TMath Namespace Reference

TMath. More...

Classes
struct	Limits

Functions
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)
	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 T >
Long64_t	BinarySearch (Long64_t n, const T *array, T value)
	Binary search in an array of n values to locate value. More...
template<typename T >
Long64_t	BinarySearch (Long64_t n, const T **array, T value)
	Binary search in an array of n values to locate value. More...
template<typename Iterator , typename Element >
Iterator	BinarySearch (Iterator first, Iterator last, Element value)
	Binary search in an array defined by its iterators. More...
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: \[ \frac{\pi}{180} \] . More...
Double_t	DiLog (Double_t x)
	Modified Struve functions of order 1. More...
constexpr Double_t	E ()
	Base of natural log: \[ e \] . 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...
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 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...
constexpr Double_t	Gcgs ()
	\( cm^{3} g^{-1} s^{-2} \) 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...
template<typename Iterator >
Double_t	GeomMean (Iterator first, Iterator last)
	Return the geometric mean of an array defined by the iterators. 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 \( J s \) \[ h \] . More...
ULong_t	Hash (const void *txt, Int_t ntxt)
	Calculates hash index from any char string. More...
ULong_t	Hash (const char *str)
constexpr Double_t	Hbar ()
	\( \hbar \) in \( J s \) \[ \hbar = \frac{h}{2\pi} \] 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 \( J K^{-1} \) \[ k \] . 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 T >
Long64_t	LocMax (Long64_t n, const T *a)
	Return index of array with the maximum element. More...
template<typename Iterator >
Iterator	LocMax (Iterator first, Iterator last)
	Return index of array with the maximum element. More...
template<typename T >
Long64_t	LocMin (Long64_t n, const T *a)
	Return index of array with the minimum element. More...
template<typename Iterator >
Iterator	LocMin (Iterator first, Iterator last)
	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...
template<typename T >
T	MaxElement (Long64_t n, const T *a)
	Return maximum of array a of length n. 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 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	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...
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...
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...
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 >
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	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 ()
	\[ \pi\] More...
constexpr Double_t	PiOver2 ()
	\[ \frac{\pi}{2} \] More...
constexpr Double_t	PiOver4 ()
	\[ \frac{\pi}{4} \] More...
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...
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: \[ \frac{180}{\pi} \] . More...
constexpr Double_t	Rgair ()
	Dry Air Gas Constant (R / MWair) in \( J kg^{-1} K^{-1} \) 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...
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...
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. More...
constexpr Double_t	RUncertainty ()
	Universal gas constant uncertainty. More...
constexpr Double_t	Sigma ()
	Stefan-Boltzmann constant in \( W m^{-2} K^{-4}\) \[ \sigma \] . More...
constexpr Double_t	SigmaUncertainty ()
	Stefan-Boltzmann constant uncertainty. More...
Double_t	SignalingNaN ()
	Returns a signaling NaN as defined by IEEE 754](http://en.wikipedia.org/wiki/NaN#Signaling_NaN) More...
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)
	Sort the n elements of the array a of generic templated type Element. More...
template<typename Iterator , typename IndexIterator >
void	SortItr (Iterator first, Iterator last, IndexIterator index, Bool_t down=kTRUE)
	Sort the n1 elements of the Short_t array defined by its iterators. More...
Double_t	Sq (Double_t x)
Double_t	Sqrt (Double_t x)
constexpr Double_t	Sqrt2 ()
	\[ \sqrt{2} \] More...
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)
	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 ()
	\[ 2\pi\] More...
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

ACos()

Double_t TMath::ACos ( Double_t  x )

inline

Definition at line 667 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 416 of file TMath.h.

ASin()

Double_t TMath::ASin ( Double_t  x )

inline

Definition at line 660 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 674 of file TMath.h.

ATan2()

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

inline

Definition at line 678 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 1572 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 1408 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 1476 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 1616 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 1651 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 1543 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 1442 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 1511 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 1687 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 1721 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 1991 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 2000 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 2051 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 2069 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 2082 of file TMath.cxx.

BinarySearch() [1/3]

template<typename T >

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

Binary search in an array of n values to locate value.

Array is supposed to be sorted prior to this call. If match is found, function returns position of element. If no match found, function gives nearest element smaller than value.

Definition at line 1221 of file TMath.h.

BinarySearch() [2/3]

template<typename T >

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

Binary search in an array of n values to locate value.

Array is supposed to be sorted prior to this call. If match is found, function returns position of element. If no match found, function gives nearest element smaller than value.

Definition at line 1237 of file TMath.h.

BinarySearch() [3/3]

template<typename Iterator , typename Element >

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

Binary search in an array defined by its iterators.

The values in the iterators range are supposed to be sorted prior to this call. If match is found, function returns position of element. If no match found, function gives nearest element smaller than value.

Definition at line 1205 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 2090 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 2115 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 1296 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 1335 of file TMath.cxx.

C()

constexpr Double_t TMath::C

(

)

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 2148 of file TMath.cxx.

Ccgs()

constexpr Double_t TMath::Ccgs

(

)

\( cm s^{-1} \)

Definition at line 125 of file TMath.h.

Ceil()

Double_t TMath::Ceil ( Double_t  x )

inline

Definition at line 694 of file TMath.h.

CeilNint()

Int_t TMath::CeilNint ( Double_t  x )

inline

Definition at line 698 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 2164 of file TMath.cxx.

Cos()

Double_t TMath::Cos ( Double_t  x )

inline

Definition at line 640 of file TMath.h.

CosH()

Double_t TMath::CosH ( Double_t  x )

inline

Definition at line 652 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 1295 of file TMath.h.

CUncertainty()

constexpr Double_t TMath::CUncertainty

(

)

Speed of light uncertainty.

Definition at line 132 of file TMath.h.

DegToRad()

constexpr Double_t TMath::DegToRad

(

)

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

(

)

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

(

)

Euler-Mascheroni Constant.

Definition at line 329 of file TMath.h.

Exp()

Double_t TMath::Exp ( Double_t  x )

inline

Definition at line 726 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 2247 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 2266 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 770 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 799 of file TMath.h.

Floor()

Double_t TMath::Floor ( Double_t  x )

inline

Definition at line 702 of file TMath.h.

FloorNint()

Int_t TMath::FloorNint ( Double_t  x )

inline

Definition at line 706 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

(

)

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

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.

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 .

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

Author

NvE 14-nov-1998 UU-SAP Utrecht

Definition at line 364 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 2315 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

(

)

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

Definition at line 146 of file TMath.h.

GeomMean() [1/2]

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 1130 of file TMath.h.

GeomMean() [2/2]

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 1111 of file TMath.h.

GhbarC()

constexpr Double_t TMath::GhbarC

(

)

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

Definition at line 160 of file TMath.h.

GhbarCUncertainty()

constexpr Double_t TMath::GhbarCUncertainty

(

)

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

Definition at line 167 of file TMath.h.

Gn()

constexpr Double_t TMath::Gn

(

)

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

Definition at line 174 of file TMath.h.

GnUncertainty()

constexpr Double_t TMath::GnUncertainty

(

)

Standard acceleration of gravity uncertainty.

Definition at line 181 of file TMath.h.

GUncertainty()

constexpr Double_t TMath::GUncertainty

(

)

Gravitational constant uncertainty.

Definition at line 153 of file TMath.h.

H()

constexpr Double_t TMath::H

(

)

Planck's constant in \( J s \)

\[ h \]

.

Definition at line 189 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 1390 of file TMath.cxx.

Hash() [2/2]

ULong_t TMath::Hash

(

const char *

str

)

Definition at line 1398 of file TMath.cxx.

Hbar()

constexpr Double_t TMath::Hbar

(

)

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

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

Definition at line 211 of file TMath.h.

Hbarcgs()

constexpr Double_t TMath::Hbarcgs

(

)

\( erg s \)

Definition at line 218 of file TMath.h.

HbarUncertainty()

constexpr Double_t TMath::HbarUncertainty

(

)

\( \hbar \) uncertainty.

Definition at line 225 of file TMath.h.

HC()

constexpr Double_t TMath::HC

(

)

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

Definition at line 232 of file TMath.h.

HCcgs()

constexpr Double_t TMath::HCcgs

(

)

\( erg cm \)

Definition at line 239 of file TMath.h.

Hcgs()

constexpr Double_t TMath::Hcgs

(

)

\( erg s \)

Definition at line 196 of file TMath.h.

HUncertainty()

constexpr Double_t TMath::HUncertainty

(

)

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 913 of file TMath.h.

InvPi()

constexpr Double_t TMath::InvPi

(

)

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

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 1329 of file TMath.h.

IsNaN() [1/2]

Bool_t TMath::IsNaN ( Double_t  x )

inline

Definition at line 891 of file TMath.h.

IsNaN() [2/2]

Bool_t TMath::IsNaN ( Float_t  x )

inline

Definition at line 892 of file TMath.h.

K()

constexpr Double_t TMath::K

(

)

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

\[ k \]

.

Definition at line 247 of file TMath.h.

Kcgs()

constexpr Double_t TMath::Kcgs

(

)

\( erg K^{-1} \)

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:

• \( z = dn \sqrt{n} \), 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 663 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 789 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 1453 of file TMath.h.

KUncertainty()

constexpr Double_t TMath::KUncertainty

(

)

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 2803 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 2332 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 2348 of file TMath.cxx.

Ldexp()

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

inline

Definition at line 730 of file TMath.h.

Ln10()

constexpr Double_t TMath::Ln10

(

)

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 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 999 of file TMath.h.

LocMax() [2/2]

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 1016 of file TMath.h.

LocMin() [1/2]

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 971 of file TMath.h.

LocMin() [2/2]

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 988 of file TMath.h.

Log()

Double_t TMath::Log ( Double_t  x )

inline

Definition at line 759 of file TMath.h.

Log10()

Double_t TMath::Log10 ( Double_t  x )

inline

Definition at line 763 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

(

)

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 2404 of file TMath.cxx.

MaxElement()

template<typename T >

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

Return maximum of array a of length n.

Definition at line 958 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
	)

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

Definition at line 1098 of file TMath.h.

Mean() [2/3]

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 1051 of file TMath.h.

Mean() [3/3]

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 1070 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 1367 of file TMath.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 951 of file TMath.h.

MWair()

constexpr Double_t TMath::MWair

(

)

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

(

)

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

Definition at line 283 of file TMath.h.

NaUncertainty()

constexpr Double_t TMath::NaUncertainty

(

)

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 712 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 1309 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 495 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 512 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 943 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 2423 of file TMath.cxx.

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 2524 of file TMath.cxx.

Pi()

constexpr Double_t TMath::Pi

(

)

\[ \pi\]

Definition at line 38 of file TMath.h.

PiOver2()

constexpr Double_t TMath::PiOver2

(

)

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

Definition at line 52 of file TMath.h.

PiOver4()

constexpr Double_t TMath::PiOver4

(

)

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

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 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 568 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 599 of file TMath.cxx.

Power() [1/5]

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

inline

Definition at line 734 of file TMath.h.

Power() [2/5]

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

inline

Definition at line 738 of file TMath.h.

Power() [3/5]

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

inline

Definition at line 742 of file TMath.h.

Power() [4/5]

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

inline

Definition at line 746 of file TMath.h.

Power() [5/5]

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

inline

Definition at line 750 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 621 of file TMath.cxx.

Qe()

constexpr Double_t TMath::Qe

(

)

Elementary charge in \( C \) .

Definition at line 336 of file TMath.h.

QeUncertainty()

constexpr Double_t TMath::QeUncertainty

(

)

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:

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 1190 of file TMath.cxx.

QuietNaN()

Double_t TMath::QuietNaN ( )

inline

Returns a quiet NaN as defined by IEEE 754

Definition at line 900 of file TMath.h.

R()

constexpr Double_t TMath::R

(

)

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

(

)

Conversion from radian to degree:

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

.

Definition at line 74 of file TMath.h.

Rgair()

constexpr Double_t TMath::Rgair

(

)

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 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 1192 of file TMath.h.

RMS() [2/3]

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 1144 of file TMath.h.

RMS() [3/3]

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 1168 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 1091 of file TMath.cxx.

RUncertainty()

constexpr Double_t TMath::RUncertainty

(

)

Universal gas constant uncertainty.

Definition at line 306 of file TMath.h.

Sigma()

constexpr Double_t TMath::Sigma

(

)

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

\[ \sigma \]

.

Definition at line 269 of file TMath.h.

SigmaUncertainty()

constexpr Double_t TMath::SigmaUncertainty

(

)

Stefan-Boltzmann constant uncertainty.

Definition at line 276 of file TMath.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 907 of file TMath.h.

Sin()

Double_t TMath::Sin ( Double_t  x )

inline

Definition at line 636 of file TMath.h.

SinH()

Double_t TMath::SinH ( Double_t  x )

inline

Definition at line 648 of file TMath.h.

Sort()

template<typename Element , typename Index >

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

Sort the n elements of the array a of generic templated type Element.

In output the array index of type Index contains the indices of the sorted array. If down is false sort in increasing order (default is decreasing order).

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

Definition at line 1282 of file TMath.h.

SortItr()

template<typename Iterator , typename IndexIterator >

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

Sort the n1 elements of the Short_t array defined by its iterators.

In output the array index contains the indices of the sorted array. If down is false sort in increasing order (default is decreasing order).

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

Definition at line 1256 of file TMath.h.

Sq()

Double_t TMath::Sq ( Double_t  x )

inline

Definition at line 686 of file TMath.h.

Sqrt()

Double_t TMath::Sqrt ( Double_t  x )

inline

Definition at line 690 of file TMath.h.

Sqrt2()

constexpr Double_t TMath::Sqrt2

(

)

\[ \sqrt{2} \]

Definition at line 89 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 527 of file TMath.h.

StdDev() [2/3]

template<typename Iterator >

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

Definition at line 528 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 529 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 1759 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 1828 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 1904 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 1950 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 2590 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 2612 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 2640 of file TMath.cxx.

Tan()

Double_t TMath::Tan ( Double_t  x )

inline

Definition at line 644 of file TMath.h.

TanH()

Double_t TMath::TanH ( Double_t  x )

inline

Definition at line 656 of file TMath.h.

TwoPi()

constexpr Double_t TMath::TwoPi

(

)

\[ 2\pi\]

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 2741 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 3113 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 2774 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 2812 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 882 of file TMath.cxx.