Namespaces
namespace	ROOTDict

Classes
struct	Limits

Functions
Double_t	Abs (Double_t d)
	Returns the absolute value of parameter `Double_t d`.

Float_t	Abs (Float_t d)
	Returns the absolute value of parameter `Float_t d`.

Int_t	Abs (Int_t d)
	Returns the absolute value of parameter `Int_t d`.

Long64_t	Abs (Long64_t d)
	Returns the absolute value of parameter `Long64_t d`.

Long_t	Abs (Long_t d)
	Returns the absolute value of parameter `Long_t d`.

LongDouble_t	Abs (LongDouble_t d)
	Returns the absolute value of parameter `LongDouble_t d`.

Short_t	Abs (Short_t d)
	Returns the absolute value of parameter `Short_t d`.

Double_t	ACos (Double_t)
	Returns the principal value of the arc cosine of `x`, expressed in radians.

Double_t	ACosH (Double_t)
	Returns the nonnegative area hyperbolic cosine of `x`.

Bool_t	AreEqualAbs (Double_t af, Double_t bf, Double_t epsilon)
	Comparing floating points.

Bool_t	AreEqualRel (Double_t af, Double_t bf, Double_t relPrec)
	Comparing floating points.

Double_t	ASin (Double_t)
	Returns the principal value of the arc sine of `x`, expressed in radians.

Double_t	ASinH (Double_t)
	Returns the area hyperbolic sine of `x`.

Double_t	ATan (Double_t)
	Returns the principal value of the arc tangent of `x`, expressed in radians.

Double_t	ATan2 (Double_t y, Double_t x)
	Returns the principal value of the arc tangent of `y/x`, expressed in radians.

Double_t	ATanH (Double_t)
	Returns the area hyperbolic tangent of `x`.

Double_t	BesselI (Int_t n, Double_t x)
	Computes the Integer Order Modified Bessel function I_n(x) for n=0,1,2,... and any real x.

Double_t	BesselI0 (Double_t x)
	Integer order modified Bessel function K_n(x)

Double_t	BesselI1 (Double_t x)
	Modified Bessel function K_0(x)

Double_t	BesselJ0 (Double_t x)
	Modified Bessel function K_1(x)

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

Double_t	BesselK (Int_t n, Double_t x)
	Integer order modified Bessel function I_n(x)

Double_t	BesselK0 (Double_t x)
	Modified Bessel function I_0(x)

Double_t	BesselK1 (Double_t x)
	Modified Bessel function I_1(x)

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

Double_t	BesselY1 (Double_t x)
	Bessel function Y0(x) for positive x.

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

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.

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

Double_t	BetaDistI (Double_t x, Double_t p, Double_t q)
	Computes the cumulative distribution function of the Beta distribution, i.e.

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

template<typename Iterator , typename Element >
Iterator	BinarySearch (Iterator first, Iterator last, Element value)
	Binary search in an array defined by its iterators.

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.

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.

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

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:

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

Double_t	BreitWignerRelativistic (Double_t x, Double_t median=0, Double_t gamma=1)
	Calculates a Relativistic Breit Wigner function with median and gamma.

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.

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

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

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)

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

Double_t	Ceil (Double_t x)
	Rounds `x` upward, returning the smallest integral value that is not less than `x`.

Int_t	CeilNint (Double_t x)
	Returns the nearest integer of `TMath::Ceil(x)`.

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

Double_t	Cos (Double_t)
	Returns the cosine of an angle of `x` radians.

Double_t	CosH (Double_t)
	Returns the hyperbolic cosine of `x`.

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

constexpr Double_t	CUncertainty ()
	Speed of light uncertainty.

constexpr Double_t	DegToRad ()
	Conversion from degree to radian: \( \frac{\pi}{180} \).

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

constexpr Double_t	E ()
	Base of natural log: \( e \).

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

Double_t	Erfc (Double_t x)
	Computes the complementary error function erfc(x).

Double_t	ErfcInverse (Double_t x)
	Returns the inverse of the complementary error function.

Double_t	ErfInverse (Double_t x)
	Returns the inverse error function.

constexpr Double_t	EulerGamma ()
	Euler-Mascheroni Constant.

Bool_t	Even (Long_t a)
	Returns `true` if `a` is even.

Double_t	Exp (Double_t x)
	Returns the base-e exponential function of x, which is e raised to the power `x`.

Double_t	Factorial (Int_t i)
	Computes factorial(n).

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).

Double_t	FDistI (Double_t F, Double_t N, Double_t M)
	Calculates the cumulative distribution function of F-distribution (see ROOT::Math::fdistribution_cdf).

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

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

Double_t	Floor (Double_t x)
	Rounds `x` downward, returning the largest integral value that is not greater than `x`.

Int_t	FloorNint (Double_t x)
	Returns the nearest integer of `TMath::Floor(x)`.

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

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

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

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 .

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

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.

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

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

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

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

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

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

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

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

constexpr Double_t	GnUncertainty ()
	Standard acceleration of gravity uncertainty.

constexpr Double_t	GUncertainty ()
	Gravitational constant uncertainty.

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

ULong_t	Hash (const char *str)

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

constexpr Double_t	Hbar ()
	\( \hbar \) in \( J s \): \( \hbar = \frac{h}{2\pi} \)

constexpr Double_t	Hbarcgs ()
	\( erg s \)

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

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

constexpr Double_t	HCcgs ()
	\( erg cm \)

constexpr Double_t	Hcgs ()
	\( erg s \)

constexpr Double_t	HUncertainty ()
	Planck's constant uncertainty.

Double_t	Hypot (Double_t x, Double_t y)
	Returns `sqrt(xx + yy)`

Long_t	Hypot (Long_t x, Long_t y)
	Returns `sqrt(xx + yy)`

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

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

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.

Bool_t	IsNaN (Double_t x)

Bool_t	IsNaN (Float_t x)

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

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

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

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.

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).

constexpr Double_t	KUncertainty ()
	Boltzmann's constant uncertainty.

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

Double_t	LandauI (Double_t x)
	Returns the cumulative (lower tail integral) of the Landau distribution function at point x.

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.

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

Double_t	Ldexp (Double_t x, Int_t exp)
	Returns the result of multiplying `x` (the significant) by 2 raised to the power of `exp` (the exponent).

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

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

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

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

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

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

Double_t	Log (Double_t x)
	Returns the natural logarithm of `x`.

Double_t	Log10 (Double_t x)
	Returns the common (base-10) logarithm of `x`.

Double_t	Log2 (Double_t x)
	Returns the binary (base-2) logarithm of `x`.

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

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.

Double_t	Max (Double_t a, Double_t b)
	Returns the largest of `a` and `b`.

Float_t	Max (Float_t a, Float_t b)
	Returns the largest of `a` and `b`.

Int_t	Max (Int_t a, Int_t b)
	Returns the largest of `a` and `b`.

Long64_t	Max (Long64_t a, Long64_t b)
	Returns the largest of `a` and `b`.

Long_t	Max (Long_t a, Long_t b)
	Returns the largest of `a` and `b`.

Short_t	Max (Short_t a, Short_t b)
	Returns the largest of `a` and `b`.

UInt_t	Max (UInt_t a, UInt_t b)
	Returns the largest of `a` and `b`.

ULong64_t	Max (ULong64_t a, ULong64_t b)
	Returns the largest of `a` and `b`.

ULong_t	Max (ULong_t a, ULong_t b)
	Returns the largest of `a` and `b`.

UShort_t	Max (UShort_t a, UShort_t b)
	Returns the largest of `a` and `b`.

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

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

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

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

template<typename T >
Double_t	Median (Long64_t n, const T a, const Double_t w=nullptr, Long64_t *work=nullptr)
	Same as RMS.

Double_t	Min (Double_t a, Double_t b)
	Returns the smallest of `a` and `b`.

Float_t	Min (Float_t a, Float_t b)
	Returns the smallest of `a` and `b`.

Int_t	Min (Int_t a, Int_t b)
	Returns the smallest of `a` and `b`.

Long64_t	Min (Long64_t a, Long64_t b)
	Returns the smallest of `a` and `b`.

Long_t	Min (Long_t a, Long_t b)
	Returns the smallest of `a` and `b`.

Short_t	Min (Short_t a, Short_t b)
	Returns the smallest of `a` and `b`.

UInt_t	Min (UInt_t a, UInt_t b)
	Returns the smallest of `a` and `b`.

ULong64_t	Min (ULong64_t a, ULong64_t b)
	Returns the smallest of `a` and `b`.

ULong_t	Min (ULong_t a, ULong_t b)
	Returns the smallest of `a` and `b`.

UShort_t	Min (UShort_t a, UShort_t b)
	Returns the smallest of `a` and `b`.

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

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

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

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

Long_t	NextPrime (Long_t x)

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

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

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

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

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

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

Bool_t	Odd (Long_t a)
	Returns `true` if `a` is odd.

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.

constexpr Double_t	Pi ()
	\( \pi\)

constexpr Double_t	PiOver2 ()
	\( \frac{\pi}{2} \)

constexpr Double_t	PiOver4 ()
	\( \frac{\pi}{4} \)

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

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

Double_t	Power (Double_t x, Double_t y)
	Returns `x` raised to the power `y`.

Double_t	Power (Double_t x, Int_t y)
	Returns `x` raised to the power `y`.

LongDouble_t	Power (Long64_t x, Long64_t y)
	Returns `x` raised to the power `y`.

LongDouble_t	Power (LongDouble_t x, Long64_t y)
	Returns `x` raised to the power `y`.

LongDouble_t	Power (LongDouble_t x, LongDouble_t y)
	Returns `x` raised to the power `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).

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

constexpr Double_t	QeUncertainty ()
	Elementary charge uncertainty.

void	Quantiles (Int_t n, Int_t nprob, Double_t x, Double_t quantiles, Double_t prob, Bool_t isSorted=kTRUE, Int_t index=nullptr, Int_t type=7)
	Computes sample quantiles, corresponding to the given probabilities.

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

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

constexpr Double_t	RadToDeg ()
	Conversion from radian to degree: \( \frac{180}{\pi} \).

Double_t	Range (Double_t lb, Double_t ub, Double_t x)
	Returns `x` if `lb < x < up`, `lb` if `x < lb` and `ub` if `x > ub`.

Int_t	Range (Int_t lb, Int_t ub, Int_t x)
	Returns `x` if `lb < x < up`, `lb` if `x < lb` and `ub` if `x > ub`.

Long_t	Range (Long_t lb, Long_t ub, Long_t x)
	Returns `x` if `lb < x < up`, `lb` if `x < lb` and `ub` if `x > ub`.

Short_t	Range (Short_t lb, Short_t ub, Short_t x)
	Returns `x` if `lb < x < up`, `lb` if `x < lb` and `ub` if `x > ub`.

ULong_t	Range (ULong_t lb, ULong_t ub, ULong_t x)
	Returns `x` if `lb < x < up`, `lb` if `x < lb` and `ub` if `x > ub`.

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

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

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

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

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.

constexpr Double_t	RUncertainty ()
	Universal gas constant uncertainty.

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

constexpr Double_t	SigmaUncertainty ()
	Stefan-Boltzmann constant uncertainty.

Double_t	Sign (Double_t a, Double_t b)
	Returns a value with the magnitude of `a` and the sign of `b`.

Float_t	Sign (Float_t a, Float_t b)
	Returns a value with the magnitude of `a` and the sign of `b`.

LongDouble_t	Sign (LongDouble_t a, LongDouble_t b)
	Returns a value with the magnitude of `a` and the sign of `b`.

template<typename T1 , typename T2 >
T1	Sign (T1 a, T2 b)
	Returns a value with the magnitude of `a` and the sign of `b`.

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

Bool_t	SignBit (Double_t a)
	Returns whether the sign of `Double_t a` is negative.

Bool_t	SignBit (Float_t a)
	Returns whether the sign of `Float_t a` is negative.

template<typename Integer >
Bool_t	SignBit (Integer a)
	Returns whether the sign of `Integer a` is negative.

Bool_t	SignBit (LongDouble_t a)
	Returns whether the sign of `LongDouble_t a` is negative.

Double_t	Sin (Double_t)
	Returns the sine of an angle of `x` radians.

Double_t	SinH (Double_t)
	Returns the hyperbolic sine of `x.

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.

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.

Double_t	Sq (Double_t x)
	Returns `x*x`.

Double_t	Sqrt (Double_t x)
	Returns the square root of x.

constexpr Double_t	Sqrt2 ()
	\( \sqrt{2} \)

template<typename Iterator >
Double_t	StdDev (Iterator first, Iterator last)
	Same as RMS.

template<typename Iterator , typename WeightIterator >
Double_t	StdDev (Iterator first, Iterator last, WeightIterator wfirst)
	Same as RMS.

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

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

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

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

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

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).

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.

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.

Double_t	Tan (Double_t)
	Returns the tangent of an angle of `x` radians.

Double_t	TanH (Double_t)
	Returns the hyperbolic tangent of `x`.

constexpr Double_t	TwoPi ()
	\( 2\pi\)

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

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

Double_t	VavilovI (Double_t x, Double_t kappa, Double_t beta2)
	Returns the value of the Vavilov cumulative distribution function (lower tail integral of the probability distribution function)

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.

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

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

Returns the absolute value of parameter Double_t d.

Definition at line 143 of file TMathBase.h.

◆ Abs() [2/7]

Float_t TMath::Abs ( Float_t d )

inline

Returns the absolute value of parameter Float_t d.

Definition at line 139 of file TMathBase.h.

◆ Abs() [3/7]

Int_t TMath::Abs ( Int_t d )

inline

Returns the absolute value of parameter Int_t d.

Definition at line 127 of file TMathBase.h.

◆ Abs() [4/7]

Long64_t TMath::Abs ( Long64_t d )

inline

Returns the absolute value of parameter Long64_t d.

Definition at line 135 of file TMathBase.h.

◆ Abs() [5/7]

Long_t TMath::Abs ( Long_t d )

inline

Returns the absolute value of parameter Long_t d.

Definition at line 131 of file TMathBase.h.

◆ Abs() [6/7]

LongDouble_t TMath::Abs ( LongDouble_t d )

inline

Returns the absolute value of parameter LongDouble_t d.

Definition at line 147 of file TMathBase.h.

◆ Abs() [7/7]

Short_t TMath::Abs ( Short_t d )

inline

Returns the absolute value of parameter Short_t d.

Definition at line 123 of file TMathBase.h.

◆ ACos()

Double_t TMath::ACos ( Double_t x )

inline

Returns the principal value of the arc cosine of x, expressed in radians.

Definition at line 632 of file TMath.h.

◆ ACosH()

Double_t TMath::ACosH ( Double_t x )

Returns the nonnegative area hyperbolic cosine of x.

Definition at line 81 of file TMath.cxx.

◆ AreEqualAbs()

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

inline

Comparing floating points.

Returns kTRUE if the absolute difference between af and bf is less than epsilon.

Definition at line 418 of file TMath.h.

◆ AreEqualRel()

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

inline

Comparing floating points.

Returns kTRUE if the relative difference between af and bf is less than relPrec.

Definition at line 426 of file TMath.h.

◆ ASin()

Double_t TMath::ASin ( Double_t x )

inline

Returns the principal value of the arc sine of x, expressed in radians.

Definition at line 624 of file TMath.h.

◆ ASinH()

Double_t TMath::ASinH ( Double_t x )

Returns the area hyperbolic sine of x.

Definition at line 67 of file TMath.cxx.

◆ ATan()

Double_t TMath::ATan ( Double_t x )

inline

Returns the principal value of the arc tangent of x, expressed in radians.

Definition at line 640 of file TMath.h.

◆ ATan2()

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

inline

Returns the principal value of the arc tangent of y/x, expressed in radians.

Definition at line 646 of file TMath.h.

◆ ATanH()

Double_t TMath::ATanH ( Double_t x )

Returns the area hyperbolic tangent of x.

Definition at line 95 of file TMath.cxx.

◆ BesselI()

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

Computes the Integer Order Modified Bessel function I_n(x) for n=0,1,2,... and any real x.

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

Definition at line 1590 of file TMath.cxx.

◆ BesselI0()

Double_t TMath::BesselI0 ( Double_t x )

Integer order modified Bessel function K_n(x)

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

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

Definition at line 1426 of file TMath.cxx.

◆ BesselI1()

Double_t TMath::BesselI1 ( Double_t x )

Modified Bessel function K_0(x)

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

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

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

Definition at line 1494 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 1634 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 1669 of file TMath.cxx.

◆ BesselK()

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

Integer order modified Bessel function I_n(x)

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

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

Definition at line 1561 of file TMath.cxx.

◆ BesselK0()

Double_t TMath::BesselK0 ( Double_t x )

Modified Bessel function I_0(x)

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

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

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

Definition at line 1460 of file TMath.cxx.

◆ BesselK1()

Double_t TMath::BesselK1 ( Double_t x )

Modified Bessel function I_1(x)

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

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

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

Definition at line 1529 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 1705 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 1739 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 2011 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 2020 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 cumulative distribution function is computed in BetaDistI).

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

Definition at line 2071 of file TMath.cxx.

◆ BetaDistI()

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

Computes the cumulative distribution function of the Beta distribution, i.e.

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

Definition at line 2090 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 2103 of file TMath.cxx.

◆ BinarySearch() [1/3]

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 332 of file TMathBase.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 362 of file TMathBase.h.

◆ BinarySearch() [3/3]

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 347 of file TMathBase.h.

◆ Binomial()

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

Calculates the binomial coefficient n over k.

Definition at line 2111 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)*TMath::Power (1-p, n-j)

p

winID h TVirtualViewer3D TVirtualGLPainter p

Definition TGWin32VirtualGLProxy.cxx:51

n

const Int_t n

Definition legend1.C:16

TMath::Binomial

Double_t Binomial(Int_t n, Int_t k)

Calculates the binomial coefficient n over k.

Definition TMath.cxx:2111

TMath::Power

LongDouble_t Power(LongDouble_t x, LongDouble_t y)

Returns x raised to the power y.

Definition TMath.h:721

For n larger than 12 BetaIncomplete is a much better way to evaluate the sum than would be the straightforward sum calculation for n smaller than 12 either method is acceptable ("Numerical Recipes")

Note this function is not exactly implementing the cumulative or the complement of the cumulative of the Binomial distrinution. It is equivalent to ROOT::Math::binomial_cdf_c(k-1,p,n)

Author: Anna Kreshuk

Definition at line 2141 of file TMath.cxx.

◆ BreitWigner()

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

Calculates a Breit Wigner function with mean and gamma.

Definition at line 442 of file TMath.cxx.

◆ BreitWignerRelativistic()

Double_t TMath::BreitWignerRelativistic	(	Double_t	x,
		Double_t	median = `0`,
		Double_t	gamma = `1`
	)

Calculates a Relativistic Breit Wigner function with median and gamma.

Definition at line 452 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]	Narr	number of array elements
[in]	*arr1	is unchanged;
[in]	*arr2	is the array of indices corresponding to the descending value of arr1 with arr2[0] corresponding to the largest arr1 value and arr2[Narr] the smallest.

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

Definition at line 1314 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 1353 of file TMath.cxx.

◆ C()

constexpr Double_t TMath::C ( )

constexpr

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

Definition at line 114 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

Example:

TF1* fc = new TF1("fc", "TMath::CauchyDist(x, [0], [1])", -5, 5);
fc->SetParameters(0, 1);
fc->Draw();

Author: Anna Kreshuk

Definition at line 2175 of file TMath.cxx.

◆ Ccgs()

constexpr Double_t TMath::Ccgs ( )

constexpr

\( cm s^{-1} \)

Definition at line 121 of file TMath.h.

◆ Ceil()

Double_t TMath::Ceil ( Double_t x )

inline

Rounds x upward, returning the smallest integral value that is not less than x.

Definition at line 668 of file TMath.h.

◆ CeilNint()

Int_t TMath::CeilNint ( Double_t x )

inline

Returns the nearest integer of TMath::Ceil(x).

Definition at line 674 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.35Incorporates 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

Author: Anna Kreshuk

Definition at line 2193 of file TMath.cxx.

◆ Cos()

Double_t TMath::Cos ( Double_t x )

inline

Returns the cosine of an angle of x radians.

Definition at line 594 of file TMath.h.

◆ CosH()

Double_t TMath::CosH ( Double_t x )

inline

Returns the hyperbolic cosine of x.

Definition at line 612 of file TMath.h.

◆ Cross()

template<typename T >

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

Calculates the Cross Product of two vectors: out = [v1 x v2].

Definition at line 1197 of file TMath.h.

◆ CUncertainty()

constexpr Double_t TMath::CUncertainty ( )

constexpr

Speed of light uncertainty.

Definition at line 128 of file TMath.h.

◆ DegToRad()

constexpr Double_t TMath::DegToRad ( )

constexpr

Conversion from degree to radian: \( \frac{\pi}{180} \).

Definition at line 79 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 116 of file TMath.cxx.

◆ E()

constexpr Double_t TMath::E ( )

constexpr

Base of natural log: \( e \).

Definition at line 93 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 190 of file TMath.cxx.

◆ Erfc()

Double_t TMath::Erfc ( Double_t x )

Computes the complementary error function erfc(x).

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

Definition at line 199 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 242 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 208 of file TMath.cxx.

◆ EulerGamma()

constexpr Double_t TMath::EulerGamma ( )

constexpr

Euler-Mascheroni Constant.

Definition at line 332 of file TMath.h.

◆ Even()

Bool_t TMath::Even ( Long_t a )

inline

Returns true if a is even.

Definition at line 113 of file TMathBase.h.

◆ Exp()

Double_t TMath::Exp ( Double_t x )

inline

Returns the base-e exponential function of x, which is e raised to the power x.

Definition at line 709 of file TMath.h.

◆ Factorial()

Double_t TMath::Factorial ( Int_t i )

Computes factorial(n).

Definition at line 252 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.

Author: Anna Kreshuk

Definition at line 2277 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 (see ROOT::Math::fdistribution_cdf).

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

Author: Anna Kreshuk

Definition at line 2297 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 800 of file TMath.h.

◆ Floor()

Double_t TMath::Floor ( Double_t x )

inline

Rounds x downward, returning the largest integral value that is not greater than x.

Definition at line 680 of file TMath.h.

◆ FloorNint()

Int_t TMath::FloorNint ( Double_t x )

inline

Returns the nearest integer of TMath::Floor(x).

Definition at line 686 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 270 of file TMath.cxx.

◆ G()

constexpr Double_t TMath::G ( )

constexpr

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

Definition at line 135 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 380 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 369 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 353 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]	x	evaluation point
[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 2347 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 417 of file TMath.cxx.

◆ Gaus()

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

Calculates 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 471 of file TMath.cxx.

◆ Gcgs()

constexpr Double_t TMath::Gcgs ( )

constexpr

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

Definition at line 143 of file TMath.h.

◆ GeomMean() [1/2]

template<typename Iterator >

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

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

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

Definition at line 1103 of file TMath.h.

◆ GeomMean() [2/2]

template<typename T >

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

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

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

Definition at line 1123 of file TMath.h.

◆ GhbarC()

constexpr Double_t TMath::GhbarC ( )

constexpr

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

Definition at line 158 of file TMath.h.

◆ GhbarCUncertainty()

constexpr Double_t TMath::GhbarCUncertainty ( )

constexpr

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

Definition at line 166 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 150 of file TMath.h.

◆ H()

constexpr Double_t TMath::H ( )

constexpr

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

Definition at line 188 of file TMath.h.

◆ Hash() [1/2]

ULong_t TMath::Hash ( const char * str )

Definition at line 1416 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 1408 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 233 of file TMath.h.

◆ HCcgs()

constexpr Double_t TMath::HCcgs ( )

constexpr

\( erg cm \)

Definition at line 240 of file TMath.h.

◆ Hcgs()

constexpr Double_t TMath::Hcgs ( )

constexpr

\( erg s \)

Definition at line 195 of file TMath.h.

◆ HUncertainty()

constexpr Double_t TMath::HUncertainty ( )

constexpr

Planck's constant uncertainty.

Definition at line 202 of file TMath.h.

◆ Hypot() [1/2]

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

Returns sqrt(x*x + y*y)

Definition at line 59 of file TMath.cxx.

◆ Hypot() [2/2]

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

Returns sqrt(x*x + y*y)

Definition at line 51 of file TMath.cxx.

◆ Infinity()

Double_t TMath::Infinity ( )

inline

Returns an infinity as defined by the IEEE standard.

Definition at line 917 of file TMath.h.

◆ InvPi()

constexpr Double_t TMath::InvPi ( )

constexpr

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

Definition at line 65 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 1233 of file TMath.h.

◆ IsNaN() [1/2]

Bool_t TMath::IsNaN ( Double_t x )

inline

Definition at line 892 of file TMath.h.

◆ IsNaN() [2/2]

Bool_t TMath::IsNaN ( Float_t x )

inline

Definition at line 893 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#647, and - \_form#648 is the maximum deviation between a hypothetical distribution function and an experimental distribution with - \_form#354 events NOTE: To compare two experimental distributions with m and n events, use \_form#649 Accuracy: The function is far too accurate for any imaginable application. Probabilities less than \_form#650 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 679 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 (JADetwiler@lbl.gov)

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

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 805 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 1359 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 492 of file TMath.cxx.

◆ LandauI()

Double_t TMath::LandauI ( Double_t x )

Returns the cumulative (lower tail integral) of the Landau distribution function at point x.

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

Definition at line 2845 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 2364 of file TMath.cxx.

◆ LaplaceDistI()

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

Computes the cumulative distribution function (lower tail integral) of Laplace distribution at point x, with location parameter alpha and shape parameter beta.

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

Definition at line 2380 of file TMath.cxx.

◆ Ldexp()

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

inline

Returns the result of multiplying x (the significant) by 2 raised to the power of exp (the exponent).

Definition at line 715 of file TMath.h.

◆ Ln10()

constexpr Double_t TMath::Ln10 ( )

constexpr

Natural log of 10 (to convert log to ln)

Definition at line 100 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 509 of file TMath.cxx.

◆ LocMax() [1/2]

template<typename Iterator >

Iterator TMath::LocMax	(	Iterator	first,
		Iterator	last
	)

Returns index of array with the maximum element.

If more than one element is maximum returns first found.

Definition at line 1030 of file TMath.h.

◆ LocMax() [2/2]

template<typename T >

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

Returns 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 1012 of file TMath.h.

◆ LocMin() [1/2]

template<typename Iterator >

Iterator TMath::LocMin	(	Iterator	first,
		Iterator	last
	)

Returns index of array with the minimum element.

If more than one element is minimum returns first found.

Definition at line 1000 of file TMath.h.

◆ LocMin() [2/2]

template<typename T >

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

Returns 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 982 of file TMath.h.

◆ Log()

Double_t TMath::Log ( Double_t x )

inline

Returns the natural logarithm of x.

Definition at line 756 of file TMath.h.

◆ Log10()

Double_t TMath::Log10 ( Double_t x )

inline

Returns the common (base-10) logarithm of x.

Definition at line 762 of file TMath.h.

◆ Log2()

Double_t TMath::Log2 ( Double_t x )

Returns the binary (base-2) logarithm of x.

Definition at line 107 of file TMath.cxx.

◆ LogE()

constexpr Double_t TMath::LogE ( )

constexpr

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

Definition at line 107 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]	x	is the evaluation point
[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 2437 of file TMath.cxx.