Classes
struct	Limits

Functions
Double_t	Abs (Double_t d)

Float_t	Abs (Float_t d)

Int_t	Abs (Int_t d)

Long64_t	Abs (Long64_t d)

Long_t	Abs (Long_t d)

LongDouble_t	Abs (LongDouble_t d)

Short_t	Abs (Short_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 y, Double_t x)

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,... and any real x. More...

Double_t	BesselI0 (Double_t x)
	integer order modified Bessel function K_n(x) More...

Double_t	BesselI1 (Double_t x)
	modified Bessel function K_0(x) More...

Double_t	BesselJ0 (Double_t x)
	modified Bessel function K_1(x) More...

Double_t	BesselJ1 (Double_t x)
	Bessel function J0(x) for any real x. More...

Double_t	BesselK (Int_t n, Double_t x)
	integer order modified Bessel function I_n(x) More...

Double_t	BesselK0 (Double_t x)
	modified Bessel function I_0(x) More...

Double_t	BesselK1 (Double_t x)
	modified Bessel function I_1(x) More...

Double_t	BesselY0 (Double_t x)
	Bessel function J1(x) for any real x. More...

Double_t	BesselY1 (Double_t x)
	Bessel function Y0(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 Iterator , typename Element >
Iterator	BinarySearch (Iterator first, Iterator last, Element value)

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)

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...

constexpr Double_t	C ()
	Velocity of light in \( m s^{-1} \). More...

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) More...

constexpr Double_t	Ccgs ()
	\( cm s^{-1} \) More...

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])
	Calculate the Cross Product of two vectors: out = [v1 x v2]. More...

constexpr Double_t	CUncertainty ()
	Speed of light uncertainty. More...

constexpr Double_t	DegToRad ()
	Conversion from degree to radian: More...

Double_t	DiLog (Double_t x)
	Modified Struve functions of order 1. More...

constexpr Double_t	E ()
	Base of natural log: More...

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)
	returns the inverse of the complementary error function x must be 0<x<2 implement using the quantile of the normal distribution instead of ErfInverse for better numerical precision for large x More...

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

constexpr Double_t	EulerGamma ()
	Euler-Mascheroni Constant. More...

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)
	Check if it is finite with a mask in order to be consistent in presence of fast math. More...

Int_t	Finite (Float_t x)
	Check if it is finite with a mask in order to be consistent in presence of fast math. More...

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...

constexpr Double_t	G ()
	Gravitational constant in: \( m^{3} kg^{-1} s^{-2} \). More...

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

constexpr Double_t	Gcgs ()
	\( cm^{3} g^{-1} s^{-2} \) More...

template<typename Iterator >
Double_t	GeomMean (Iterator first, Iterator last)
	Return the geometric mean of an array defined by the iterators. More...

template<typename T >
Double_t	GeomMean (Long64_t n, const T *a)
	Return the geometric mean of an array a of size n. More...

constexpr Double_t	GhbarC ()
	\( \frac{G}{\hbar C} \) in \( (GeV/c^{2})^{-2} \) More...

constexpr Double_t	GhbarCUncertainty ()
	\( \frac{G}{\hbar C} \) uncertainty. More...

constexpr Double_t	Gn ()
	Standard acceleration of gravity in \( m s^{-2} \). More...

constexpr Double_t	GnUncertainty ()
	Standard acceleration of gravity uncertainty. More...

constexpr Double_t	GUncertainty ()
	Gravitational constant uncertainty. More...

constexpr Double_t	H ()
	Planck's constant in \( J s \). More...

ULong_t	Hash (const char *str)

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

constexpr Double_t	Hbar ()
	\( \hbar \) in \( J s \) More...

constexpr Double_t	Hbarcgs ()
	\( erg s \) More...

constexpr Double_t	HbarUncertainty ()
	\( \hbar \) uncertainty. More...

constexpr Double_t	HC ()
	\( hc \) in \( J m \) More...

constexpr Double_t	HCcgs ()
	\( erg cm \) More...

constexpr Double_t	Hcgs ()
	\( erg s \) More...

constexpr Double_t	HUncertainty ()
	Planck's constant uncertainty. More...

Double_t	Hypot (Double_t x, Double_t y)

Long_t	Hypot (Long_t x, Long_t y)

Double_t	Infinity ()
	Returns an infinity as defined by the IEEE standard. More...

constexpr Double_t	InvPi ()
	\( \frac{1.}{\pi}\) More...

template<typename T >
Bool_t	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. More...

Bool_t	IsNaN (Double_t x)

Bool_t	IsNaN (Float_t x)

constexpr Double_t	K ()
	Boltzmann's constant in \( J K^{-1} \). More...

constexpr Double_t	Kcgs ()
	\( erg K^{-1} \) More...

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)
	Returns k_th order statistic of the array a of size n (k_th smallest element out of n elements). More...

constexpr Double_t	KUncertainty ()
	Boltzmann's constant uncertainty. More...

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)

constexpr Double_t	Ln10 ()
	Natural log of 10 (to convert log to ln) More...

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

template<typename Iterator >
Iterator	LocMax (Iterator first, Iterator last)
	Return index of array with the maximum element. More...

template<typename T >
Long64_t	LocMax (Long64_t n, const T *a)
	Return index of array with the maximum element. More...

template<typename Iterator >
Iterator	LocMin (Iterator first, Iterator last)
	Return index of array with the minimum element. More...

template<typename T >
Long64_t	LocMin (Long64_t n, const T *a)
	Return index of array with the minimum element. More...

Double_t	Log (Double_t x)

Double_t	Log10 (Double_t x)

Double_t	Log2 (Double_t x)

constexpr Double_t	LogE ()
	Base-10 log of e (to convert ln to log) More...

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...

Double_t	Max (Double_t a, Double_t b)

Float_t	Max (Float_t a, Float_t b)

Int_t	Max (Int_t a, Int_t b)

Long64_t	Max (Long64_t a, Long64_t b)

Long_t	Max (Long_t a, Long_t b)

Short_t	Max (Short_t a, Short_t b)

UInt_t	Max (UInt_t a, UInt_t b)

ULong64_t	Max (ULong64_t a, ULong64_t b)

ULong_t	Max (ULong_t a, ULong_t b)

UShort_t	Max (UShort_t a, UShort_t b)

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

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

template<typename Iterator , typename WeightIterator >
Double_t	Mean (Iterator first, Iterator last, WeightIterator wfirst)
	Return the weighted mean of an array defined by the first and last iterators. More...

template<typename T >
Double_t	Mean (Long64_t n, const T a, const Double_t w=0)
	Return the weighted mean of an array a with length n. More...

template<typename T >
Double_t	Median (Long64_t n, const T a, const Double_t w=0, Long64_t *work=0)
	Return the median of the array a where each entry i has weight w[i] . More...

Double_t	Min (Double_t a, Double_t b)

Float_t	Min (Float_t a, Float_t b)

Int_t	Min (Int_t a, Int_t b)

Long64_t	Min (Long64_t a, Long64_t b)

Long_t	Min (Long_t a, Long_t b)

Short_t	Min (Short_t a, Short_t b)

UInt_t	Min (UInt_t a, UInt_t b)

ULong64_t	Min (ULong64_t a, ULong64_t b)

ULong_t	Min (ULong_t a, ULong_t b)

UShort_t	Min (UShort_t a, UShort_t b)

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

constexpr Double_t	MWair ()
	Molecular weight of dry air 1976 US Standard Atmosphere in \( kg kmol^{-1} \) or \( gm mol^{-1} \) More...

constexpr Double_t	Na ()
	Avogadro constant (Avogadro's Number) in \( mol^{-1} \). More...

constexpr Double_t	NaUncertainty ()
	Avogadro constant (Avogadro's Number) uncertainty. More...

Long_t	NextPrime (Long_t x)
	TMath Base functions. More...

template<typename T >
Int_t	Nint (T x)
	Round to nearest integer. Rounds half integers to the nearest even integer. More...

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

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

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

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

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

constexpr Double_t	Pi ()

constexpr Double_t	PiOver2 ()

constexpr Double_t	PiOver4 ()

Double_t	Poisson (Double_t x, Double_t par)
	Compute the Poisson distribution function for (x,par). More...

Double_t	PoissonI (Double_t x, Double_t par)
	Compute the Discrete Poisson distribution function for (x,par). More...

Double_t	Power (Double_t x, Double_t y)

Double_t	Power (Double_t x, Int_t y)

LongDouble_t	Power (Long64_t x, Long64_t y)

LongDouble_t	Power (LongDouble_t x, Long64_t y)

LongDouble_t	Power (LongDouble_t x, LongDouble_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...

constexpr Double_t	Qe ()
	Elementary charge in \( C \) . More...

constexpr Double_t	QeUncertainty ()
	Elementary charge uncertainty. More...

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. More...

Double_t	QuietNaN ()
	Returns a quiet NaN as defined by IEEE 754 More...

constexpr Double_t	R ()
	Universal gas constant ( \( Na K \)) in \( J K^{-1} mol^{-1} \) More...

constexpr Double_t	RadToDeg ()
	Conversion from radian to degree: More...

Double_t	Range (Double_t lb, Double_t ub, Double_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)

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

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

constexpr Double_t	Rgair ()
	Dry Air Gas Constant (R / MWair) in \( J kg^{-1} K^{-1} \) More...

template<typename Iterator >
Double_t	RMS (Iterator first, Iterator last)
	Return the Standard Deviation of an array defined by the iterators. More...

template<typename Iterator , typename WeightIterator >
Double_t	RMS (Iterator first, Iterator last, WeightIterator wfirst)
	Return the weighted Standard Deviation of an array defined by the iterators. More...

template<typename T >
Double_t	RMS (Long64_t n, const T a, const Double_t w=0)
	Return the Standard Deviation of an array a with length n. More...

Bool_t	RootsCubic (const Double_t coef[4], Double_t &a, Double_t &b, Double_t &c)
	Calculates roots of polynomial of 3rd order ax^3 + bx^2 + c*x + d, where. More...

constexpr Double_t	RUncertainty ()
	Universal gas constant uncertainty. More...

constexpr Double_t	Sigma ()
	Stefan-Boltzmann constant in \( W m^{-2} K^{-4}\). More...

constexpr Double_t	SigmaUncertainty ()
	Stefan-Boltzmann constant uncertainty. More...

Double_t	Sign (Double_t a, Double_t b)

Float_t	Sign (Float_t a, Float_t b)

LongDouble_t	Sign (LongDouble_t a, LongDouble_t b)

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

Double_t	SignalingNaN ()
	Returns a signaling NaN as defined by IEEE 754](http://en.wikipedia.org/wiki/NaN#Signaling_NaN) More...

Bool_t	SignBit (Double_t a)

Bool_t	SignBit (Float_t a)

template<typename Integer >
Bool_t	SignBit (Integer 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)

constexpr Double_t	Sqrt2 ()

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

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

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

Double_t	StruveH0 (Double_t x)
	Bessel function Y1(x) for positive x. More...

Double_t	StruveH1 (Double_t x)
	Struve functions of order 0. More...

Double_t	StruveL0 (Double_t x)
	Struve functions of order 1. More...

Double_t	StruveL1 (Double_t x)
	Modified Struve functions of order 0. 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. More...

Double_t	Tan (Double_t)

Double_t	TanH (Double_t)

constexpr Double_t	TwoPi ()

Double_t	Vavilov (Double_t x, Double_t kappa, Double_t beta2)
	Returns the value of the Vavilov density function. 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. 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...

Detailed Description

TMath.

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

Function Documentation

◆ Abs() [1/7]

Double_t TMath::Abs ( Double_t d )

inline

Definition at line 139 of file TMathBase.h.

◆ Abs() [2/7]

Float_t TMath::Abs ( Float_t d )

inline

Definition at line 136 of file TMathBase.h.

◆ Abs() [3/7]

Int_t TMath::Abs ( Int_t d )

inline

Definition at line 123 of file TMathBase.h.

◆ Abs() [4/7]

Long64_t TMath::Abs ( Long64_t d )

inline

Definition at line 129 of file TMathBase.h.

◆ Abs() [5/7]

Long_t TMath::Abs ( Long_t d )

inline

Definition at line 126 of file TMathBase.h.

◆ Abs() [6/7]

LongDouble_t TMath::Abs ( LongDouble_t d )

inline

Definition at line 142 of file TMathBase.h.

◆ Abs() [7/7]

Short_t TMath::Abs ( Short_t d )

inline

Definition at line 120 of file TMathBase.h.

◆ ACos()

Double_t TMath::ACos ( Double_t x )

inline

Definition at line 658 of file TMath.h.

◆ ACosH()

Double_t TMath::ACosH ( Double_t x )

Definition at line 77 of file TMath.cxx.

◆ AreEqualAbs()

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

inline

Definition at line 412 of file TMath.h.

◆ AreEqualRel()

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

inline

Definition at line 418 of file TMath.h.

◆ ASin()

Double_t TMath::ASin ( Double_t x )

inline

Definition at line 651 of file TMath.h.

◆ ASinH()

Double_t TMath::ASinH ( Double_t x )

Definition at line 64 of file TMath.cxx.

◆ ATan()

Double_t TMath::ATan ( Double_t x )

inline

Definition at line 665 of file TMath.h.

◆ ATan2()

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

inline

Definition at line 669 of file TMath.h.

◆ ATanH()

Double_t TMath::ATanH ( Double_t x )

Definition at line 90 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.

Author: NvE 12-mar-2000 UU-SAP Utrecht

Definition at line 1565 of file TMath.cxx.

◆ BesselI0()

Double_t TMath::BesselI0 ( Double_t x )

integer order modified Bessel function K_n(x)

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

Author: NvE 12-mar-2000 UU-SAP Utrecht

Definition at line 1401 of file TMath.cxx.

◆ BesselI1()

Double_t TMath::BesselI1 ( Double_t x )

modified Bessel function K_0(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.

Author: NvE 12-mar-2000 UU-SAP Utrecht

Definition at line 1469 of file TMath.cxx.

◆ BesselJ0()

Double_t TMath::BesselJ0 ( Double_t x )

modified Bessel function K_1(x)

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

Definition at line 1609 of file TMath.cxx.

◆ BesselJ1()

Double_t TMath::BesselJ1 ( Double_t x )

Bessel function J0(x) for any real x.

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

Definition at line 1644 of file TMath.cxx.

◆ BesselK()

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

integer order modified Bessel function I_n(x)

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

Author: NvE 12-mar-2000 UU-SAP Utrecht

Definition at line 1536 of file TMath.cxx.

◆ BesselK0()

Double_t TMath::BesselK0 ( Double_t x )

modified Bessel function I_0(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.

Author: NvE 12-mar-2000 UU-SAP Utrecht

Definition at line 1435 of file TMath.cxx.

◆ BesselK1()

Double_t TMath::BesselK1 ( Double_t x )

modified Bessel function I_1(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.

Author: NvE 12-mar-2000 UU-SAP Utrecht

Definition at line 1504 of file TMath.cxx.

◆ BesselY0()

Double_t TMath::BesselY0 ( Double_t x )

Bessel function J1(x) for any real x.

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

Definition at line 1680 of file TMath.cxx.

◆ BesselY1()

Double_t TMath::BesselY1 ( Double_t x )

Bessel function Y0(x) for positive x.

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

Definition at line 1714 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 1984 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 1993 of file TMath.cxx.

◆ BetaDist()

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 2044 of file TMath.cxx.

◆ BetaDistI()

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 2062 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 2075 of file TMath.cxx.

◆ BinarySearch() [1/3]

template<typename Iterator , typename Element >

Iterator TMath::BinarySearch	(	Iterator	first,
		Iterator	last,
		Element	value
	)

Definition at line 260 of file TMathBase.h.

◆ BinarySearch() [2/3]

template<typename T >

Long64_t TMath::BinarySearch	(	Long64_t	n,
		const T **	array,
		T	value
	)

Definition at line 294 of file TMathBase.h.

◆ BinarySearch() [3/3]

template<typename T >

Long64_t TMath::BinarySearch	(	Long64_t	n,
		const T *	array,
		T	value
	)

Definition at line 278 of file TMathBase.h.

◆ Binomial()

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

Calculate the binomial coefficient n over k.

Definition at line 2083 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 2108 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 437 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.

Parameters

[in]	*arr1	is unchanged;
[in]	*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.

Author: Adrian Bevan (bevan.nosp@m.@sla.nosp@m.c.sta.nosp@m.nfor.nosp@m.d.edu)

Definition at line 1289 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.

Author: Adrian Bevan (bevan.nosp@m.@sla.nosp@m.c.sta.nosp@m.nfor.nosp@m.d.edu)

Definition at line 1328 of file TMath.cxx.

◆ C()

constexpr Double_t TMath::C ( )

constexpr

Velocity of light in \( m s^{-1} \).

Definition at line 118 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 2141 of file TMath.cxx.

◆ Ccgs()

constexpr Double_t TMath::Ccgs ( )

constexpr

\( cm s^{-1} \)

Definition at line 125 of file TMath.h.

◆ Ceil()

Double_t TMath::Ceil ( Double_t x )

inline

Definition at line 685 of file TMath.h.

◆ CeilNint()

Int_t TMath::CeilNint ( Double_t x )

inline

Definition at line 689 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

[in]	p	the probability value, at which the quantile is computed
[in]	ndf	number of degrees of freedom

Definition at line 2157 of file TMath.cxx.

◆ Cos()

Double_t TMath::Cos ( Double_t x )

inline

Definition at line 631 of file TMath.h.

◆ CosH()

Double_t TMath::CosH ( Double_t x )

inline

Definition at line 643 of file TMath.h.

◆ Cross()

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].

Definition at line 1165 of file TMath.h.

◆ CUncertainty()

constexpr Double_t TMath::CUncertainty ( )

constexpr

Speed of light uncertainty.

Definition at line 132 of file TMath.h.

◆ DegToRad()

constexpr Double_t TMath::DegToRad ( )

constexpr

Conversion from degree to radian:

\[ \frac{\pi}{180} \]

Definition at line 82 of file TMath.h.

◆ DiLog()

Double_t TMath::DiLog ( Double_t x )

Modified Struve functions of order 1.

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

Definition at line 110 of file TMath.cxx.

◆ E()

constexpr Double_t TMath::E ( )

constexpr

Base of natural log:

\[ e \]

Definition at line 97 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 184 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 194 of file TMath.cxx.

◆ ErfcInverse()

Double_t TMath::ErfcInverse ( Double_t x )

returns the inverse of the complementary error function x must be 0<x<2 implement using the quantile of the normal distribution instead of ErfInverse for better numerical precision for large x

Definition at line 237 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 203 of file TMath.cxx.

◆ EulerGamma()

constexpr Double_t TMath::EulerGamma ( )

constexpr

Euler-Mascheroni Constant.

Definition at line 329 of file TMath.h.

◆ Even()

Bool_t TMath::Even ( Long_t a )

inline

Definition at line 112 of file TMathBase.h.

◆ Exp()

Double_t TMath::Exp ( Double_t x )

inline

Definition at line 717 of file TMath.h.

◆ Factorial()

Double_t TMath::Factorial ( Int_t i )

Compute factorial(n).

Definition at line 247 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 2240 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 2259 of file TMath.cxx.

◆ Finite() [1/2]

Int_t TMath::Finite ( Double_t x )

inline

Check if it is finite with a mask in order to be consistent in presence of fast math.

Inspired from the CMSSW FWCore/Utilities package

Definition at line 761 of file TMath.h.

◆ Finite() [2/2]

Int_t TMath::Finite ( Float_t x )

inline

Check if it is finite with a mask in order to be consistent in presence of fast math.

Inspired from the CMSSW FWCore/Utilities package

Definition at line 790 of file TMath.h.

◆ Floor()

Double_t TMath::Floor ( Double_t x )

inline

Definition at line 693 of file TMath.h.

◆ FloorNint()

Int_t TMath::FloorNint ( Double_t x )

inline

Definition at line 697 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 265 of file TMath.cxx.

◆ G()

constexpr Double_t TMath::G ( )

constexpr

Gravitational constant in: \( m^{3} kg^{-1} s^{-2} \).

Definition at line 139 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.

Author: NvE 14-nov-1998 UU-SAP Utrecht

Definition at line 375 of file TMath.cxx.

◆ Gamma() [1/2]

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 .

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

Author: NvE 14-nov-1998 UU-SAP Utrecht

Definition at line 364 of file TMath.cxx.

◆ Gamma() [2/2]

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 348 of file TMath.cxx.

◆ GammaDist()

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.

Parameters

[in]	gamma	shape parameter
[in]	mu	location parameter
[in]	beta	scale parameter

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

Definition at line 2308 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.

Author: NvE 14-nov-1998 UU-SAP Utrecht

Definition at line 412 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 448 of file TMath.cxx.

◆ Gcgs()

constexpr Double_t TMath::Gcgs ( )

constexpr

\( cm^{3} g^{-1} s^{-2} \)

Definition at line 146 of file TMath.h.

◆ GeomMean() [1/2]

template<typename Iterator >

Double_t TMath::GeomMean	(	Iterator	first,
		Iterator	last
	)

Return the geometric mean of an array defined by the iterators.

\[ GeomMean = (\prod_{i=0}^{n-1} |a[i]|)^{1/n} \]

Definition at line 1076 of file TMath.h.

◆ GeomMean() [2/2]

template<typename T >

Double_t TMath::GeomMean	(	Long64_t	n,
		const T *	a
	)

Return the geometric mean of an array a of size n.

\[ GeomMean = (\prod_{i=0}^{n-1} |a[i]|)^{1/n} \]

Definition at line 1095 of file TMath.h.

◆ GhbarC()

constexpr Double_t TMath::GhbarC ( )

constexpr

\( \frac{G}{\hbar C} \) in \( (GeV/c^{2})^{-2} \)

Definition at line 160 of file TMath.h.

◆ GhbarCUncertainty()

constexpr Double_t TMath::GhbarCUncertainty ( )

constexpr

\( \frac{G}{\hbar C} \) uncertainty.

Definition at line 167 of file TMath.h.

◆ Gn()

constexpr Double_t TMath::Gn ( )

constexpr

Standard acceleration of gravity in \( m s^{-2} \).

Definition at line 174 of file TMath.h.

◆ GnUncertainty()

constexpr Double_t TMath::GnUncertainty ( )

constexpr

Standard acceleration of gravity uncertainty.

Definition at line 181 of file TMath.h.

◆ GUncertainty()

constexpr Double_t TMath::GUncertainty ( )

constexpr

Gravitational constant uncertainty.

Definition at line 153 of file TMath.h.

◆ H()

constexpr Double_t TMath::H ( )

constexpr

Planck's constant in \( J s \).

\[ h \]

Definition at line 189 of file TMath.h.

◆ Hash() [1/2]

ULong_t TMath::Hash ( const char * str )

Definition at line 1391 of file TMath.cxx.

◆ Hash() [2/2]

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 1383 of file TMath.cxx.

◆ Hbar()

constexpr Double_t TMath::Hbar ( )

constexpr

\( \hbar \) in \( J s \)

\[ \hbar = \frac{h}{2\pi} \]

Definition at line 211 of file TMath.h.

◆ Hbarcgs()

constexpr Double_t TMath::Hbarcgs ( )

constexpr

\( erg s \)

Definition at line 218 of file TMath.h.

◆ HbarUncertainty()

constexpr Double_t TMath::HbarUncertainty ( )

constexpr

\( \hbar \) uncertainty.

Definition at line 225 of file TMath.h.

◆ HC()

constexpr Double_t TMath::HC ( )

constexpr

\( hc \) in \( J m \)

Definition at line 232 of file TMath.h.

◆ HCcgs()

constexpr Double_t TMath::HCcgs ( )

constexpr

\( erg cm \)

Definition at line 239 of file TMath.h.

◆ Hcgs()

constexpr Double_t TMath::Hcgs ( )

constexpr

\( erg s \)

Definition at line 196 of file TMath.h.

◆ HUncertainty()

constexpr Double_t TMath::HUncertainty ( )

constexpr

Planck's constant uncertainty.

Definition at line 203 of file TMath.h.

◆ Hypot() [1/2]

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

Definition at line 57 of file TMath.cxx.

◆ Hypot() [2/2]

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

Definition at line 50 of file TMath.cxx.

◆ Infinity()

Double_t TMath::Infinity ( )

inline

Returns an infinity as defined by the IEEE standard.

Definition at line 904 of file TMath.h.

◆ InvPi()

constexpr Double_t TMath::InvPi ( )

constexpr

\( \frac{1.}{\pi}\)

Definition at line 66 of file TMath.h.

◆ IsInside()

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.

Definition at line 1199 of file TMath.h.

◆ IsNaN() [1/2]

Bool_t TMath::IsNaN ( Double_t x )

inline

Definition at line 882 of file TMath.h.

◆ IsNaN() [2/2]

Bool_t TMath::IsNaN ( Float_t x )

inline

Definition at line 883 of file TMath.h.

◆ K()

constexpr Double_t TMath::K ( )

constexpr

Boltzmann's constant in \( J K^{-1} \).

\[ k \]

Definition at line 247 of file TMath.h.

◆ Kcgs()

constexpr Double_t TMath::Kcgs ( )

constexpr

\( erg K^{-1} \)

Definition at line 254 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: - \_form#603, and - \_form#604 is the maximum deviation between a hypothetical distribution function and an experimental distribution with - \_form#318 events NOTE: To compare two experimental distributions with m and n events, use \_form#605 Accuracy: The function is far too accurate for any imaginable application. Probabilities less than \_form#606 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 656 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;}
}

Note:

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 782 of file TMath.cxx.

◆ KOrdStat()

template<class Element , typename Size >

Element TMath::KOrdStat	(	Size	n,
		const Element *	a,
		Size	k,
		Size *	work = `0`
	)

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. Note that the work index array will not contain the sorted indices but all indices of the smaller element in arbitrary order in work[0,...,k-1] and all indices of the larger element in arbitrary order in work[k+1,..,n-1] work[k] will contain instead the index of the returned element.

Taken from "Numerical Recipes in C++" without the index array implemented by Anna Khreshuk.

See also the declarations at the top of this file

Definition at line 1323 of file TMath.h.

◆ KUncertainty()

constexpr Double_t TMath::KUncertainty ( )

constexpr

Boltzmann's constant uncertainty.

Definition at line 261 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 469 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 2796 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 2325 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 2341 of file TMath.cxx.

◆ Ldexp()

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

inline

Definition at line 721 of file TMath.h.

◆ Ln10()

constexpr Double_t TMath::Ln10 ( )

constexpr

Natural log of 10 (to convert log to ln)

Definition at line 104 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.

Author: NvE 14-nov-1998 UU-SAP Utrecht

Definition at line 486 of file TMath.cxx.

◆ LocMax() [1/2]

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.

Definition at line 1007 of file TMath.h.

◆ LocMax() [2/2]

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)

Definition at line 990 of file TMath.h.

◆ LocMin() [1/2]

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.

Definition at line 979 of file TMath.h.

◆ LocMin() [2/2]

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

Definition at line 962 of file TMath.h.

◆ Log()

Double_t TMath::Log ( Double_t x )

inline

Definition at line 750 of file TMath.h.

◆ Log10()

Double_t TMath::Log10 ( Double_t x )

inline

Definition at line 754 of file TMath.h.

◆ Log2()

Double_t TMath::Log2 ( Double_t x )

Definition at line 101 of file TMath.cxx.

◆ LogE()

constexpr Double_t TMath::LogE ( )

constexpr

Base-10 log of e (to convert ln to log)

Definition at line 111 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

Parameters

[in]	sigma	is the shape parameter
[in]	theta	is the location parameter
[in]	m	is the scale parameter

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

Definition at line 2397 of file TMath.cxx.

◆ Max() [1/10]

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

inline

Definition at line 239 of file TMathBase.h.

◆ Max() [2/10]

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

inline

Definition at line 236 of file TMathBase.h.

◆ Max() [3/10]

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

inline

Definition at line 218 of file TMathBase.h.

◆ Max() [4/10]

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

inline

Definition at line 230 of file TMathBase.h.

◆ Max() [5/10]

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

inline

Definition at line 224 of file TMathBase.h.

◆ Max() [6/10]

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

inline

Definition at line 212 of file TMathBase.h.

◆ Max() [7/10]

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

inline

Definition at line 221 of file TMathBase.h.

◆ Max() [8/10]

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

inline

Definition at line 233 of file TMathBase.h.

◆ Max() [9/10]

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

inline

Definition at line 227 of file TMathBase.h.

◆ Max() [10/10]

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

inline

Definition at line 215 of file TMathBase.h.

◆ MaxElement()

template<typename T >

T TMath::MaxElement	(	Long64_t	n,
		const T *	a
	)

Return maximum of array a of length n.

Definition at line 949 of file TMath.h.

◆ Mean() [1/3]

template<typename Iterator >

Double_t TMath::Mean	(	Iterator	first,
		Iterator	last
	)

Return the weighted mean of an array defined by the iterators.

Definition at line 1016 of file TMath.h.

◆ Mean() [2/3]

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.

Definition at line 1035 of file TMath.h.

◆ Mean() [3/3]

template<typename T >

Double_t TMath::Mean	(	Long64_t	n,
		const T *	a,
		const Double_t *	w = `0`
	)

Return the weighted mean of an array a with length n.

Definition at line 1063 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`
	)

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 .

Definition at line 1237 of file TMath.h.

◆ Min() [1/10]

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

inline

Definition at line 207 of file TMathBase.h.

◆ Min() [2/10]

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

inline

Definition at line 204 of file TMathBase.h.

◆ Min() [3/10]

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

inline

Definition at line 186 of file TMathBase.h.

◆ Min() [4/10]

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

inline

Definition at line 198 of file TMathBase.h.

◆ Min() [5/10]

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

inline

Definition at line 192 of file TMathBase.h.

◆ Min() [6/10]

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

inline

Definition at line 180 of file TMathBase.h.

◆ Min() [7/10]

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

inline

Definition at line 189 of file TMathBase.h.

◆ Min() [8/10]

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

inline

Definition at line 201 of file TMathBase.h.

◆ Min() [9/10]

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

inline

Definition at line 195 of file TMathBase.h.

◆ Min() [10/10]

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

inline

Definition at line 183 of file TMathBase.h.

◆ MinElement()

template<typename T >

T TMath::MinElement	(	Long64_t	n,
		const T *	a
	)

Return minimum of array a of length n.

Definition at line 942 of file TMath.h.

◆ MWair()

constexpr Double_t TMath::MWair ( )

constexpr

Molecular weight of dry air 1976 US Standard Atmosphere in \( kg kmol^{-1} \) or \( gm mol^{-1} \)

Definition at line 314 of file TMath.h.

◆ Na()

constexpr Double_t TMath::Na ( )

constexpr

Avogadro constant (Avogadro's Number) in \( mol^{-1} \).

Definition at line 283 of file TMath.h.

◆ NaUncertainty()

constexpr Double_t TMath::NaUncertainty ( )

constexpr

Avogadro constant (Avogadro's Number) uncertainty.

Definition at line 290 of file TMath.h.

◆ Nint()

template<typename T >

Int_t TMath::Nint ( T x )

inline

Round to nearest integer. Rounds half integers to the nearest even integer.

Definition at line 703 of file TMath.h.

◆ Normal2Plane()

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.

Parameters

[in]	p1,p2,p3	3 3D points belonged the plane to define it.
[out]	normal	Pointer to 3D normal vector (normalized)

Definition at line 1179 of file TMath.h.

◆ Normalize() [1/2]

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 512 of file TMath.cxx.

◆ Normalize() [2/2]

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

Normalize a vector v in place.

Returns the norm of the original vector.

Definition at line 495 of file TMath.cxx.

◆ NormCross()

template<typename T >

T TMath::NormCross	(	const T	v1[3],
		const T	v2[3],
		T	out[3]
	)

inline

Calculate the Normalized Cross Product of two vectors.

Definition at line 934 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 2416 of file TMath.cxx.

◆ Odd()

Bool_t TMath::Odd ( Long_t a )

inline

Definition at line 115 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 2517 of file TMath.cxx.

◆ Pi()

constexpr Double_t TMath::Pi ( )

constexpr

\[ \pi\]

Definition at line 38 of file TMath.h.

◆ PiOver2()

constexpr Double_t TMath::PiOver2 ( )

constexpr

\[ \frac{\pi}{2} \]

Definition at line 52 of file TMath.h.

◆ PiOver4()

constexpr Double_t TMath::PiOver4 ( )

constexpr

\[ \frac{\pi}{4} \]

Definition at line 59 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 the correct Poisson distribution. BUT for non-integer x values, it IS NOT equal to the Poisson distribution !

Definition at line 564 of file TMath.cxx.

◆ PoissonI()

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

Compute the Discrete Poisson distribution function for (x,par).

This is a discrete and a non-smooth function. This function is equivalent to ROOT::Math::poisson_pdf

Definition at line 592 of file TMath.cxx.

◆ Power() [1/5]

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

inline

Definition at line 737 of file TMath.h.

◆ Power() [2/5]

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

inline

Definition at line 741 of file TMath.h.

◆ Power() [3/5]

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

inline

Definition at line 733 of file TMath.h.

◆ Power() [4/5]

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

inline

Definition at line 729 of file TMath.h.

◆ Power() [5/5]

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

inline

Definition at line 725 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.

Author: NvE 14-nov-1998 UU-SAP Utrecht

Definition at line 614 of file TMath.cxx.

◆ Qe()

constexpr Double_t TMath::Qe ( )

constexpr

Elementary charge in \( C \) .

Definition at line 336 of file TMath.h.

◆ QeUncertainty()

constexpr Double_t TMath::QeUncertainty ( )

constexpr

Elementary charge uncertainty.

Definition at line 343 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

[in]	x	the data sample
[in]	n	its size
[out]	quantiles	computed quantiles are returned in there
[in]	prob	probabilities where to compute quantiles
[in]	nprob	size of prob array
[in]	isSorted	is the input array x sorted ?
[in]	type	method to compute (from 1 to 9).

NOTE:

When the input is not sorted, an array of integers of size n needs to be allocated. It can be passed by the user in parameter index, or, if not passed, it will be allocated inside the function

Following types are provided:

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

References:

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

Definition at line 1183 of file TMath.cxx.

◆ QuietNaN()

Double_t TMath::QuietNaN ( )

inline

Returns a quiet NaN as defined by IEEE 754

Definition at line 891 of file TMath.h.

◆ R()

constexpr Double_t TMath::R ( )

constexpr

Universal gas constant ( \( Na K \)) in \( J K^{-1} mol^{-1} \)

Definition at line 299 of file TMath.h.

◆ RadToDeg()

constexpr Double_t TMath::RadToDeg ( )

constexpr

Conversion from radian to degree:

\[ \frac{180}{\pi} \]

Definition at line 74 of file TMath.h.

◆ Range() [1/5]

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

inline

Definition at line 256 of file TMathBase.h.

◆ Range() [2/5]

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

inline

Definition at line 247 of file TMathBase.h.

◆ Range() [3/5]

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

inline

Definition at line 250 of file TMathBase.h.

◆ Range() [4/5]

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

inline

Definition at line 244 of file TMathBase.h.

◆ Range() [5/5]

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

inline

Definition at line 253 of file TMathBase.h.

◆ Rgair()

constexpr Double_t TMath::Rgair ( )

constexpr

Dry Air Gas Constant (R / MWair) in \( J kg^{-1} K^{-1} \)

Definition at line 322 of file TMath.h.

◆ RMS() [1/3]

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.

Use the two pass algorithm, which is slower (! a factor of 2) but much more precise. Since we have a vector the 2 pass algorithm is still faster than the Welford algorithm. (See also ROOT-5545)

Definition at line 1109 of file TMath.h.

◆ RMS() [2/3]

template<typename Iterator , typename WeightIterator >

Double_t TMath::RMS	(	Iterator	first,
		Iterator	last,
		WeightIterator	w
	)

Return the weighted 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.

As in the unweighted case use the two pass algorithm

Definition at line 1133 of file TMath.h.

◆ RMS() [3/3]

template<typename T >

Double_t TMath::RMS	(	Long64_t	n,
		const T *	a,
		const Double_t *	w = `0`
	)

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.

Definition at line 1157 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 1084 of file TMath.cxx.

◆ RUncertainty()

constexpr Double_t TMath::RUncertainty ( )

constexpr

Universal gas constant uncertainty.

Definition at line 306 of file TMath.h.

◆ Sigma()

constexpr Double_t TMath::Sigma ( )

constexpr

Stefan-Boltzmann constant in \( W m^{-2} K^{-4}\).

\[ \sigma \]

Definition at line 269 of file TMath.h.

◆ SigmaUncertainty()

constexpr Double_t TMath::SigmaUncertainty ( )

constexpr

Stefan-Boltzmann constant uncertainty.

Definition at line 276 of file TMath.h.

◆ Sign() [1/4]

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

inline

Definition at line 171 of file TMathBase.h.

◆ Sign() [2/4]

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

inline

Definition at line 168 of file TMathBase.h.

◆ Sign() [3/4]

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

inline

Definition at line 174 of file TMathBase.h.

◆ Sign() [4/4]

template<typename T1 , typename T2 >

T1 TMath::Sign	(	T1	a,
		T2	b
	)

inline

Definition at line 165 of file TMathBase.h.

◆ SignalingNaN()

Double_t TMath::SignalingNaN ( )

inline

Returns a signaling NaN as defined by IEEE 754](http://en.wikipedia.org/wiki/NaN#Signaling_NaN)

Definition at line 898 of file TMath.h.

◆ SignBit() [1/4]

Bool_t TMath::SignBit ( Double_t a )

inline

Definition at line 155 of file TMathBase.h.

◆ SignBit() [2/4]

Bool_t TMath::SignBit ( Float_t a )

inline

Definition at line 152 of file TMathBase.h.

◆ SignBit() [3/4]

template<typename Integer >

Bool_t TMath::SignBit ( Integer a )

inline

Definition at line 149 of file TMathBase.h.

◆ SignBit() [4/4]

Bool_t TMath::SignBit ( LongDouble_t a )

inline

Definition at line 158 of file TMathBase.h.

◆ Sin()

Double_t TMath::Sin ( Double_t x )

inline

Definition at line 627 of file TMath.h.

◆ SinH()

Double_t TMath::SinH ( Double_t x )

inline

Definition at line 639 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 362 of file TMathBase.h.

◆ SortItr()

template<typename Iterator , typename IndexIterator >

void TMath::SortItr	(	Iterator	first,
		Iterator	last,
		IndexIterator	index,
		Bool_t	down = `kTRUE`
	)

Definition at line 337 of file TMathBase.h.

◆ Sq()

Double_t TMath::Sq ( Double_t x )

inline

Definition at line 677 of file TMath.h.

◆ Sqrt()

Double_t TMath::Sqrt ( Double_t x )

inline

Definition at line 681 of file TMath.h.

◆ Sqrt2()

constexpr Double_t TMath::Sqrt2 ( )

constexpr

\[ \sqrt{2} \]

Definition at line 89 of file TMath.h.

◆ StdDev() [1/3]

template<typename Iterator >

Double_t TMath::StdDev	(	Iterator	first,
		Iterator	last
	)

Definition at line 519 of file TMath.h.

◆ StdDev() [2/3]

template<typename Iterator , typename WeightIterator >

Double_t TMath::StdDev	(	Iterator	first,
		Iterator	last,
		WeightIterator	wfirst
	)

Definition at line 520 of file TMath.h.

◆ StdDev() [3/3]

template<typename T >

Double_t TMath::StdDev	(	Long64_t	n,
		const T *	a,
		const Double_t *	w = `0`
	)

Definition at line 518 of file TMath.h.

◆ StruveH0()

Double_t TMath::StruveH0 ( Double_t x )

Bessel function Y1(x) for positive x.

Struve Functions of Order 0.

Converted from CERNLIB M342 by Rene Brun.

Definition at line 1752 of file TMath.cxx.

◆ StruveH1()

Double_t TMath::StruveH1 ( Double_t x )

Struve functions of order 0.

Struve Functions of Order 1.

Converted from CERNLIB M342 by Rene Brun.

Definition at line 1821 of file TMath.cxx.

◆ StruveL0()

Double_t TMath::StruveL0 ( Double_t x )

Struve functions of order 1.

Modified Struve Function of Order 0.

By Kirill Filimonov.

Definition at line 1897 of file TMath.cxx.

◆ StruveL1()

Double_t TMath::StruveL1 ( Double_t x )

Modified Struve functions of order 0.

Modified Struve Function of Order 1.

By Kirill Filimonov.

Definition at line 1943 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 2583 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 2605 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 2633 of file TMath.cxx.

◆ Tan()

Double_t TMath::Tan ( Double_t x )

inline

Definition at line 635 of file TMath.h.

◆ TanH()

Double_t TMath::TanH ( Double_t x )

inline

Definition at line 647 of file TMath.h.

◆ TwoPi()

constexpr Double_t TMath::TwoPi ( )

constexpr

\[ 2\pi\]

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

[in]	x	the point were the density function is evaluated
[in]	kappa	value of kappa (distribution parameter)
[in]	beta2	value of beta2 (distribution parameter)

The algorithm was taken from the CernLib function vavden(G115) Reference: A.Rotondi and P.Montagna, Fast Calculation of Vavilov distribution Nucl.Instr. and Meth. B47(1990), 215-224

Accuracy: quote from the reference above:

"The results of our code have been compared with the values of the Vavilov density function computed numerically in an accurate way: our approximation shows a difference of less than 3% around the peak of the density function, slowly increasing going towards the extreme tails to the right and to the left"

Definition at line 2734 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 3106 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

[in]	x	the point were the density function is evaluated
[in]	kappa	value of kappa (distribution parameter)
[in]	beta2	value of beta2 (distribution parameter)

The algorithm was taken from the CernLib function vavden(G115)

Reference: A.Rotondi and P.Montagna, Fast Calculation of Vavilov distribution Nucl.Instr. and Meth. B47(1990), 215-224

Accuracy: quote from the reference above:

"The results of our code have been compared with the values of the Vavilov density function computed numerically in an accurate way: our approximation shows a difference of less than 3% around the peak of the density function, slowly increasing going towards the extreme tails to the right and to the left"

Definition at line 2767 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 2805 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 the two functions:

\[ gauss(xx) = \frac{1}{(\sqrt{2\pi} sigma)} e^{\frac{xx^{2}}{(2 sigma{^2})}} \]

and

\[ lorentz(xx) = \frac{ \frac{1}{\pi} \frac{lg}{2} }{ (xx^{2} + \frac{lg^{2}}{4}) } \]

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.

Definition at line 875 of file TMath.cxx.

Classes

Functions

Detailed Description

Function Documentation

◆ Abs() [1/7]

◆ Abs() [2/7]

◆ Abs() [3/7]

◆ Abs() [4/7]

◆ Abs() [5/7]

◆ Abs() [6/7]

◆ Abs() [7/7]

◆ ACos()

◆ ACosH()

◆ AreEqualAbs()

◆ AreEqualRel()

◆ ASin()

◆ ASinH()

◆ ATan()

◆ ATan2()

◆ ATanH()

◆ BesselI()

◆ BesselI0()

◆ BesselI1()

◆ BesselJ0()

◆ BesselJ1()

◆ BesselK()

◆ BesselK0()

◆ BesselK1()

◆ BesselY0()

◆ BesselY1()

◆ Beta()

◆ BetaCf()

◆ BetaDist()

◆ BetaDistI()

◆ BetaIncomplete()

◆ BinarySearch() [1/3]

◆ BinarySearch() [2/3]

◆ BinarySearch() [3/3]

◆ Binomial()

◆ BinomialI()

◆ BreitWigner()

◆ BubbleHigh()

◆ BubbleLow()

◆ C()

◆ CauchyDist()

◆ Ccgs()

◆ Ceil()

◆ CeilNint()

◆ ChisquareQuantile()

◆ Cos()

◆ CosH()

◆ Cross()

◆ CUncertainty()

◆ DegToRad()

◆ DiLog()

◆ E()

◆ Erf()

◆ Erfc()

◆ ErfcInverse()

◆ ErfInverse()

◆ EulerGamma()

◆ Even()

◆ Exp()

◆ Factorial()

◆ FDist()

About F distribution:

◆ FDistI()

◆ Finite() [1/2]

◆ Finite() [2/2]

◆ Floor()

◆ FloorNint()

◆ Freq()

◆ G()

◆ GamCf()

◆ Gamma() [1/2]

◆ Gamma() [2/2]

◆ GammaDist()

◆ GamSer()

◆ Gaus()

◆ Gcgs()