ROOT   6.08/07 Reference Guide
TMath Namespace Reference

struct  Limits

## Functions

Short_t Abs (Short_t d)

Int_t Abs (Int_t d)

Long_t Abs (Long_t d)

Long64_t Abs (Long64_t d)

Float_t Abs (Float_t d)

Double_t Abs (Double_t d)

LongDouble_t Abs (LongDouble_t d)

Double_t ACos (Double_t)

Double_t ACosH (Double_t)

Bool_t AreEqualAbs (Double_t af, Double_t bf, Double_t epsilon)

Bool_t AreEqualRel (Double_t af, Double_t bf, Double_t relPrec)

Double_t ASin (Double_t)

Double_t ASinH (Double_t)

Double_t ATan (Double_t)

Double_t ATan2 (Double_t, Double_t)

Double_t ATanH (Double_t)

Double_t BesselI (Int_t n, Double_t x)
Compute the Integer Order Modified Bessel function I_n(x) for n=0,1,2,... More...

Double_t BesselI0 (Double_t x)
Compute the modified Bessel function I_0(x) for any real x. More...

Double_t BesselI1 (Double_t x)
Compute the modified Bessel function I_1(x) for any real x. More...

Double_t BesselJ0 (Double_t x)
Returns the Bessel function J0(x) for any real x. More...

Double_t BesselJ1 (Double_t x)
Returns the Bessel function J1(x) for any real x. More...

Double_t BesselK (Int_t n, Double_t x)
Compute the Integer Order Modified Bessel function K_n(x) for n=0,1,2,... More...

Double_t BesselK0 (Double_t x)
Compute the modified Bessel function K_0(x) for positive real x. More...

Double_t BesselK1 (Double_t x)
Compute the modified Bessel function K_1(x) for positive real x. More...

Double_t BesselY0 (Double_t x)
Returns the Bessel function Y0(x) for positive x. More...

Double_t BesselY1 (Double_t x)
Returns the Bessel function Y1(x) for positive x. More...

Double_t Beta (Double_t p, Double_t q)
Calculates Beta-function Gamma(p)*Gamma(q)/Gamma(p+q). More...

Double_t BetaCf (Double_t x, Double_t a, Double_t b)
Continued fraction evaluation by modified Lentz's method used in calculation of incomplete Beta function. More...

Double_t BetaDist (Double_t x, Double_t p, Double_t q)
Computes the probability density function of the Beta distribution (the distribution function is computed in BetaDistI). More...

Double_t BetaDistI (Double_t x, Double_t p, Double_t q)
Computes the distribution function of the Beta distribution. More...

Double_t BetaIncomplete (Double_t x, Double_t a, Double_t b)
Calculates the incomplete Beta-function. More...

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)

Double_t Binomial (Int_t n, Int_t k)
Calculate the binomial coefficient n over k. More...

Double_t BinomialI (Double_t p, Int_t n, Int_t k)
Suppose an event occurs with probability p per trial Then the probability P of its occurring k or more times in n trials is termed a cumulative binomial probability the formula is P = sum_from_j=k_to_n(TMath::Binomial(n, j)* *TMath::Power(p, j)*TMathPower(1-p, n-j) For n larger than 12 BetaIncomplete is a much better way to evaluate the sum than would be the straightforward sum calculation for n smaller than 12 either method is acceptable ("Numerical Recipes") –implementation by Anna Kreshuk. More...

Double_t BreitWigner (Double_t x, Double_t mean=0, Double_t gamma=1)
Calculate a Breit Wigner function with mean and gamma. More...

void BubbleHigh (Int_t Narr, Double_t *arr1, Int_t *arr2)
Bubble sort variant to obtain the order of an array's elements into an index in order to do more useful things than the standard built in functions. More...

void BubbleLow (Int_t Narr, Double_t *arr1, Int_t *arr2)
Opposite ordering of the array arr2[] to that of BubbleHigh. More...

Double_t C ()

Double_t CauchyDist (Double_t x, Double_t t=0, Double_t s=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. More...

Double_t Ccgs ()

Double_t Ceil (Double_t x)

Int_t CeilNint (Double_t x)

Double_t ChisquareQuantile (Double_t p, Double_t ndf)
Evaluate the quantiles of the chi-squared probability distribution function. More...

Double_t Cos (Double_t)

Double_t CosH (Double_t)

template<typename T >
T * Cross (const T v1[3], const T v2[3], T out[3])

Double_t CUncertainty ()

Double_t DiLog (Double_t x)
The DiLogarithm function Code translated by R.Brun from CERNLIB DILOG function C332. More...

Double_t E ()

Double_t Erf (Double_t x)
Computation of the error function erf(x). More...

Double_t Erfc (Double_t x)
Compute the complementary error function erfc(x). More...

Double_t ErfcInverse (Double_t x)

Double_t ErfInverse (Double_t x)
returns the inverse error function x must be <-1<x<1 More...

Double_t EulerGamma ()

Bool_t Even (Long_t a)

Double_t Exp (Double_t x)

Double_t Factorial (Int_t i)
Compute factorial(n). More...

Double_t FDist (Double_t F, Double_t N, Double_t M)
Computes the density function of F-distribution (probability function, integral of density, is computed in FDistI). More...

Double_t FDistI (Double_t F, Double_t N, Double_t M)
Calculates the cumulative distribution function of F-distribution, this function occurs in the statistical test of whether two observed samples have the same variance. More...

Int_t Finite (Double_t x)

Double_t Floor (Double_t x)

Int_t FloorNint (Double_t x)

Double_t Freq (Double_t x)
Computation of the normal frequency function freq(x). More...

Double_t G ()

Double_t GamCf (Double_t a, Double_t x)
Computation of the incomplete gamma function P(a,x) via its continued fraction representation. More...

Double_t Gamma (Double_t z)
Computation of gamma(z) for all z. More...

Double_t Gamma (Double_t a, Double_t x)
Computation of the normalized lower incomplete gamma function P(a,x) as defined in the Handbook of Mathematical Functions by Abramowitz and Stegun, formula 6.5.1 on page 260 . More...

Double_t GammaDist (Double_t x, Double_t gamma, Double_t mu=0, Double_t beta=1)
Computes the density function of Gamma distribution at point x. More...

Double_t GamSer (Double_t a, Double_t x)
Computation of the incomplete gamma function P(a,x) via its series representation. More...

Double_t Gaus (Double_t x, Double_t mean=0, Double_t sigma=1, Bool_t norm=kFALSE)
Calculate a gaussian function with mean and sigma. More...

Double_t Gcgs ()

template<typename T >
Double_t GeomMean (Long64_t n, const T *a)

template<typename Iterator >
Double_t GeomMean (Iterator first, Iterator last)

Double_t GhbarC ()

Double_t GhbarCUncertainty ()

Double_t Gn ()

Double_t GnUncertainty ()

Double_t GUncertainty ()

Double_t H ()

ULong_t Hash (const void *txt, Int_t ntxt)
Calculates hash index from any char string. More...

ULong_t Hash (const char *str)
Return a case-sensitive hash value (endian independent). More...

Double_t Hbar ()

Double_t Hbarcgs ()

Double_t HbarUncertainty ()

Double_t HC ()

Double_t HCcgs ()

Double_t Hcgs ()

Double_t HUncertainty ()

Double_t Hypot (Double_t x, Double_t y)

Long_t Hypot (Long_t x, Long_t y)

Double_t Infinity ()

Double_t InvPi ()

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

Int_t IsNaN (Double_t x)

Double_t K ()

Double_t Kcgs ()

Double_t KolmogorovProb (Double_t z)
Calculates the Kolmogorov distribution function,. More...

Double_t KolmogorovTest (Int_t na, const Double_t *a, Int_t nb, const Double_t *b, Option_t *option)
Statistical test whether two one-dimensional sets of points are compatible with coming from the same parent distribution, using the Kolmogorov test. More...

template<class Element , typename Size >
Element KOrdStat (Size n, const Element *a, Size k, Size *work=0)

Double_t KUncertainty ()

Double_t Landau (Double_t x, Double_t mpv=0, Double_t sigma=1, Bool_t norm=kFALSE)
The LANDAU function. More...

Double_t LandauI (Double_t x)
Returns the value of the Landau distribution function at point x. More...

Double_t LaplaceDist (Double_t x, Double_t alpha=0, Double_t beta=1)
Computes the probability density function of Laplace distribution at point x, with location parameter alpha and shape parameter beta. More...

Double_t LaplaceDistI (Double_t x, Double_t alpha=0, Double_t beta=1)
Computes the distribution function of Laplace distribution at point x, with location parameter alpha and shape parameter beta. More...

Double_t Ldexp (Double_t x, Int_t exp)

Double_t Ln10 ()

Double_t LnGamma (Double_t z)
Computation of ln[gamma(z)] for all z. More...

template<typename T >
Long64_t LocMax (Long64_t n, const T *a)

template<typename Iterator >
Iterator LocMax (Iterator first, Iterator last)

template<typename T >
Long64_t LocMin (Long64_t n, const T *a)

template<typename Iterator >
Iterator LocMin (Iterator first, Iterator last)

Double_t Log (Double_t x)

Double_t Log10 (Double_t x)

Double_t Log2 (Double_t x)

Double_t LogE ()

Double_t LogNormal (Double_t x, Double_t sigma, Double_t theta=0, Double_t m=1)
Computes the density of LogNormal distribution at point x. More...

Short_t Max (Short_t a, Short_t b)

UShort_t Max (UShort_t a, UShort_t b)

Int_t Max (Int_t a, Int_t b)

UInt_t Max (UInt_t a, UInt_t b)

Long_t Max (Long_t a, Long_t b)

ULong_t Max (ULong_t a, ULong_t b)

Long64_t Max (Long64_t a, Long64_t b)

ULong64_t Max (ULong64_t a, ULong64_t b)

Float_t Max (Float_t a, Float_t b)

Double_t Max (Double_t a, Double_t b)

template<typename T >
MaxElement (Long64_t n, const T *a)

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 wfirst)

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

Short_t Min (Short_t a, Short_t b)

UShort_t Min (UShort_t a, UShort_t b)

Int_t Min (Int_t a, Int_t b)

UInt_t Min (UInt_t a, UInt_t b)

Long_t Min (Long_t a, Long_t b)

ULong_t Min (ULong_t a, ULong_t b)

Long64_t Min (Long64_t a, Long64_t b)

ULong64_t Min (ULong64_t a, ULong64_t b)

Float_t Min (Float_t a, Float_t b)

Double_t Min (Double_t a, Double_t b)

template<typename T >
MinElement (Long64_t n, const T *a)

Double_t MWair ()

Double_t Na ()

Double_t NaUncertainty ()

Long_t NextPrime (Long_t x)
TMath Base functionsDefine the functions Min, Max, Abs, Sign, Range for all types. More...

template<typename T >
Int_t Nint (T x)

template<typename T >
T * Normal2Plane (const T v1[3], const T v2[3], const T v3[3], T normal[3])

Float_t Normalize (Float_t v[3])
Normalize a vector v in place. More...

Double_t Normalize (Double_t v[3])
Normalize a vector v in place. More...

template<typename T >
NormCross (const T v1[3], const T v2[3], T out[3])

Double_t NormQuantile (Double_t p)
Computes quantiles for standard normal distribution N(0, 1) at probability p ALGORITHM AS241 APPL. More...

Bool_t Odd (Long_t a)

Bool_t Permute (Int_t n, Int_t *a)
Simple recursive algorithm to find the permutations of n natural numbers, not necessarily all distinct adapted from CERNLIB routine PERMU. More...

Double_t Pi ()

Double_t PiOver2 ()

Double_t PiOver4 ()

Double_t Poisson (Double_t x, Double_t par)
Compute the Poisson distribution function for (x,par) The Poisson PDF is implemented by means of Euler's Gamma-function (for the factorial), so for any x integer argument it is correct. More...

Double_t PoissonI (Double_t x, Double_t par)
compute the Poisson distribution function for (x,par) This is a non-smooth function. More...

LongDouble_t Power (LongDouble_t x, LongDouble_t y)

LongDouble_t Power (LongDouble_t x, Long64_t y)

LongDouble_t Power (Long64_t x, Long64_t y)

Double_t Power (Double_t x, Double_t y)

Double_t Power (Double_t x, Int_t y)

Double_t Prob (Double_t chi2, Int_t ndf)
Computation of the probability for a certain Chi-squared (chi2) and number of degrees of freedom (ndf). More...

Double_t Qe ()

Double_t QeUncertainty ()

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)
Computes sample quantiles, corresponding to the given probabilities Parameters: x -the data sample n - its size quantiles - computed quantiles are returned in there prob - probabilities where to compute quantiles nprob - size of prob array isSorted - is the input array x sorted? NOTE, that when the input is not sorted, an array of integers of size n needs to be allocated. More...

Double_t QuietNaN ()

Double_t R ()

Short_t Range (Short_t lb, Short_t ub, Short_t x)

Int_t Range (Int_t lb, Int_t ub, Int_t x)

Long_t Range (Long_t lb, Long_t ub, Long_t x)

ULong_t Range (ULong_t lb, ULong_t ub, ULong_t x)

Double_t Range (Double_t lb, Double_t ub, Double_t x)

Double_t Rgair ()

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

template<typename Iterator >
Double_t RMS (Iterator first, Iterator last)

template<typename Iterator , typename WeightIterator >
Double_t RMS (Iterator first, Iterator last, WeightIterator wfirst)

Bool_t RootsCubic (const Double_t coef[4], Double_t &a, Double_t &b, Double_t &c)
Calculates roots of polynomial of 3rd order a*x^3 + b*x^2 + c*x + d, where a == coef[3], b == coef[2], c == coef[1], d == coef[0] coef[3] must be different from 0 If the boolean returned by the method is false: ==> there are 3 real roots a,b,c If the boolean returned by the method is true: ==> there is one real root a and 2 complex conjugates roots (b+i*c,b-i*c) Author: Francois-Xavier Gentit. More...

Double_t RUncertainty ()

Double_t Sigma ()

Double_t SigmaUncertainty ()

template<typename T1 , typename T2 >
T1 Sign (T1 a, T2 b)

Float_t Sign (Float_t a, Float_t b)

Double_t Sign (Double_t a, Double_t b)

LongDouble_t Sign (LongDouble_t a, LongDouble_t b)

Double_t SignalingNaN ()

template<typename Integer >
Bool_t SignBit (Integer a)

Bool_t SignBit (Float_t a)

Bool_t SignBit (Double_t a)

Bool_t SignBit (LongDouble_t a)

Double_t Sin (Double_t)

Double_t SinH (Double_t)

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)

Double_t Sq (Double_t x)

Double_t Sqrt (Double_t x)

Double_t Sqrt2 ()

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

template<typename Iterator >
Double_t StdDev (Iterator first, Iterator last)

template<typename Iterator , typename WeightIterator >
Double_t StdDev (Iterator first, Iterator last, WeightIterator wfirst)

Double_t StruveH0 (Double_t x)
Struve Functions of Order 0. More...

Double_t StruveH1 (Double_t x)
Struve Functions of Order 1. More...

Double_t StruveL0 (Double_t x)
Modified Struve Function of Order 0. More...

Double_t StruveL1 (Double_t x)
Modified Struve Function of Order 1. More...

Double_t Student (Double_t T, Double_t ndf)
Computes density function for Student's t- distribution (the probability function (integral of density) is computed in StudentI). More...

Double_t StudentI (Double_t T, Double_t ndf)
Calculates the cumulative distribution function of Student's t-distribution second parameter stands for number of degrees of freedom, not for the number of samples if x has Student's t-distribution, the function returns the probability of x being less than T. More...

Double_t StudentQuantile (Double_t p, Double_t ndf, Bool_t lower_tail=kTRUE)
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. More...

Double_t Tan (Double_t)

Double_t TanH (Double_t)

Double_t TwoPi ()

Double_t Vavilov (Double_t x, Double_t kappa, Double_t beta2)
Returns the value of the Vavilov density function Parameters: 1st - the point were the density function is evaluated 2nd - value of kappa (distribution parameter) 3rd - value of beta2 (distribution parameter) The algorithm was taken from the CernLib function vavden(G115) Reference: A.Rotondi and P.Montagna, Fast Calculation of Vavilov distribution Nucl.Instr. More...

Double_t VavilovDenEval (Double_t rlam, Double_t *AC, Double_t *HC, Int_t itype)
Internal function, called by Vavilov and VavilovSet. More...

Double_t VavilovI (Double_t x, Double_t kappa, Double_t beta2)
Returns the value of the Vavilov distribution function Parameters: 1st - the point were the density function is evaluated 2nd - value of kappa (distribution parameter) 3rd - value of beta2 (distribution parameter) The algorithm was taken from the CernLib function vavden(G115) Reference: A.Rotondi and P.Montagna, Fast Calculation of Vavilov distribution Nucl.Instr. More...

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)
Internal function, called by Vavilov and VavilovI. More...

Double_t Voigt (Double_t x, Double_t sigma, Double_t lg, Int_t r=4)
Computation of Voigt function (normalised). More...

## ◆ Abs() [1/7]

 Short_t TMath::Abs ( Short_t d )
inline

Definition at line 110 of file TMathBase.h.

## ◆ Abs() [2/7]

 Int_t TMath::Abs ( Int_t d )
inline

Definition at line 113 of file TMathBase.h.

## ◆ Abs() [3/7]

 Long_t TMath::Abs ( Long_t d )
inline

Definition at line 116 of file TMathBase.h.

## ◆ Abs() [4/7]

 Long64_t TMath::Abs ( Long64_t d )
inline

Definition at line 119 of file TMathBase.h.

## ◆ Abs() [5/7]

 Float_t TMath::Abs ( Float_t d )
inline

Definition at line 126 of file TMathBase.h.

## ◆ Abs() [6/7]

 Double_t TMath::Abs ( Double_t d )
inline

Definition at line 129 of file TMathBase.h.

## ◆ Abs() [7/7]

 LongDouble_t TMath::Abs ( LongDouble_t d )
inline

Definition at line 132 of file TMathBase.h.

## ◆ ACos()

 Double_t TMath::ACos ( Double_t x )
inline

Definition at line 445 of file TMath.h.

## ◆ ACosH()

 Double_t TMath::ACosH ( Double_t x )

Definition at line 80 of file TMath.cxx.

## ◆ AreEqualAbs()

 Bool_t TMath::AreEqualAbs ( Double_t af, Double_t bf, Double_t epsilon )
inline

Definition at line 192 of file TMath.h.

## ◆ AreEqualRel()

 Bool_t TMath::AreEqualRel ( Double_t af, Double_t bf, Double_t relPrec )
inline

Definition at line 196 of file TMath.h.

## ◆ ASin()

 Double_t TMath::ASin ( Double_t x )
inline

Definition at line 439 of file TMath.h.

## ◆ ASinH()

 Double_t TMath::ASinH ( Double_t x )

Definition at line 67 of file TMath.cxx.

## ◆ ATan()

 Double_t TMath::ATan ( Double_t x )
inline

Definition at line 451 of file TMath.h.

## ◆ ATan2()

 Double_t TMath::ATan2 ( Double_t y, Double_t x )
inline

Definition at line 454 of file TMath.h.

## ◆ ATanH()

 Double_t TMath::ATanH ( Double_t x )

Definition at line 93 of file TMath.cxx.

## ◆ BesselI()

 Double_t TMath::BesselI ( Int_t n, Double_t x )

Compute the Integer Order Modified Bessel function I_n(x) for n=0,1,2,...

and any real x.

— NvE 12-mar-2000 UU-SAP Utrecht

Definition at line 1557 of file TMath.cxx.

## ◆ BesselI0()

 Double_t TMath::BesselI0 ( Double_t x )

Compute the modified Bessel function I_0(x) for any real x.

— NvE 12-mar-2000 UU-SAP Utrecht

Definition at line 1393 of file TMath.cxx.

## ◆ BesselI1()

 Double_t TMath::BesselI1 ( Double_t x )

Compute the modified Bessel function I_1(x) for any real x.

M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions, Applied Mathematics Series vol. 55 (1964), Washington.

— NvE 12-mar-2000 UU-SAP Utrecht

Definition at line 1461 of file TMath.cxx.

## ◆ BesselJ0()

 Double_t TMath::BesselJ0 ( Double_t x )

Returns the Bessel function J0(x) for any real x.

Definition at line 1601 of file TMath.cxx.

## ◆ BesselJ1()

 Double_t TMath::BesselJ1 ( Double_t x )

Returns the Bessel function J1(x) for any real x.

Definition at line 1636 of file TMath.cxx.

## ◆ BesselK()

 Double_t TMath::BesselK ( Int_t n, Double_t x )

Compute the Integer Order Modified Bessel function K_n(x) for n=0,1,2,...

and positive real x.

— NvE 12-mar-2000 UU-SAP Utrecht

Definition at line 1528 of file TMath.cxx.

## ◆ BesselK0()

 Double_t TMath::BesselK0 ( Double_t x )

Compute the modified Bessel function K_0(x) for positive real x.

M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions, Applied Mathematics Series vol. 55 (1964), Washington.

— NvE 12-mar-2000 UU-SAP Utrecht

Definition at line 1427 of file TMath.cxx.

## ◆ BesselK1()

 Double_t TMath::BesselK1 ( Double_t x )

Compute the modified Bessel function K_1(x) for positive real x.

M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions, Applied Mathematics Series vol. 55 (1964), Washington.

— NvE 12-mar-2000 UU-SAP Utrecht

Definition at line 1496 of file TMath.cxx.

## ◆ BesselY0()

 Double_t TMath::BesselY0 ( Double_t x )

Returns the Bessel function Y0(x) for positive x.

Definition at line 1672 of file TMath.cxx.

## ◆ BesselY1()

 Double_t TMath::BesselY1 ( Double_t x )

Returns the Bessel function Y1(x) for positive x.

Definition at line 1706 of file TMath.cxx.

## ◆ Beta()

 Double_t TMath::Beta ( Double_t p, Double_t q )

Calculates Beta-function Gamma(p)*Gamma(q)/Gamma(p+q).

Definition at line 1977 of file TMath.cxx.

## ◆ BetaCf()

 Double_t TMath::BetaCf ( Double_t x, Double_t a, Double_t b )

Continued fraction evaluation by modified Lentz's method used in calculation of incomplete Beta function.

Definition at line 1986 of file TMath.cxx.

 Double_t TMath::BetaDist ( Double_t x, Double_t p, Double_t q )

Computes the probability density function of the Beta distribution (the distribution function is computed in BetaDistI).

The first argument is the point, where the function will be computed, second and third are the function parameters. Since the Beta distribution is bounded on both sides, it's often used to represent processes with natural lower and upper limits.

Definition at line 2037 of file TMath.cxx.

 Double_t TMath::BetaDistI ( Double_t x, Double_t p, Double_t q )

Computes the distribution function of the Beta distribution.

The first argument is the point, where the function will be computed, second and third are the function parameters. Since the Beta distribution is bounded on both sides, it's often used to represent processes with natural lower and upper limits.

Definition at line 2055 of file TMath.cxx.

## ◆ BetaIncomplete()

 Double_t TMath::BetaIncomplete ( Double_t x, Double_t a, Double_t b )

Calculates the incomplete Beta-function.

Definition at line 2068 of file TMath.cxx.

## ◆ BinarySearch() [1/3]

template<typename T >
 Long64_t TMath::BinarySearch ( Long64_t n, const T * array, T value )

Definition at line 931 of file TMath.h.

## ◆ BinarySearch() [2/3]

template<typename T >
 Long64_t TMath::BinarySearch ( Long64_t n, const T ** array, T value )

Definition at line 947 of file TMath.h.

## ◆ BinarySearch() [3/3]

template<typename Iterator , typename Element >
 Iterator TMath::BinarySearch ( Iterator first, Iterator last, Element value )

Definition at line 913 of file TMath.h.

## ◆ Binomial()

 Double_t TMath::Binomial ( Int_t n, Int_t k )

Calculate the binomial coefficient n over k.

Definition at line 2076 of file TMath.cxx.

## ◆ BinomialI()

 Double_t TMath::BinomialI ( Double_t p, Int_t n, Int_t k )

Suppose an event occurs with probability p per trial Then the probability P of its occurring k or more times in n trials is termed a cumulative binomial probability the formula is P = sum_from_j=k_to_n(TMath::Binomial(n, j)* *TMath::Power(p, j)*TMathPower(1-p, n-j) For n larger than 12 BetaIncomplete is a much better way to evaluate the sum than would be the straightforward sum calculation for n smaller than 12 either method is acceptable ("Numerical Recipes") –implementation by Anna Kreshuk.

Definition at line 2101 of file TMath.cxx.

## ◆ BreitWigner()

 Double_t TMath::BreitWigner ( Double_t x, Double_t mean = 0, Double_t gamma = 1 )

Calculate a Breit Wigner function with mean and gamma.

Definition at line 441 of file TMath.cxx.

## ◆ BubbleHigh()

 void TMath::BubbleHigh ( Int_t Narr, Double_t * arr1, Int_t * arr2 )

Bubble sort variant to obtain the order of an array's elements into an index in order to do more useful things than the standard built in functions.

*arr1 is unchanged; *arr2 is the array of indicies corresponding to the descending value of arr1 with arr2[0] corresponding to the largest arr1 value and arr2[Narr] the smallest.

Definition at line 1281 of file TMath.cxx.

## ◆ BubbleLow()

 void TMath::BubbleLow ( Int_t Narr, Double_t * arr1, Int_t * arr2 )

Opposite ordering of the array arr2[] to that of BubbleHigh.

Definition at line 1320 of file TMath.cxx.

## ◆ C()

 Double_t TMath::C ( )
inline

Definition at line 63 of file TMath.h.

## ◆ CauchyDist()

 Double_t TMath::CauchyDist ( Double_t x, Double_t t = 0, Double_t s = 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 Implementation by Anna Kreshuk. Example: TF1* fc = new TF1("fc", "TMath::CauchyDist(x, [0], [1])", -5, 5); fc->SetParameters(0, 1); fc->Draw();

Definition at line 2129 of file TMath.cxx.

## ◆ Ccgs()

 Double_t TMath::Ccgs ( )
inline

Definition at line 64 of file TMath.h.

## ◆ Ceil()

 Double_t TMath::Ceil ( Double_t x )
inline

Definition at line 467 of file TMath.h.

## ◆ CeilNint()

 Int_t TMath::CeilNint ( Double_t x )
inline

Definition at line 470 of file TMath.h.

## ◆ ChisquareQuantile()

 Double_t TMath::ChisquareQuantile ( Double_t p, Double_t ndf )

Evaluate the quantiles of the chi-squared probability distribution function.

Algorithm AS 91 Appl. Statist. (1975) Vol.24, P.35 implemented by Anna Kreshuk. Incorporates the suggested changes in AS R85 (vol.40(1), pp.233-5, 1991) Parameters: p - the probability value, at which the quantile is computed ndf - number of degrees of freedom

Definition at line 2145 of file TMath.cxx.

## ◆ Cos()

 Double_t TMath::Cos ( Double_t x )
inline

Definition at line 424 of file TMath.h.

## ◆ CosH()

 Double_t TMath::CosH ( Double_t x )
inline

Definition at line 433 of file TMath.h.

## ◆ Cross()

template<typename T >
 T * TMath::Cross ( const T v1[3], const T v2[3], T out[3] )

Definition at line 1007 of file TMath.h.

## ◆ CUncertainty()

 Double_t TMath::CUncertainty ( )
inline

Definition at line 65 of file TMath.h.

inline

Definition at line 50 of file TMath.h.

## ◆ DiLog()

 Double_t TMath::DiLog ( Double_t x )

The DiLogarithm function Code translated by R.Brun from CERNLIB DILOG function C332.

Definition at line 113 of file TMath.cxx.

## ◆ E()

 Double_t TMath::E ( )
inline

Definition at line 54 of file TMath.h.

## ◆ Erf()

 Double_t TMath::Erf ( Double_t x )

Computation of the error function erf(x).

Erf(x) = (2/sqrt(pi)) Integral(exp(-t^2))dt between 0 and x

Definition at line 187 of file TMath.cxx.

## ◆ Erfc()

 Double_t TMath::Erfc ( Double_t x )

Compute the complementary error function erfc(x).

Erfc(x) = (2/sqrt(pi)) Integral(exp(-t^2))dt between x and infinity

Definition at line 197 of file TMath.cxx.

## ◆ ErfcInverse()

 Double_t TMath::ErfcInverse ( Double_t x )

Definition at line 233 of file TMath.cxx.

## ◆ ErfInverse()

 Double_t TMath::ErfInverse ( Double_t x )

returns the inverse error function x must be <-1<x<1

Definition at line 206 of file TMath.cxx.

## ◆ EulerGamma()

 Double_t TMath::EulerGamma ( )
inline

Definition at line 122 of file TMath.h.

## ◆ Even()

 Bool_t TMath::Even ( Long_t a )
inline

Definition at line 102 of file TMathBase.h.

## ◆ Exp()

 Double_t TMath::Exp ( Double_t x )
inline

Definition at line 495 of file TMath.h.

## ◆ Factorial()

 Double_t TMath::Factorial ( Int_t i )

Compute factorial(n).

Definition at line 250 of file TMath.cxx.

## ◆ FDist()

 Double_t TMath::FDist ( Double_t F, Double_t N, Double_t M )

Computes the density function of F-distribution (probability function, integral of density, is computed in FDistI).

Parameters N and M stand for degrees of freedom of chi-squares mentioned above parameter F is the actual variable x of the density function p(x) and the point at which the density function is calculated.

About F distribution: F-distribution arises in testing whether two random samples have the same variance. It is the ratio of two chi-square distributions, with N and M degrees of freedom respectively, where each chi-square is first divided by it's number of degrees of freedom. Implementation by Anna Kreshuk.

Definition at line 2228 of file TMath.cxx.

## ◆ FDistI()

 Double_t TMath::FDistI ( Double_t F, Double_t N, Double_t M )

Calculates the cumulative distribution function of F-distribution, this function occurs in the statistical test of whether two observed samples have the same variance.

For this test a certain statistic F, the ratio of observed dispersion of the first sample to that of the second sample, is calculated. N and M stand for numbers of degrees of freedom in the samples 1-FDistI() is the significance level at which the hypothesis "1 has smaller variance than 2" can be rejected. A small numerical value of 1 - FDistI() implies a very significant rejection, in turn implying high confidence in the hypothesis "1 has variance greater than 2". Implementation by Anna Kreshuk.

Definition at line 2246 of file TMath.cxx.

## ◆ Finite()

 Int_t TMath::Finite ( Double_t x )
inline

Definition at line 532 of file TMath.h.

## ◆ Floor()

 Double_t TMath::Floor ( Double_t x )
inline

Definition at line 473 of file TMath.h.

## ◆ FloorNint()

 Int_t TMath::FloorNint ( Double_t x )
inline

Definition at line 476 of file TMath.h.

## ◆ Freq()

 Double_t TMath::Freq ( Double_t x )

Computation of the normal frequency function freq(x).

Freq(x) = (1/sqrt(2pi)) Integral(exp(-t^2/2))dt between -infinity and x.

Translated from CERNLIB C300 by Rene Brun.

Definition at line 268 of file TMath.cxx.

## ◆ G()

 Double_t TMath::G ( )
inline

Definition at line 68 of file TMath.h.

## ◆ GamCf()

 Double_t TMath::GamCf ( Double_t a, Double_t x )

Computation of the incomplete gamma function P(a,x) via its continued fraction representation.

— Nve 14-nov-1998 UU-SAP Utrecht

Definition at line 379 of file TMath.cxx.

## ◆ Gamma() [1/2]

 Double_t TMath::Gamma ( Double_t z )

Computation of gamma(z) for all z.

C.Lanczos, SIAM Journal of Numerical Analysis B1 (1964), 86.

Definition at line 352 of file TMath.cxx.

## ◆ Gamma() [2/2]

 Double_t TMath::Gamma ( Double_t a, Double_t x )

Computation of the normalized lower incomplete gamma function P(a,x) as defined in the Handbook of Mathematical Functions by Abramowitz and Stegun, formula 6.5.1 on page 260 .

Its normalization is such that TMath::Gamma(a,+infinity) = 1 .

$P(a, x) = \frac{1}{\Gamma(a)} \int_{0}^{x} t^{a-1} e^{-t} dt$

— Nve 14-nov-1998 UU-SAP Utrecht

Definition at line 368 of file TMath.cxx.

 Double_t TMath::GammaDist ( Double_t x, Double_t gamma, Double_t mu = 0, Double_t beta = 1 )

Computes the density function of Gamma distribution at point x.

gamma - shape parameter mu - location parameter beta - scale parameter

The definition can be found in "Engineering Statistics Handbook" on site http://www.itl.nist.gov/div898/handbook/eda/section3/eda366b.htm use now implementation in ROOT::Math::gamma_pdf

Definition at line 2294 of file TMath.cxx.

## ◆ GamSer()

 Double_t TMath::GamSer ( Double_t a, Double_t x )

Computation of the incomplete gamma function P(a,x) via its series representation.

— Nve 14-nov-1998 UU-SAP Utrecht

Definition at line 416 of file TMath.cxx.

## ◆ Gaus()

 Double_t TMath::Gaus ( Double_t x, Double_t mean = 0, Double_t sigma = 1, Bool_t norm = kFALSE )

Calculate a gaussian function with mean and sigma.

If norm=kTRUE (default is kFALSE) the result is divided by sqrt(2*Pi)*sigma.

Definition at line 452 of file TMath.cxx.

## ◆ Gcgs()

 Double_t TMath::Gcgs ( )
inline

Definition at line 69 of file TMath.h.

## ◆ GeomMean() [1/2]

template<typename T >
 Double_t TMath::GeomMean ( Long64_t n, const T * a )

Definition at line 842 of file TMath.h.

## ◆ GeomMean() [2/2]

template<typename Iterator >
 Double_t TMath::GeomMean ( Iterator first, Iterator last )

Definition at line 823 of file TMath.h.

## ◆ GhbarC()

 Double_t TMath::GhbarC ( )
inline

Definition at line 73 of file TMath.h.

## ◆ GhbarCUncertainty()

 Double_t TMath::GhbarCUncertainty ( )
inline

Definition at line 74 of file TMath.h.

## ◆ Gn()

 Double_t TMath::Gn ( )
inline

Definition at line 77 of file TMath.h.

## ◆ GnUncertainty()

 Double_t TMath::GnUncertainty ( )
inline

Definition at line 78 of file TMath.h.

## ◆ GUncertainty()

 Double_t TMath::GUncertainty ( )
inline

Definition at line 70 of file TMath.h.

## ◆ H()

 Double_t TMath::H ( )
inline

Definition at line 81 of file TMath.h.

## ◆ Hash() [1/2]

 ULong_t TMath::Hash ( const void * txt, Int_t ntxt )

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

Definition at line 1375 of file TMath.cxx.

## ◆ Hash() [2/2]

 ULong_t TMath::Hash ( const char * str )

Return a case-sensitive hash value (endian independent).

Definition at line 1383 of file TMath.cxx.

## ◆ Hbar()

 Double_t TMath::Hbar ( )
inline

Definition at line 86 of file TMath.h.

## ◆ Hbarcgs()

 Double_t TMath::Hbarcgs ( )
inline

Definition at line 87 of file TMath.h.

## ◆ HbarUncertainty()

 Double_t TMath::HbarUncertainty ( )
inline

Definition at line 88 of file TMath.h.

## ◆ HC()

 Double_t TMath::HC ( )
inline

Definition at line 91 of file TMath.h.

## ◆ HCcgs()

 Double_t TMath::HCcgs ( )
inline

Definition at line 92 of file TMath.h.

## ◆ Hcgs()

 Double_t TMath::Hcgs ( )
inline

Definition at line 82 of file TMath.h.

## ◆ HUncertainty()

 Double_t TMath::HUncertainty ( )
inline

Definition at line 83 of file TMath.h.

## ◆ Hypot() [1/2]

 Double_t TMath::Hypot ( Double_t x, Double_t y )

Definition at line 60 of file TMath.cxx.

## ◆ Hypot() [2/2]

 Long_t TMath::Hypot ( Long_t x, Long_t y )

Definition at line 53 of file TMath.cxx.

## ◆ Infinity()

 Double_t TMath::Infinity ( )
inline

Definition at line 635 of file TMath.h.

## ◆ InvPi()

 Double_t TMath::InvPi ( )
inline

Definition at line 48 of file TMath.h.

## ◆ IsInside()

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

Definition at line 1043 of file TMath.h.

## ◆ IsNaN()

 Int_t TMath::IsNaN ( Double_t x )
inline

Definition at line 613 of file TMath.h.

## ◆ K()

 Double_t TMath::K ( )
inline

Definition at line 95 of file TMath.h.

## ◆ Kcgs()

 Double_t TMath::Kcgs ( )
inline

Definition at line 96 of file TMath.h.

## ◆ KolmogorovProb()

 Double_t TMath::KolmogorovProb ( Double_t z )

Calculates the Kolmogorov distribution function,.

$P(z) = 2 \sum_{j=1}^{\infty} (-1)^{j-1} e^{-2 j^2 z^2}$

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: z = dn*sqrt(n), and dn is the maximum deviation between a hypothetical distribution function and an experimental distribution with n events

NOTE: To compare two experimental distributions with m and n events, use z = sqrt(m*n/(m+n))*dn

Accuracy: The function is far too accurate for any imaginable application. Probabilities less than 10^-15 are returned as zero. However, remember that the formula is only valid for "large" n. Theta function inversion formula is used for z <= 1

This function was translated by Rene Brun from PROBKL in CERNLIB.

Definition at line 665 of file TMath.cxx.

## ◆ KolmogorovTest()

 Double_t TMath::KolmogorovTest ( Int_t na, const Double_t * a, Int_t nb, const Double_t * b, Option_t * option )

Statistical test whether two one-dimensional sets of points are compatible with coming from the same parent distribution, using the Kolmogorov test.

That is, it is used to compare two experimental distributions of unbinned data.

Input: a,b: One-dimensional arrays of length na, nb, respectively. The elements of a and b must be given in ascending order. option is a character string to specify options "D" Put out a line of "Debug" printout "M" Return the Maximum Kolmogorov distance instead of prob

Output: The returned value prob is a calculated confidence level which gives a statistical test for compatibility of a and b. Values of prob close to zero are taken as indicating a small probability of compatibility. For two point sets drawn randomly from the same parent distribution, the value of prob should be uniformly distributed between zero and one. in case of error the function return -1 If the 2 sets have a different number of points, the minimum of the two sets is used.

Method: The Kolmogorov test is used. The test statistic is the maximum deviation between the two integrated distribution functions, multiplied by the normalizing factor (rdmax*sqrt(na*nb/(na+nb)).

Code adapted by Rene Brun from CERNLIB routine TKOLMO (Fred James) (W.T. Eadie, D. Drijard, F.E. James, M. Roos and B. Sadoulet, Statistical Methods in Experimental Physics, (North-Holland, Amsterdam 1971) 269-271)

## Method Improvement by Jason A Detwiler (JADet.nosp@m.wile.nosp@m.r@lbl.nosp@m..gov)

The nuts-and-bolts of the TMath::KolmogorovTest() algorithm is a for-loop over the two sorted arrays a and b representing empirical distribution functions. The for-loop handles 3 cases: when the next points to be evaluated satisfy a>b, a<b, or a=b:

for (Int_t i=0;i<na+nb;i++) { if (a[ia-1] < b[ib-1]) { rdiff -= sa; ia++; if (ia > na) {ok = kTRUE; break;} } else if (a[ia-1] > b[ib-1]) { rdiff += sb; ib++; if (ib > nb) {ok = kTRUE; break;} } else { rdiff += sb - sa; ia++; ib++; if (ia > na) {ok = kTRUE; break;} if (ib > nb) {ok = kTRUE; break;} } rdmax = TMath::Max(rdmax,TMath::Abs(rdiff)); }

For the last case, a=b, the algorithm advances each array by one index in an attempt to move through the equality. However, this is incorrect when one or the other of a or b (or both) have a repeated value, call it x. For the KS statistic to be computed properly, rdiff needs to be calculated after all of the a and b at x have been tallied (this is due to the definition of the empirical distribution function; another way to convince yourself that the old CERNLIB method is wrong is that it implies that the function defined as the difference between a and b is multi-valued at x – besides being ugly, this would invalidate Kolmogorov's theorem).

The solution is to just add while-loops into the equality-case handling to perform the tally:

  } else {
double x = a[ia-1];
while(a[ia-1] == x && ia <= na) {
rdiff -= sa;
ia++;
}
while(b[ib-1] == x && ib <= nb) {
rdiff += sb;
ib++;
}
if (ia > na) {ok = kTRUE; break;}
if (ib > nb) {ok = kTRUE; break;}
}


NOTE1 A good description of the Kolmogorov test can be seen at: http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm

Definition at line 788 of file TMath.cxx.

## ◆ KOrdStat()

template<class Element , typename Size >
 Element TMath::KOrdStat ( Size n, const Element * a, Size k, Size * work = 0 )

Definition at line 1154 of file TMath.h.

## ◆ KUncertainty()

 Double_t TMath::KUncertainty ( )
inline

Definition at line 97 of file TMath.h.

## ◆ Landau()

 Double_t TMath::Landau ( Double_t x, Double_t mu = 0, Double_t sigma = 1, Bool_t norm = kFALSE )

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

Definition at line 472 of file TMath.cxx.

## ◆ LandauI()

 Double_t TMath::LandauI ( Double_t x )

Returns the value of the Landau distribution function at point x.

The algorithm was taken from the Cernlib function dislan(G110) Reference: K.S.Kolbig and B.Schorr, "A program package for the Landau distribution", Computer Phys.Comm., 31(1984), 97-111

Definition at line 2765 of file TMath.cxx.

## ◆ LaplaceDist()

 Double_t TMath::LaplaceDist ( Double_t x, Double_t alpha = 0, Double_t beta = 1 )

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

Definition at line 2311 of file TMath.cxx.

## ◆ LaplaceDistI()

 Double_t TMath::LaplaceDistI ( Double_t x, Double_t alpha = 0, Double_t beta = 1 )

Computes the distribution function of Laplace distribution at point x, with location parameter alpha and shape parameter beta.

By default, alpha=0, beta=1 This distribution is known under different names, most common is double exponential distribution, but it also appears as the two-tailed exponential or the bilateral exponential distribution

Definition at line 2327 of file TMath.cxx.

## ◆ Ldexp()

 Double_t TMath::Ldexp ( Double_t x, Int_t exp )
inline

Definition at line 498 of file TMath.h.

## ◆ Ln10()

 Double_t TMath::Ln10 ( )
inline

Definition at line 57 of file TMath.h.

## ◆ LnGamma()

 Double_t TMath::LnGamma ( Double_t z )

Computation of ln[gamma(z)] for all z.

C.Lanczos, SIAM Journal of Numerical Analysis B1 (1964), 86.

The accuracy of the result is better than 2e-10.

— Nve 14-nov-1998 UU-SAP Utrecht

Definition at line 489 of file TMath.cxx.

## ◆ LocMax() [1/2]

template<typename T >
 Long64_t TMath::LocMax ( Long64_t n, const T * a )

Definition at line 711 of file TMath.h.

## ◆ LocMax() [2/2]

template<typename Iterator >
 Iterator TMath::LocMax ( Iterator first, Iterator last )

Definition at line 730 of file TMath.h.

## ◆ LocMin() [1/2]

template<typename T >
 Long64_t TMath::LocMin ( Long64_t n, const T * a )

Definition at line 682 of file TMath.h.

## ◆ LocMin() [2/2]

template<typename Iterator >
 Iterator TMath::LocMin ( Iterator first, Iterator last )

Definition at line 704 of file TMath.h.

## ◆ Log()

 Double_t TMath::Log ( Double_t x )
inline

Definition at line 526 of file TMath.h.

## ◆ Log10()

 Double_t TMath::Log10 ( Double_t x )
inline

Definition at line 529 of file TMath.h.

## ◆ Log2()

 Double_t TMath::Log2 ( Double_t x )

Definition at line 104 of file TMath.cxx.

## ◆ LogE()

 Double_t TMath::LogE ( )
inline

Definition at line 60 of file TMath.h.

## ◆ LogNormal()

 Double_t TMath::LogNormal ( Double_t x, Double_t sigma, Double_t theta = 0, Double_t m = 1 )

Computes the density of LogNormal distribution at point x.

Variable X has lognormal distribution if Y=Ln(X) has normal distribution

• sigma is the shape parameter
• theta is the location parameter
• m is the scale parameter

The formula was taken from "Engineering Statistics Handbook" on site http://www.itl.nist.gov/div898/handbook/eda/section3/eda3669.htm Implementation using ROOT::Math::lognormal_pdf

Definition at line 2382 of file TMath.cxx.

## ◆ Max() [1/10]

 Short_t TMath::Max ( Short_t a, Short_t b )
inline

Definition at line 202 of file TMathBase.h.

## ◆ Max() [2/10]

 UShort_t TMath::Max ( UShort_t a, UShort_t b )
inline

Definition at line 205 of file TMathBase.h.

## ◆ Max() [3/10]

 Int_t TMath::Max ( Int_t a, Int_t b )
inline

Definition at line 208 of file TMathBase.h.

## ◆ Max() [4/10]

 UInt_t TMath::Max ( UInt_t a, UInt_t b )
inline

Definition at line 211 of file TMathBase.h.

## ◆ Max() [5/10]

 Long_t TMath::Max ( Long_t a, Long_t b )
inline

Definition at line 214 of file TMathBase.h.

## ◆ Max() [6/10]

 ULong_t TMath::Max ( ULong_t a, ULong_t b )
inline

Definition at line 217 of file TMathBase.h.

## ◆ Max() [7/10]

 Long64_t TMath::Max ( Long64_t a, Long64_t b )
inline

Definition at line 220 of file TMathBase.h.

## ◆ Max() [8/10]

 ULong64_t TMath::Max ( ULong64_t a, ULong64_t b )
inline

Definition at line 223 of file TMathBase.h.

## ◆ Max() [9/10]

 Float_t TMath::Max ( Float_t a, Float_t b )
inline

Definition at line 226 of file TMathBase.h.

## ◆ Max() [10/10]

 Double_t TMath::Max ( Double_t a, Double_t b )
inline

Definition at line 229 of file TMathBase.h.

## ◆ MaxElement()

template<typename T >
 T TMath::MaxElement ( Long64_t n, const T * a )

Definition at line 675 of file TMath.h.

## ◆ Mean() [1/3]

template<typename T >
 Double_t TMath::Mean ( Long64_t n, const T * a, const Double_t * w = 0 )

Definition at line 811 of file TMath.h.

## ◆ Mean() [2/3]

template<typename Iterator >
 Double_t TMath::Mean ( Iterator first, Iterator last )

Definition at line 765 of file TMath.h.

## ◆ Mean() [3/3]

template<typename Iterator , typename WeightIterator >
 Double_t TMath::Mean ( Iterator first, Iterator last, WeightIterator wfirst )

Definition at line 782 of file TMath.h.

## ◆ Median()

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

Definition at line 1064 of file TMath.h.

## ◆ Min() [1/10]

 Short_t TMath::Min ( Short_t a, Short_t b )
inline

Definition at line 170 of file TMathBase.h.

## ◆ Min() [2/10]

 UShort_t TMath::Min ( UShort_t a, UShort_t b )
inline

Definition at line 173 of file TMathBase.h.

## ◆ Min() [3/10]

 Int_t TMath::Min ( Int_t a, Int_t b )
inline

Definition at line 176 of file TMathBase.h.

## ◆ Min() [4/10]

 UInt_t TMath::Min ( UInt_t a, UInt_t b )
inline

Definition at line 179 of file TMathBase.h.

## ◆ Min() [5/10]

 Long_t TMath::Min ( Long_t a, Long_t b )
inline

Definition at line 182 of file TMathBase.h.

## ◆ Min() [6/10]

 ULong_t TMath::Min ( ULong_t a, ULong_t b )
inline

Definition at line 185 of file TMathBase.h.

## ◆ Min() [7/10]

 Long64_t TMath::Min ( Long64_t a, Long64_t b )
inline

Definition at line 188 of file TMathBase.h.

## ◆ Min() [8/10]

 ULong64_t TMath::Min ( ULong64_t a, ULong64_t b )
inline

Definition at line 191 of file TMathBase.h.

## ◆ Min() [9/10]

 Float_t TMath::Min ( Float_t a, Float_t b )
inline

Definition at line 194 of file TMathBase.h.

## ◆ Min() [10/10]

 Double_t TMath::Min ( Double_t a, Double_t b )
inline

Definition at line 197 of file TMathBase.h.

## ◆ MinElement()

template<typename T >
 T TMath::MinElement ( Long64_t n, const T * a )

Definition at line 668 of file TMath.h.

## ◆ MWair()

 Double_t TMath::MWair ( )
inline

Definition at line 115 of file TMath.h.

## ◆ Na()

 Double_t TMath::Na ( )
inline

Definition at line 104 of file TMath.h.

## ◆ NaUncertainty()

 Double_t TMath::NaUncertainty ( )
inline

Definition at line 105 of file TMath.h.

## ◆ Nint()

template<typename T >
 Int_t TMath::Nint ( T x )
inline

Definition at line 480 of file TMath.h.

## ◆ Normal2Plane()

template<typename T >
 T * TMath::Normal2Plane ( const T v1[3], const T v2[3], const T v3[3], T normal[3] )

Definition at line 1019 of file TMath.h.

## ◆ Normalize() [1/2]

 Float_t TMath::Normalize ( Float_t v[3] )

Normalize a vector v in place.

Returns the norm of the original vector.

Definition at line 498 of file TMath.cxx.

## ◆ Normalize() [2/2]

 Double_t TMath::Normalize ( Double_t v[3] )

Normalize a vector v in place.

Returns the norm of the original vector. This implementation (thanks Kevin Lynch krlyn.nosp@m.ch@b.nosp@m.u.edu) is protected against possible overflows.

Definition at line 515 of file TMath.cxx.

## ◆ NormCross()

template<typename T >
 T TMath::NormCross ( const T v1[3], const T v2[3], T out[3] )
inline

Definition at line 661 of file TMath.h.

## ◆ NormQuantile()

 Double_t TMath::NormQuantile ( Double_t p )

Computes quantiles for standard normal distribution N(0, 1) at probability p ALGORITHM AS241 APPL.

STATIST. (1988) VOL. 37, NO. 3, 477-484.

Definition at line 2400 of file TMath.cxx.

## ◆ Odd()

 Bool_t TMath::Odd ( Long_t a )
inline

Definition at line 105 of file TMathBase.h.

## ◆ Permute()

 Bool_t TMath::Permute ( Int_t n, Int_t * a )

Simple recursive algorithm to find the permutations of n natural numbers, not necessarily all distinct adapted from CERNLIB routine PERMU.

The input array has to be initialised with a non descending sequence. The method returns kFALSE when all combinations are exhausted.

Definition at line 2501 of file TMath.cxx.

## ◆ Pi()

 Double_t TMath::Pi ( )
inline

Definition at line 44 of file TMath.h.

## ◆ PiOver2()

 Double_t TMath::PiOver2 ( )
inline

Definition at line 46 of file TMath.h.

## ◆ PiOver4()

 Double_t TMath::PiOver4 ( )
inline

Definition at line 47 of file TMath.h.

## ◆ Poisson()

 Double_t TMath::Poisson ( Double_t x, Double_t par )

Compute the Poisson distribution function for (x,par) The Poisson PDF is implemented by means of Euler's Gamma-function (for the factorial), so for any x integer argument it is correct.

BUT for non-integer x values, it IS NOT equal to the Poisson distribution. see TMath::PoissonI to get a non-smooth function. Note that for large values of par, it is better to call

TMath::Gaus(x,par,sqrt(par),kTRUE)


Definition at line 571 of file TMath.cxx.

## ◆ PoissonI()

 Double_t TMath::PoissonI ( Double_t x, Double_t par )

compute the Poisson distribution function for (x,par) This is a non-smooth function.

This function is equivalent to ROOT::Math::poisson_pdf

Definition at line 602 of file TMath.cxx.

## ◆ Power() [1/5]

 LongDouble_t TMath::Power ( LongDouble_t x, LongDouble_t y )
inline

Definition at line 501 of file TMath.h.

## ◆ Power() [2/5]

 LongDouble_t TMath::Power ( LongDouble_t x, Long64_t y )
inline

Definition at line 504 of file TMath.h.

## ◆ Power() [3/5]

 LongDouble_t TMath::Power ( Long64_t x, Long64_t y )
inline

Definition at line 507 of file TMath.h.

## ◆ Power() [4/5]

 Double_t TMath::Power ( Double_t x, Double_t y )
inline

Definition at line 514 of file TMath.h.

## ◆ Power() [5/5]

 Double_t TMath::Power ( Double_t x, Int_t y )
inline

Definition at line 517 of file TMath.h.

## ◆ Prob()

 Double_t TMath::Prob ( Double_t chi2, Int_t ndf )

Computation of the probability for a certain Chi-squared (chi2) and number of degrees of freedom (ndf).

Calculations are based on the incomplete gamma function P(a,x), where a=ndf/2 and x=chi2/2.

P(a,x) represents the probability that the observed Chi-squared for a correct model should be less than the value chi2.

The returned probability corresponds to 1-P(a,x), which denotes the probability that an observed Chi-squared exceeds the value chi2 by chance, even for a correct model.

— NvE 14-nov-1998 UU-SAP Utrecht

Definition at line 624 of file TMath.cxx.

## ◆ Qe()

 Double_t TMath::Qe ( )
inline

Definition at line 125 of file TMath.h.

## ◆ QeUncertainty()

 Double_t TMath::QeUncertainty ( )
inline

Definition at line 126 of file TMath.h.

## ◆ Quantiles()

 void TMath::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 )

Computes sample quantiles, corresponding to the given probabilities Parameters: x -the data sample n - its size quantiles - computed quantiles are returned in there prob - probabilities where to compute quantiles nprob - size of prob array isSorted - is the input array x sorted? NOTE, that when the input is not sorted, an array of integers of size n needs to be allocated.

It can be passed by the user in parameter index, or, if not passed, it will be allocated inside the function

type - method to compute (from 1 to 9). Following types are provided: Discontinuous: type=1 - inverse of the empirical distribution function type=2 - like type 1, but with averaging at discontinuities type=3 - SAS definition: nearest even order statistic Piecwise linear continuous: In this case, sample quantiles can be obtained by linear interpolation between the k-th order statistic and p(k). type=4 - linear interpolation of empirical cdf, p(k)=k/n; type=5 - a very popular definition, p(k) = (k-0.5)/n; type=6 - used by Minitab and SPSS, p(k) = k/(n+1); type=7 - used by S-Plus and R, p(k) = (k-1)/(n-1); type=8 - resulting sample quantiles are approximately median unbiased regardless of the distribution of x. p(k) = (k-1/3)/(n+1/3); type=9 - resulting sample quantiles are approximately unbiased, when the sample comes from Normal distribution. p(k)=(k-3/8)/(n+1/4);

default type = 7

References: 1) Hyndman, R.J and Fan, Y, (1996) "Sample quantiles in statistical packages" American Statistician, 50, 361-365 2) R Project documentation for the function quantile of package {stats}

Definition at line 1177 of file TMath.cxx.

## ◆ QuietNaN()

 Double_t TMath::QuietNaN ( )
inline

Definition at line 622 of file TMath.h.

## ◆ R()

 Double_t TMath::R ( )
inline

Definition at line 109 of file TMath.h.

inline

Definition at line 49 of file TMath.h.

## ◆ Range() [1/5]

 Short_t TMath::Range ( Short_t lb, Short_t ub, Short_t x )
inline

Definition at line 234 of file TMathBase.h.

## ◆ Range() [2/5]

 Int_t TMath::Range ( Int_t lb, Int_t ub, Int_t x )
inline

Definition at line 237 of file TMathBase.h.

## ◆ Range() [3/5]

 Long_t TMath::Range ( Long_t lb, Long_t ub, Long_t x )
inline

Definition at line 240 of file TMathBase.h.

## ◆ Range() [4/5]

 ULong_t TMath::Range ( ULong_t lb, ULong_t ub, ULong_t x )
inline

Definition at line 243 of file TMathBase.h.

## ◆ Range() [5/5]

 Double_t TMath::Range ( Double_t lb, Double_t ub, Double_t x )
inline

Definition at line 246 of file TMathBase.h.

## ◆ Rgair()

 Double_t TMath::Rgair ( )
inline

Definition at line 119 of file TMath.h.

## ◆ RMS() [1/3]

template<typename T >
 Double_t TMath::RMS ( Long64_t n, const T * a, const Double_t * w = 0 )

Definition at line 903 of file TMath.h.

## ◆ RMS() [2/3]

template<typename Iterator >
 Double_t TMath::RMS ( Iterator first, Iterator last )

Definition at line 851 of file TMath.h.

## ◆ RMS() [3/3]

template<typename Iterator , typename WeightIterator >
 Double_t TMath::RMS ( Iterator first, Iterator last, WeightIterator wfirst )

Definition at line 876 of file TMath.h.

## ◆ RootsCubic()

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

Calculates roots of polynomial of 3rd order a*x^3 + b*x^2 + c*x + d, where a == coef[3], b == coef[2], c == coef[1], d == coef[0] coef[3] must be different from 0 If the boolean returned by the method is false: ==> there are 3 real roots a,b,c If the boolean returned by the method is true: ==> there is one real root a and 2 complex conjugates roots (b+i*c,b-i*c) Author: Francois-Xavier Gentit.

Definition at line 1081 of file TMath.cxx.

## ◆ RUncertainty()

 Double_t TMath::RUncertainty ( )
inline

Definition at line 110 of file TMath.h.

## ◆ Sigma()

 Double_t TMath::Sigma ( )
inline

Definition at line 100 of file TMath.h.

## ◆ SigmaUncertainty()

 Double_t TMath::SigmaUncertainty ( )
inline

Definition at line 101 of file TMath.h.

## ◆ Sign() [1/4]

template<typename T1 , typename T2 >
 T1 TMath::Sign ( T1 a, T2 b )
inline

Definition at line 155 of file TMathBase.h.

## ◆ Sign() [2/4]

 Float_t TMath::Sign ( Float_t a, Float_t b )
inline

Definition at line 158 of file TMathBase.h.

## ◆ Sign() [3/4]

 Double_t TMath::Sign ( Double_t a, Double_t b )
inline

Definition at line 161 of file TMathBase.h.

## ◆ Sign() [4/4]

 LongDouble_t TMath::Sign ( LongDouble_t a, LongDouble_t b )
inline

Definition at line 164 of file TMathBase.h.

## ◆ SignalingNaN()

 Double_t TMath::SignalingNaN ( )
inline

Definition at line 629 of file TMath.h.

## ◆ SignBit() [1/4]

template<typename Integer >
 Bool_t TMath::SignBit ( Integer a )
inline

Definition at line 139 of file TMathBase.h.

## ◆ SignBit() [2/4]

 Bool_t TMath::SignBit ( Float_t a )
inline

Definition at line 142 of file TMathBase.h.

## ◆ SignBit() [3/4]

 Bool_t TMath::SignBit ( Double_t a )
inline

Definition at line 145 of file TMathBase.h.

## ◆ SignBit() [4/4]

 Bool_t TMath::SignBit ( LongDouble_t a )
inline

Definition at line 148 of file TMathBase.h.

## ◆ Sin()

 Double_t TMath::Sin ( Double_t x )
inline

Definition at line 421 of file TMath.h.

## ◆ SinH()

 Double_t TMath::SinH ( Double_t x )
inline

Definition at line 430 of file TMath.h.

## ◆ Sort()

template<typename Element , typename Index >
 void TMath::Sort ( Index n, const Element * a, Index * index, Bool_t down = kTRUE )

Definition at line 989 of file TMath.h.

## ◆ SortItr()

template<typename Iterator , typename IndexIterator >
 void TMath::SortItr ( Iterator first, Iterator last, IndexIterator index, Bool_t down = kTRUE )

Definition at line 964 of file TMath.h.

## ◆ Sq()

 Double_t TMath::Sq ( Double_t x )
inline

Definition at line 461 of file TMath.h.

## ◆ Sqrt()

 Double_t TMath::Sqrt ( Double_t x )
inline

Definition at line 464 of file TMath.h.

## ◆ Sqrt2()

 Double_t TMath::Sqrt2 ( )
inline

Definition at line 51 of file TMath.h.

## ◆ StdDev() [1/3]

template<typename T >
 Double_t TMath::StdDev ( Long64_t n, const T * a, const Double_t * w = 0 )

Definition at line 311 of file TMath.h.

## ◆ StdDev() [2/3]

template<typename Iterator >
 Double_t TMath::StdDev ( Iterator first, Iterator last )

Definition at line 312 of file TMath.h.

## ◆ StdDev() [3/3]

template<typename Iterator , typename WeightIterator >
 Double_t TMath::StdDev ( Iterator first, Iterator last, WeightIterator wfirst )

Definition at line 313 of file TMath.h.

## ◆ StruveH0()

 Double_t TMath::StruveH0 ( Double_t x )

Struve Functions of Order 0.

Converted from CERNLIB M342 by Rene Brun.

Definition at line 1744 of file TMath.cxx.

## ◆ StruveH1()

 Double_t TMath::StruveH1 ( Double_t x )

Struve Functions of Order 1.

Converted from CERNLIB M342 by Rene Brun.

Definition at line 1813 of file TMath.cxx.

## ◆ StruveL0()

 Double_t TMath::StruveL0 ( Double_t x )

Modified Struve Function of Order 0.

By Kirill Filimonov.

Definition at line 1890 of file TMath.cxx.

## ◆ StruveL1()

 Double_t TMath::StruveL1 ( Double_t x )

Modified Struve Function of Order 1.

By Kirill Filimonov.

Definition at line 1936 of file TMath.cxx.

## ◆ Student()

 Double_t TMath::Student ( Double_t T, Double_t ndf )

Computes density function for Student's t- distribution (the probability function (integral of density) is computed in StudentI).

First parameter stands for x - the actual variable of the density function p(x) and the point at which the density is calculated. Second parameter stands for number of degrees of freedom.

About Student distribution: Student's t-distribution is used for many significance tests, for example, for the Student's t-tests for the statistical significance of difference between two sample means and for confidence intervals for the difference between two population means.

Example: suppose we have a random sample of size n drawn from normal distribution with mean Mu and st.deviation Sigma. Then the variable

t = (sample_mean - Mu)/(sample_deviation / sqrt(n))

has Student's t-distribution with n-1 degrees of freedom.

NOTE that this function's second argument is number of degrees of freedom, not the sample size.

As the number of degrees of freedom grows, t-distribution approaches Normal(0,1) distribution. Implementation by Anna Kreshuk.

Definition at line 2566 of file TMath.cxx.

## ◆ StudentI()

 Double_t TMath::StudentI ( Double_t T, Double_t ndf )

Calculates the cumulative distribution function of Student's t-distribution second parameter stands for number of degrees of freedom, not for the number of samples if x has Student's t-distribution, the function returns the probability of x being less than T.

Implementation by Anna Kreshuk.

Definition at line 2587 of file TMath.cxx.

## ◆ StudentQuantile()

 Double_t TMath::StudentQuantile ( Double_t p, Double_t ndf, Bool_t lower_tail = kTRUE )

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.

Definition at line 2611 of file TMath.cxx.

## ◆ Tan()

 Double_t TMath::Tan ( Double_t x )
inline

Definition at line 427 of file TMath.h.

## ◆ TanH()

 Double_t TMath::TanH ( Double_t x )
inline

Definition at line 436 of file TMath.h.

## ◆ TwoPi()

 Double_t TMath::TwoPi ( )
inline

Definition at line 45 of file TMath.h.

## ◆ Vavilov()

 Double_t TMath::Vavilov ( Double_t x, Double_t kappa, Double_t beta2 )

Returns the value of the Vavilov density function Parameters: 1st - the point were the density function is evaluated 2nd - value of kappa (distribution parameter) 3rd - value of beta2 (distribution parameter) The algorithm was taken from the CernLib function vavden(G115) Reference: A.Rotondi and P.Montagna, Fast Calculation of Vavilov distribution Nucl.Instr.

and Meth. B47(1990), 215-224 Accuracy: quote from the reference above: "The resuls of our code have been compared with the values of the Vavilov density function computed numerically in an accurate way: our approximation shows a difference of less than 3% around the peak of the density function, slowly increasing going towards the extreme tails to the right and to the left"

Definition at line 2708 of file TMath.cxx.

## ◆ VavilovDenEval()

 Double_t TMath::VavilovDenEval ( Double_t rlam, Double_t * AC, Double_t * HC, Int_t itype )

Internal function, called by Vavilov and VavilovSet.

Definition at line 3075 of file TMath.cxx.

## ◆ VavilovI()

 Double_t TMath::VavilovI ( Double_t x, Double_t kappa, Double_t beta2 )

Returns the value of the Vavilov distribution function Parameters: 1st - the point were the density function is evaluated 2nd - value of kappa (distribution parameter) 3rd - value of beta2 (distribution parameter) The algorithm was taken from the CernLib function vavden(G115) Reference: A.Rotondi and P.Montagna, Fast Calculation of Vavilov distribution Nucl.Instr.

and Meth. B47(1990), 215-224 Accuracy: quote from the reference above: "The resuls of our code have been compared with the values of the Vavilov density function computed numerically in an accurate way: our approximation shows a difference of less than 3% around the peak of the density function, slowly increasing going towards the extreme tails to the right and to the left"

Definition at line 2736 of file TMath.cxx.

## ◆ VavilovSet()

 void TMath::VavilovSet ( Double_t rkappa, Double_t beta2, Bool_t mode, Double_t * WCM, Double_t * AC, Double_t * HC, Int_t & itype, Int_t & npt )

Internal function, called by Vavilov and VavilovI.

Definition at line 2774 of file TMath.cxx.

## ◆ Voigt()

 Double_t TMath::Voigt ( Double_t xx, Double_t sigma, Double_t lg, Int_t r = 4 )

Computation of Voigt function (normalised).

Voigt is a convolution of gauss(xx) = 1/(sqrt(2*pi)*sigma) * exp(xx*xx/(2*sigma*sigma) and lorentz(xx) = (1/pi) * (lg/2) / (xx*xx + lg*lg/4) functions.

The Voigt function is known to be the real part of Faddeeva function also called complex error function [2].

The algoritm was developed by J. Humlicek [1]. This code is based on fortran code presented by R. J. Wells [2]. Translated and adapted by Miha D. Puc

To calculate the Faddeeva function with relative error less than 10^(-r). r can be set by the the user subject to the constraints 2 <= r <= 5.

[1] J. Humlicek, JQSRT, 21, 437 (1982). [2] R.J. Wells "Rapid Approximation to the Voigt/Faddeeva Function and its Derivatives" JQSRT 62 (1999), pp 29-48. http://www-atm.physics.ox.ac.uk/user/wells/voigt.html

Definition at line 879 of file TMath.cxx.