_t x, Long_t y);     // sqrt(px*px + py*py)

   // Comparing floating points
   inline Bool_t AreEqualAbs(Double_t af, Double_t bf, Double_t epsilon) {
      //return kTRUE if absolute difference between af and bf is less than epsilon
      return TMath::Abs(af-bf) < epsilon;
   }
   inline Bool_t AreEqualRel(Double_t af, Double_t bf, Double_t relPrec) {
      //return kTRUE if relative difference between af and bf is less than relPrec
      return TMath::Abs(af-bf) <= 0.5*relPrec*(TMath::Abs(af)+TMath::Abs(bf));
   }

   /* ******************** */
   /* * Array Algorithms * */
   /* ******************** */

   // Min, Max of an array
   template <typename T> T MinElement(Long64_t n, const T *a);
   template <typename T> T MaxElement(Long64_t n, const T *a);

   // Locate Min, Max element number in an array
   template <typename T> Long64_t  LocMin(Long64_t n, const T *a);
   template <typename Iterator> Iterator LocMin(Iterator first, Iterator last);
   template <typename T> Long64_t  LocMax(Long64_t n, const T *a);
   template <typename Iterator> Iterator LocMax(Iterator first, Iterator last);

   // Binary search
   template <typename T> Long64_t BinarySearch(Long64_t n, const T  *array, T value);
   template <typename T> Long64_t BinarySearch(Long64_t n, const T **array, T value);
   template <typename Iterator, typename Element> Iterator BinarySearch(Iterator first, Iterator last, Element value);

   // Hashing
   ULong_t Hash(const void *txt, Int_t ntxt);
   ULong_t Hash(const char *str);

   // Sorting
   template <typename Element, typename Index>
   void Sort(Index n, const Element* a, Index* index, Bool_t down=kTRUE);
   template <typename Iterator, typename IndexIterator>
   void SortItr(Iterator first, Iterator last, IndexIterator index, Bool_t down=kTRUE);

   void BubbleHigh(Int_t Narr, Double_t *arr1, Int_t *arr2);
   void BubbleLow (Int_t Narr, Double_t *arr1, Int_t *arr2);

   Bool_t   Permute(Int_t n, Int_t *a); // Find permutations

   /* ************************* */
   /* * Geometrical Functions * */
   /* ************************* */

   //Sample quantiles
   void      Quantiles(Int_t n, Int_t nprob, Double_t *x, Double_t *quantiles, Double_t *prob,
                       Bool_t isSorted=kTRUE, Int_t *index = 0, Int_t type=7);

   // IsInside
   template <typename T> Bool_t IsInside(T xp, T yp, Int_t np, T *x, T *y);

   // Calculate the Cross Product of two vectors
   template <typename T> T *Cross(const T v1[3],const T v2[3], T out[3]);

   Float_t   Normalize(Float_t v[3]);  // Normalize a vector
   Double_t  Normalize(Double_t v[3]); // Normalize a vector

   //Calculate the Normalized Cross Product of two vectors
   template <typename T> inline T NormCross(const T v1[3],const T v2[3],T out[3]);

   // Calculate a normal vector of a plane
   template <typename T> T *Normal2Plane(const T v1[3],const T v2[3],const T v3[3], T normal[3]);

   /* ************************ */
   /* * Polynomial Functions * */
   /* ************************ */

   Bool_t    RootsCubic(const Double_t coef[4],Double_t &a, Double_t &b, Double_t &c);

   /* *********************** */
   /* * Statistic Functions * */
   /* *********************** */

   Double_t Binomial(Int_t n,Int_t k);  // Calculate the binomial coefficient n over k
   Double_t BinomialI(Double_t p, Int_t n, Int_t k);
   Double_t BreitWigner(Double_t x, Double_t mean=0, Double_t gamma=1);
   Double_t CauchyDist(Double_t x, Double_t t=0, Double_t s=1);
   Double_t ChisquareQuantile(Double_t p, Double_t ndf);
   Double_t FDist(Double_t F, Double_t N, Double_t M);
   Double_t FDistI(Double_t F, Double_t N, Double_t M);
   Double_t Gaus(Double_t x, Double_t mean=0, Double_t sigma=1, Bool_t norm=kFALSE);
   Double_t KolmogorovProb(Double_t z);
   Double_t KolmogorovTest(Int_t na, const Double_t *a, Int_t nb, const Double_t *b, Option_t *option);
   Double_t Landau(Double_t x, Double_t mpv=0, Double_t sigma=1, Bool_t norm=kFALSE);
   Double_t LandauI(Double_t x);
   Double_t LaplaceDist(Double_t x, Double_t alpha=0, Double_t beta=1);
   Double_t LaplaceDistI(Double_t x, Double_t alpha=0, Double_t beta=1);
   Double_t LogNormal(Double_t x, Double_t sigma, Double_t theta=0, Double_t m=1);
   Double_t NormQuantile(Double_t p);
   Double_t Poisson(Double_t x, Double_t par);
   Double_t PoissonI(Double_t x, Double_t par);
   Double_t Prob(Double_t chi2,Int_t ndf);
   Double_t Student(Double_t T, Double_t ndf);
   Double_t StudentI(Double_t T, Double_t ndf);
   Double_t StudentQuantile(Double_t p, Double_t ndf, Bool_t lower_tail=kTRUE);
   Double_t Vavilov(Double_t x, Double_t kappa, Double_t beta2);
   Double_t VavilovI(Double_t x, Double_t kappa, Double_t beta2);
   Double_t Voigt(Double_t x, Double_t sigma, Double_t lg, Int_t R = 4);

   /* ************************** */
   /* * Statistics over arrays * */
   /* ************************** */

   //Mean, Geometric Mean, Median, RMS(sigma)

   template <typename T> Double_t Mean(Long64_t n, const T *a, const Double_t *w=0);
   template <typename Iterator> Double_t Mean(Iterator first, Iterator last);
   template <typename Iterator, typename WeightIterator> Double_t Mean(Iterator first, Iterator last, WeightIterator w);

   template <typename T> Double_t GeomMean(Long64_t n, const T *a);
   template <typename Iterator> Double_t GeomMean(Iterator first, Iterator last);

   template <typename T> Double_t RMS(Long64_t n, const T *a);
   template <typename Iterator> Double_t RMS(Iterator first, Iterator last);

   template <typename T> Double_t Median(Long64_t n, const T *a,  const Double_t *w=0, Long64_t *work=0);

   //k-th order statistic
   template <class Element, typename Size> Element KOrdStat(Size n, const Element *a, Size k, Size *work = 0);

   /* ******************* */
   /* * Special Functions */
   /* ******************* */

   Double_t Beta(Double_t p, Double_t q);
   Double_t BetaCf(Double_t x, Double_t a, Double_t b);
   Double_t BetaDist(Double_t x, Double_t p, Double_t q);
   Double_t BetaDistI(Double_t x, Double_t p, Double_t q);
   Double_t BetaIncomplete(Double_t x, Double_t a, Double_t b);

   // Bessel functions
   Double_t BesselI(Int_t n,Double_t x);  // integer order modified Bessel function I_n(x)
   Double_t BesselK(Int_t n,Double_t x);  // integer order modified Bessel function K_n(x)
   Double_t BesselI0(Double_t x);         // modified Bessel function I_0(x)
   Double_t BesselK0(Double_t x);         // modified Bessel function K_0(x)
   Double_t BesselI1(Double_t x);         // modified Bessel function I_1(x)
   Double_t BesselK1(Double_t x);         // modified Bessel function K_1(x)
   Double_t BesselJ0(Double_t x);         // Bessel function J0(x) for any real x
   Double_t BesselJ1(Double_t x);         // Bessel function J1(x) for any real x
   Double_t BesselY0(Double_t x);         // Bessel function Y0(x) for positive x
   Double_t BesselY1(Double_t x);         // Bessel function Y1(x) for positive x
   Double_t StruveH0(Double_t x);         // Struve functions of order 0
   Double_t StruveH1(Double_t x);         // Struve functions of order 1
   Double_t StruveL0(Double_t x);         // Modified Struve functions of order 0
   Double_t StruveL1(Double_t x);         // Modified Struve functions of order 1

   Double_t DiLog(Double_t x);
   Double_t Erf(Double_t x);
   Double_t ErfInverse(Double_t x);
   Double_t Erfc(Double_t x);
   Double_t ErfcInverse(Double_t x);
   Double_t Freq(Double_t x);
   Double_t Gamma(Double_t z);
   Double_t Gamma(Double_t a,Double_t x);
   Double_t GammaDist(Double_t x, Double_t gamma, Double_t mu=0, Double_t beta=1);
   Double_t LnGamma(Double_t z);
}


//---- Trig and other functions ------------------------------------------------

#include <float.h>

#if defined(R__WIN32) && !defined(__CINT__)
#   ifndef finite
#      define finite _finite
#      define isnan  _isnan
#   endif
#endif
#if defined(R__AIX) || defined(R__SOLARIS_CC50) || \
    defined(R__HPUX11) || defined(R__GLIBC) || \
    (defined(R__MACOSX) && (defined(__INTEL_COMPILER) || defined(__arm__)))
// math functions are defined inline so we have to include them here
#   include <math.h>
#   ifdef R__SOLARIS_CC50
       extern "C" { int finite(double); }
#   endif
#   if defined(R__GLIBC) && defined(__STRICT_ANSI__)
#      ifndef finite
#         define finite __finite
#      endif
#      ifndef isnan
#         define isnan  __isnan
#      endif
#   endif
#else
// don't want to include complete <math.h>
extern "C" {
   extern double sin(double);
   extern double cos(double);
   extern double tan(double);
   extern double sinh(double);
   extern double cosh(double);
   extern double tanh(double);
   extern double asin(double);
   extern double acos(double);
   extern double atan(double);
   extern double atan2(double, double);
   extern double sqrt(double);
   extern double exp(double);
   extern double pow(double, double);
   extern double log(double);
   extern double log10(double);
#ifndef R__WIN32
#   if !defined(finite)
       extern int finite(double);
#   endif
#   if !defined(isnan)
       extern int isnan(double);
#   endif
   extern double ldexp(double, int);
   extern double ceil(double);
   extern double floor(double);
#else
   _CRTIMP double ldexp(double, int);
   _CRTIMP double ceil(double);
   _CRTIMP double floor(double);
#endif
}
#endif

inline Double_t TMath::Sin(Double_t x)
   { return sin(x); }

inline Double_t TMath::Cos(Double_t x)
   { return cos(x); }

inline Double_t TMath::Tan(Double_t x)
   { return tan(x); }

inline Double_t TMath::SinH(Double_t x)
   { return sinh(x); }

inline Double_t TMath::CosH(Double_t x)
   { return cosh(x); }

inline Double_t TMath::TanH(Double_t x)
   { return tanh(x); }

inline Double_t TMath::ASin(Double_t x)
   { if (x < -1.) return -TMath::Pi()/2;
     if (x >  1.) return  TMath::Pi()/2;
     return asin(x);
   }

inline Double_t TMath::ACos(Double_t x)
   { if (x < -1.) return TMath::Pi();
     if (x >  1.) return 0;
     return acos(x);
   }

inline Double_t TMath::ATan(Double_t x)
   { return atan(x); }

inline Double_t TMath::ATan2(Double_t y, Double_t x)
   { if (x != 0) return  atan2(y, x);
     if (y == 0) return  0;
     if (y >  0) return  Pi()/2;
     else        return -Pi()/2;
   }

inline Double_t TMath::Sqrt(Double_t x)
   { return sqrt(x); }

inline Double_t TMath::Ceil(Double_t x)
   { return ceil(x); }

inline Int_t TMath::CeilNint(Double_t x)
   { return TMath::Nint(ceil(x)); }

inline Double_t TMath::Floor(Double_t x)
   { return floor(x); }

inline Int_t TMath::FloorNint(Double_t x)
   { return TMath::Nint(floor(x)); }

inline Double_t TMath::Exp(Double_t x)
   { return exp(x); }

inline Double_t TMath::Ldexp(Double_t x, Int_t exp)
   { return ldexp(x, exp); }

inline Double_t TMath::Power(Double_t x, Double_t y)
   { return pow(x, y); }

inline Double_t TMath::Log(Double_t x)
   { return log(x); }

inline Double_t TMath::Log10(Double_t x)
   { return log10(x); }

inline Int_t TMath::Finite(Double_t x)
#if defined(R__HPUX11)
   { return isfinite(x); }
#elif defined(R__MACOSX) && defined(__arm__)
#ifdef isfinite
   // from math.h
   { return isfinite(x); }
#else
   // from cmath
   { return std::isfinite(x); }
#endif
#else
   { return finite(x); }
#endif

inline Int_t TMath::IsNaN(Double_t x)
#if defined(R__MACOSX) && defined(__arm__)
#ifdef isnan
   // from math.h
   { return isnan(x); }
#else
   // from cmath
   { return std::isnan(x); }
#endif
#else
   { return isnan(x); }
#endif

//-------- Advanced -------------

template <typename T> inline T TMath::NormCross(const T v1[3],const T v2[3],T out[3])
{
   // Calculate the Normalized Cross Product of two vectors
   return Normalize(Cross(v1,v2,out));
}

template <typename T>
T TMath::MinElement(Long64_t n, const T *a) {
   // Return minimum of array a of length n.

   return *std::min_element(a,a+n);
}

template <typename T>
T TMath::MaxElement(Long64_t n, const T *a) {
   // Return maximum of array a of length n.

   return *std::max_element(a,a+n);
}

template <typename T>
Long64_t TMath::LocMin(Long64_t n, const T *a) {
   // Return index of array with the minimum element.
   // If more than one element is minimum returns first found.

   // Implement here since this one is found to be faster (mainly on 64 bit machines)
   // than stl generic implementation.
   // When performing the comparison,  the STL implementation needs to de-reference both the array iterator
   // and the iterator pointing to the resulting minimum location

   if  (n <= 0 || !a) return -1;
   T xmin = a[0];
   Long64_t loc = 0;
   for  (Long64_t i = 1; i < n; i++) {
      if (xmin > a[i])  {
         xmin = a[i];
         loc = i;
      }
   }
   return loc;
}

template <typename Iterator>
Iterator TMath::LocMin(Iterator first, Iterator last) {
   // Return index of array with the minimum element.
   // If more than one element is minimum returns first found.
   return std::min_element(first, last);
}

template <typename T>
Long64_t TMath::LocMax(Long64_t n, const T *a) {
   // Return index of array with the maximum element.
   // If more than one element is maximum returns first found.

   // Implement here since it is faster (see comment in LocMin function)

   if  (n <= 0 || !a) return -1;
   T xmax = a[0];
   Long64_t loc = 0;
   for  (Long64_t i = 1; i < n; i++) {
      if (xmax < a[i])  {
         xmax = a[i];
         loc = i;
      }
   }
   return loc;
}

template <typename Iterator>
Iterator TMath::LocMax(Iterator first, Iterator last)
{
   // Return index of array with the maximum element.
   // If more than one element is maximum returns first found.

   return std::max_element(first, last);
}

template<typename T>
struct CompareDesc {

   CompareDesc(T d) : fData(d) {}

   template<typename Index>
   bool operator()(Index i1, Index i2) {
      return *(fData + i1) > *(fData + i2);
   }

   T fData;
};

template<typename T>
struct CompareAsc {

   CompareAsc(T d) : fData(d) {}

   template<typename Index>
   bool operator()(Index i1, Index i2) {
      return *(fData + i1) < *(fData + i2);
   }

   T fData;
};

template <typename Iterator>
Double_t TMath::Mean(Iterator first, Iterator last)
{
   // Return the weighted mean of an array defined by the iterators.

   Double_t sum = 0;
   Double_t sumw = 0;
   while ( first != last )
   {
      sum += *first;
      sumw += 1;
      first++;
   }

   return sum/sumw;
}

template <typename Iterator, typename WeightIterator>
Double_t TMath::Mean(Iterator first, Iterator last, WeightIterator w)
{
   // Return the weighted mean of an array defined by the first and
   // last iterators. The w iterator should point to the first element
   // of a vector of weights of the same size as the main array.

   Double_t sum = 0;
   Double_t sumw = 0;
   int i = 0;
   while ( first != last ) {
      if ( *w < 0) {
         ::Error("TMath::Mean","w[%d] = %.4e < 0 ?!",i,*w);
         return 0;
      }
      sum  += (*w) * (*first);
      sumw += (*w) ;
      ++w;
      ++first;
      ++i;
   }
   if (sumw <= 0) {
      ::Error("TMath::Mean","sum of weights == 0 ?!");
      return 0;
   }

   return sum/sumw;
}

template <typename T>
Double_t TMath::Mean(Long64_t n, const T *a, const Double_t *w)
{
   // Return the weighted mean of an array a with length n.

   if (w) {
      return TMath::Mean(a, a+n, w);
   } else {
      return TMath::Mean(a, a+n);
   }
}

template <typename Iterator>
Double_t TMath::GeomMean(Iterator first, Iterator last)
{
   // Return the geometric mean of an array defined by the iterators.
   // geometric_mean = (Prod_i=0,n-1 |a[i]|)^1/n

   Double_t logsum = 0.;
   Long64_t n = 0;
   while ( first != last ) {
      if (*first == 0) return 0.;
      Double_t absa = (Double_t) TMath::Abs(*first);
      logsum += TMath::Log(absa);
      ++first;
      ++n;
   }

   return TMath::Exp(logsum/n);
}

template <typename T>
Double_t TMath::GeomMean(Long64_t n, const T *a)
{
   // Return the geometric mean of an array a of size n.
   // geometric_mean = (Prod_i=0,n-1 |a[i]|)^1/n

   return TMath::GeomMean(a, a+n);
}

template <typename Iterator>
Double_t TMath::RMS(Iterator first, Iterator last)
{
   // Return the Standard Deviation of an array defined by the iterators.
   // Note that this function returns the sigma(standard deviation) and
   // not the root mean square of the array.

   Double_t n = 0;

   Double_t tot = 0, tot2 =0, adouble;
   while ( first != last ) {
      adouble=Double_t(*first);
      tot += adouble; tot2 += adouble*adouble;
      ++first;
      ++n;
   }
   Double_t n1 = 1./n;
   Double_t mean = tot*n1;
   Double_t rms = TMath::Sqrt(TMath::Abs(tot2*n1 -mean*mean));
   return rms;
}

template <typename T>
Double_t TMath::RMS(Long64_t n, const T *a)
{
   // Return the Standard Deviation of an array a with length n.
   // Note that this function returns the sigma(standard deviation) and
   // not the root mean square of the array.

   return TMath::RMS(a, a+n);
}

template <typename Iterator, typename Element>
Iterator TMath::BinarySearch(Iterator first, Iterator last, Element value)
{
   // Binary search in an array defined by its iterators.
   //
   // The values in the iterators range are supposed to be sorted
   // prior to this call.  If match is found, function returns
   // position of element.  If no match found, function gives nearest
   // element smaller than value.

   Iterator pind;
   pind = std::lower_bound(first, last, value);
   if ( (pind != last) && (*pind == value) )
      return pind;
   else
      return ( pind - 1);
}


template <typename T> Long64_t TMath::BinarySearch(Long64_t n, const T  *array, T value)
{
   // Binary search in an array of n values to locate value.
   //
   // Array is supposed  to be sorted prior to this call.
   // If match is found, function returns position of element.
   // If no match found, function gives nearest element smaller than value.

   const T* pind;
   pind = std::lower_bound(array, array + n, value);
   if ( (pind != array + n) && (*pind == value) )
      return (pind - array);
   else
      return ( pind - array - 1);
}

template <typename T> Long64_t TMath::BinarySearch(Long64_t n, const T **array, T value)
{
   // Binary search in an array of n values to locate value.
   //
   // Array is supposed  to be sorted prior to this call.
   // If match is found, function returns position of element.
   // If no match found, function gives nearest element smaller than value.

   const T* pind;
   pind = std::lower_bound(*array, *array + n, value);
   if ( (pind != *array + n) && (*pind == value) )
      return (pind - *array);
   else
      return ( pind - *array - 1);
}

template <typename Iterator, typename IndexIterator>
void TMath::SortItr(Iterator first, Iterator last, IndexIterator index, Bool_t down)
{
   // Sort the n1 elements of the Short_t array defined by its
   // iterators.  In output the array index contains the indices of
   // the sorted array.  If down is false sort in increasing order
   // (default is decreasing order).

   // NOTE that the array index must be created with a length bigger
   // or equal than the main array before calling this function.

   int i = 0;

   IndexIterator cindex = index;
   for ( Iterator cfirst = first; cfirst != last; ++cfirst )
   {
      *cindex = i++;
      ++cindex;
   }

   if ( down )
      std::sort(index, cindex, CompareDesc<Iterator>(first) );
   else
      std::sort(index, cindex, CompareAsc<Iterator>(first) );
}

template <typename Element, typename Index> void TMath::Sort(Index n, const Element* a, Index* index, Bool_t down)
{
   // Sort the n elements of the  array a of generic templated type Element.
   // In output the array index of type Index contains the indices of the sorted array.
   // If down is false sort in increasing order (default is decreasing order).

   // NOTE that the array index must be created with a length >= n
   // before calling this function.
   // NOTE also that the size type for n must be the same type used for the index array
   // (templated type Index)

   for(Index i = 0; i < n; i++) { index[i] = i; }
   if ( down )
      std::sort(index, index + n, CompareDesc<const Element*>(a) );
   else
      std::sort(index, index + n, CompareAsc<const Element*>(a) );
}

template <typename T> T *TMath::Cross(const T v1[3],const T v2[3], T out[3])
{
   // Calculate the Cross Product of two vectors:
   //         out = [v1 x v2]

   out[0] = v1[1] * v2[2] - v1[2] * v2[1];
   out[1] = v1[2] * v2[0] - v1[0] * v2[2];
   out[2] = v1[0] * v2[1] - v1[1] * v2[0];

   return out;
}

template <typename T> T * TMath::Normal2Plane(const T p1[3],const T p2[3],const T p3[3], T normal[3])
{
   // Calculate a normal vector of a plane.
   //
   //  Input:
   //     Float_t *p1,*p2,*p3  -  3 3D points belonged the plane to define it.
   //
   //  Return:
   //     Pointer to 3D normal vector (normalized)

   T v1[3], v2[3];

   v1[0] = p2[0] - p1[0];
   v1[1] = p2[1] - p1[1];
   v1[2] = p2[2] - p1[2];

   v2[0] = p3[0] - p1[0];
   v2[1] = p3[1] - p1[1];
   v2[2] = p3[2] - p1[2];

   NormCross(v1,v2,normal);
   return normal;
}

template <typename T> Bool_t TMath::IsInside(T xp, T yp, Int_t np, T *x, T *y)
{
   // Function which returns kTRUE if point xp,yp lies inside the
   // polygon defined by the np points in arrays x and y, kFALSE otherwise.
   // Note that the polygon may be open or closed.

   Int_t i, j = np-1 ;
   Bool_t oddNodes = kFALSE;

   for (i=0; i<np; i++) {
      if ((y[i]<yp && y[j]>=yp) || (y[j]<yp && y[i]>=yp)) {
         if (x[i]+(yp-y[i])/(y[j]-y[i])*(x[j]-x[i])<xp) {
            oddNodes = !oddNodes;
         }
      }
      j=i;
   }

   return oddNodes;
}

template <typename T> Double_t TMath::Median(Long64_t n, const T *a,  const Double_t *w, Long64_t *work)
{
   // Return the median of the array a where each entry i has weight w[i] .
   // Both arrays have a length of at least n . The median is a number obtained
   // from the sorted array a through
   //
   // median = (a[jl]+a[jh])/2.  where (using also the sorted index on the array w)
   //
   // sum_i=0,jl w[i] <= sumTot/2
   // sum_i=0,jh w[i] >= sumTot/2
   // sumTot = sum_i=0,n w[i]
   //
   // If w=0, the algorithm defaults to the median definition where it is
   // a number that divides the sorted sequence into 2 halves.
   // When n is odd or n > 1000, the median is kth element k = (n + 1) / 2.
   // when n is even and n < 1000the median is a mean of the elements k = n/2 and k = n/2 + 1.
   //
   // If the weights are supplied (w not 0) all weights must be >= 0
   //
   // If work is supplied, it is used to store the sorting index and assumed to be
   // >= n . If work=0, local storage is used, either on the stack if n < kWorkMax
   // or on the heap for n >= kWorkMax .

   const Int_t kWorkMax = 100;

   if (n <= 0 || !a) return 0;
   Bool_t isAllocated = kFALSE;
   Double_t median;
   Long64_t *ind;
   Long64_t workLocal[kWorkMax];

   if (work) {
      ind = work;
   } else {
      ind = workLocal;
      if (n > kWorkMax) {
         isAllocated = kTRUE;
         ind = new Long64_t[n];
      }
   }

   if (w) {
      Double_t sumTot2 = 0;
      for (Int_t j = 0; j < n; j++) {
         if (w[j] < 0) {
            ::Error("TMath::Median","w[%d] = %.4e < 0 ?!",j,w[j]);
            if (isAllocated)  delete [] ind;
            return 0;
         }
         sumTot2 += w[j];
      }

      sumTot2 /= 2.;

      Sort(n, a, ind, kFALSE);

      Double_t sum = 0.;
      Int_t jl;
      for (jl = 0; jl < n; jl++) {
         sum += w[ind[jl]];
         if (sum >= sumTot2) break;
      }

      Int_t jh;
      sum = 2.*sumTot2;
      for (jh = n-1; jh >= 0; jh--) {
         sum -= w[ind[jh]];
         if (sum <= sumTot2) break;
      }

      median = 0.5*(a[ind[jl]]+a[ind[jh]]);

   } else {

      if (n%2 == 1)
         median = KOrdStat(n, a,n/2, ind);
      else {
         median = 0.5*(KOrdStat(n, a, n/2 -1, ind)+KOrdStat(n, a, n/2, ind));
      }
   }

   if (isAllocated)
      delete [] ind;
   return median;
}




template <class Element, typename Size>
Element TMath::KOrdStat(Size n, const Element *a, Size k, Size *work)
{
   // Returns k_th order statistic of the array a of size n
   // (k_th smallest element out of n elements).
   //
   // C-convention is used for array indexing, so if you want
   // the second smallest element, call KOrdStat(n, a, 1).
   //
   // If work is supplied, it is used to store the sorting index and
   // assumed to be >= n. If work=0, local storage is used, either on
   // the stack if n < kWorkMax or on the heap for n >= kWorkMax.
   //
   // Taken from "Numerical Recipes in C++" without the index array
   // implemented by Anna Khreshuk.
   //
   // See also the declarations at the top of this file

   const Int_t kWorkMax = 100;

   typedef Size Index;

   Bool_t isAllocated = kFALSE;
   Size i, ir, j, l, mid;
   Index arr;
   Index *ind;
   Index workLocal[kWorkMax];
   Index temp;

   if (work) {
      ind = work;
   } else {
      ind = workLocal;
      if (n > kWorkMax) {
         isAllocated = kTRUE;
         ind = new Index[n];
      }
   }

   for (Size ii=0; ii<n; ii++) {
      ind[ii]=ii;
   }
   Size rk = k;
   l=0;
   ir = n-1;
   for(;;) {
      if (ir<=l+1) { //active partition contains 1 or 2 elements
         if (ir == l+1 && a[ind[ir]]<a[ind[l]])
            {temp = ind[l]; ind[l]=ind[ir]; ind[ir]=temp;}
         Element tmp = a[ind[rk]];
         if (isAllocated)
            delete [] ind;
         return tmp;
      } else {
         mid = (l+ir) >> 1; //choose median of left, center and right
         {temp = ind[mid]; ind[mid]=ind[l+1]; ind[l+1]=temp;}//elements as partitioning element arr.
         if (a[ind[l]]>a[ind[ir]])  //also rearrange so that a[l]<=a[l+1]
            {temp = ind[l]; ind[l]=ind[ir]; ind[ir]=temp;}

         if (a[ind[l+1]]>a[ind[ir]])
            {temp=ind[l+1]; ind[l+1]=ind[ir]; ind[ir]=temp;}

         if (a[ind[l]]>a[ind[l+1]])
            {temp = ind[l]; ind[l]=ind[l+1]; ind[l+1]=temp;}

         i=l+1;        //initialize pointers for partitioning
         j=ir;
         arr = ind[l+1];
         for (;;){
            do i++; while (a[ind[i]]<a[arr]);
            do j--; while (a[ind[j]]>a[arr]);
            if (j<i) break;  //pointers crossed, partitioning complete
               {temp=ind[i]; ind[i]=ind[j]; ind[j]=temp;}
         }
         ind[l+1]=ind[j];
         ind[j]=arr;
         if (j>=rk) ir = j-1; //keep active the partition that
         if (j<=rk) l=i;      //contains the k_th element
      }
   }
}

#endif

---