Class describing a Vavilov distribution.

The probability density function of the Vavilov distribution as function of Landau's parameter is given by:

\[ p(\lambda_L; \kappa, \beta^2) = \frac{1}{2 \pi i}\int_{c-i\infty}^{c+i\infty} \phi(s) e^{\lambda_L s} ds\]

where $\phi(s) = e^{C} e^{\psi(s)}$ with $ C = \kappa (1+\beta^2 \gamma )$ and $\psi(s)= s \ln \kappa + (s+\beta^2 \kappa) \cdot \left ( \int \limits_{0}^{1} \frac{1 - e^{\frac{-st}{\kappa}}}{t} \,d t- \gamma \right ) - \kappa \, e^{\frac{-s}{\kappa}}$. $ \gamma = 0.5772156649\dots$ is Euler's constant.

For the class VavilovAccurate, Pdf returns the Vavilov distribution as function of Landau's parameter $\lambda_L = \lambda_V/\kappa - \ln \kappa$, which is the convention used in the CERNLIB routines, and in the tables by S.M. Seltzer and M.J. Berger: Energy loss stragglin of protons and mesons: Tabulation of the Vavilov distribution, pp 187-203 in: National Research Council (U.S.), Committee on Nuclear Science: Studies in penetration of charged particles in matter, Nat. Akad. Sci. Publication 1133, Nucl. Sci. Series Report No. 39, Washington (Nat. Akad. Sci.) 1964, 388 pp. Available from Google books

Therefore, for small values of $\kappa < 0.01$, pdf approaches the Landau distribution.

For values $\kappa > 10$, the Gauss approximation should be used with $\mu$ and $\sigma$ given by Vavilov::mean(kappa, beta2) and sqrt(Vavilov::variance(kappa, beta2).

The original Vavilov pdf is obtained by v.Pdf(lambdaV/kappa-log(kappa))/kappa.

For detailed description see B. Schorr, Programs for the Landau and the Vavilov distributions and the corresponding random numbers, Computer Phys. Comm. 7 (1974) 215-224, which has been implemented in CERNLIB (G116).

The class stores coefficients needed to calculate $p(\lambda; \kappa, \beta^2)$ for fixed values of $\kappa$ and $\beta^2$. Changing these values is computationally expensive.

The parameter $\kappa$ should be in the range $0.01 \le \kappa \le 10$. In contrast to the CERNLIB implementation, all values of $\kappa \ge 0.001$ may be used, but may result in slower running and/or inaccurate results.

The parameter $\beta^2$ must be in the range $0 \le \beta^2 \le 1$.

Two parameters which are fixed in the CERNLIB implementation may be set by the user:

epsilonPM corresponds to $\epsilon^+ = \epsilon^-$ in Eqs. (2.1) and (2.2) of Schorr's paper. epsilonPM gives an estimate on the integral of the cumulative distribution function outside the range $\lambda_{min} \le \lambda \le \lambda_{max}$ where the approximation is valid. Thus, it determines the support of the approximation used here (called $T_0 - T_1$ in the paper). Schorr recommends $\epsilon^+ = \epsilon^- = 5\cdot 10^{-4}$. The code from CERNLIB has been extended such that also smaller values are possible.
epsilon corresponds to $\epsilon$ in Eq. (4.10) of Schorr's paper. It determines the accuracy of the series expansion. Schorr recommends $\epsilon = 10^{-5}$.

For the quantile calculation, the algorithm given by Schorr is not used, because it turns out to be very slow and still inaccurate. Instead, an initial estimate is calculated based on a pre-calculated table, which is subsequently improved by Newton iterations.

While the CERNLIB implementation calculates at most 156 terms in the series expansion for the pdf and cdf calculation, this class calculates up to 500 terms, depending on the values of epsilonPM and epsilon.

Average times on a Pentium Core2 Duo P8400 2.26GHz:

38us per call to SetKappaBeta2 or constructor
0.49us per call to Pdf, Cdf
8.2us per first call to Quantile after SetKappaBeta2 or constructor
0.83us per subsequent call to Quantile

Benno List, June 2010

Definition at line 131 of file VavilovAccurate.h.

Public Member Functions
	VavilovAccurate (double kappa=1, double beta2=1, double epsilonPM=5E-4, double epsilon=1E-5)
	Initialize an object to calculate the Vavilov distribution. More...

virtual	~VavilovAccurate ()
	Destructor. More...

double	Cdf (double x) const
	Evaluate the Vavilov cumulative probability density function. More...

double	Cdf (double x, double kappa, double beta2)
	Evaluate the Vavilov cumulative probability density function, and set kappa and beta2, if necessary. More...

double	Cdf_c (double x) const
	Evaluate the Vavilov complementary cumulative probability density function. More...

double	Cdf_c (double x, double kappa, double beta2)
	Evaluate the Vavilov complementary cumulative probability density function, and set kappa and beta2, if necessary. More...

virtual double	GetBeta2 () const
	Return the current value of $\beta^2$. More...

double	GetEpsilon () const
	Return the current value of $\epsilon$. More...

double	GetEpsilonPM () const
	Return the current value of $\epsilon^+ = \epsilon^-$. More...

virtual double	GetKappa () const
	Return the current value of $\kappa$. More...

virtual double	GetLambdaMax () const
	Return the maximum value of $\lambda$ for which $p(\lambda; \kappa, \beta^2)$ is nonzero in the current approximation. More...

virtual double	GetLambdaMin () const
	Return the minimum value of $\lambda$ for which $p(\lambda; \kappa, \beta^2)$ is nonzero in the current approximation. More...

double	GetNTerms () const
	Return the number of terms used in the series expansion. More...

virtual double	Mode () const
	Return the value of $\lambda$ where the pdf is maximal. More...

virtual double	Mode (double kappa, double beta2)
	Return the value of $\lambda$ where the pdf is maximal function, and set kappa and beta2, if necessary. More...

double	Pdf (double x) const
	Evaluate the Vavilov probability density function. More...

double	Pdf (double x, double kappa, double beta2)
	Evaluate the Vavilov probability density function, and set kappa and beta2, if necessary. More...

double	Quantile (double z) const
	Evaluate the inverse of the Vavilov cumulative probability density function. More...

double	Quantile (double z, double kappa, double beta2)
	Evaluate the inverse of the Vavilov cumulative probability density function, and set kappa and beta2, if necessary. More...

double	Quantile_c (double z) const
	Evaluate the inverse of the complementary Vavilov cumulative probability density function. More...

double	Quantile_c (double z, double kappa, double beta2)
	Evaluate the inverse of the complementary Vavilov cumulative probability density function, and set kappa and beta2, if necessary. More...

void	Set (double kappa, double beta2, double epsilonPM=5E-4, double epsilon=1E-5)
	(Re)Initialize the object More...

virtual void	SetKappaBeta2 (double kappa, double beta2)
	Change $\kappa$ and $\beta^2$ and recalculate coefficients if necessary. More...

Public Member Functions inherited from ROOT::Math::Vavilov
	Vavilov ()
	Default constructor. More...

virtual	~Vavilov ()
	Destructor. More...

virtual double	Kurtosis () const
	Return the theoretical kurtosis $\gamma_2 = \frac{1/3 - \beta^2/4}{\kappa^3 \sigma^4}$. More...

virtual double	Mean () const
	Return the theoretical mean $\mu = \gamma-1- \ln \kappa - \beta^2$, where $\gamma = 0.5772\dots$ is Euler's constant. More...

virtual double	Skewness () const
	Return the theoretical skewness $\gamma_1 = \frac{1/2 - \beta^2/3}{\kappa^2 \sigma^3} $. More...

virtual double	Variance () const
	Return the theoretical variance $\sigma^2 = \frac{1 - \beta^2/2}{\kappa}$. More...

Static Public Member Functions
static VavilovAccurate *	GetInstance ()
	Returns a static instance of class VavilovFast. More...

static VavilovAccurate *	GetInstance (double kappa, double beta2)
	Returns a static instance of class VavilovFast, and sets the values of kappa and beta2. More...

Static Public Member Functions inherited from ROOT::Math::Vavilov
static double	Kurtosis (double kappa, double beta2)
	Return the theoretical kurtosis $\gamma_2 = \frac{1/3 - \beta^2/4}{\kappa^3 \sigma^4}$. More...

static double	Mean (double kappa, double beta2)
	Return the theoretical Mean $\mu = \gamma-1- \ln \kappa - \beta^2$. More...

static double	Skewness (double kappa, double beta2)
	Return the theoretical skewness $\gamma_1 = \frac{1/2 - \beta^2/3}{\kappa^2 \sigma^3} $. More...

static double	Variance (double kappa, double beta2)
	Return the theoretical Variance $\sigma^2 = \frac{1 - \beta^2/2}{\kappa}$. More...

Private Types
enum	{ MAXTERMS =500 }

enum	{ kNquantMax =32 }

Private Member Functions
double	G116f1 (double x) const

double	G116f2 (double x) const

void	InitQuantile () const

int	Rzero (double a, double b, double &x0, double eps, int mxf, double(VavilovAccurate::*f)(double) const) const

Static Private Member Functions
static double	E1plLog (double x)

Private Attributes
double	fA_cdf [MAXTERMS+1]

double	fA_pdf [MAXTERMS+1]

double	fB_cdf [MAXTERMS+1]

double	fB_pdf [MAXTERMS+1]

double	fBeta2

double	fEpsilon

double	fEpsilonPM

double	fH [8]

double	fKappa

double	fLambda [kNquantMax]

int	fNQuant

double	fOmega

double	fQuant [kNquantMax]

bool	fQuantileInit

double	fT

double	fT0

double	fT1

double	fX0

Static Private Attributes
static VavilovAccurate *	fgInstance = 0

#include <Math/VavilovAccurate.h>

Inheritance diagram for ROOT::Math::VavilovAccurate:

[legend]

Member Enumeration Documentation

◆ anonymous enum

anonymous enum

private

Enumerator
MAXTERMS

Definition at line 332 of file VavilovAccurate.h.

◆ anonymous enum

anonymous enum

private

Enumerator
kNquantMax

Definition at line 339 of file VavilovAccurate.h.

Constructor & Destructor Documentation

◆ VavilovAccurate()

ROOT::Math::VavilovAccurate::VavilovAccurate	(	double	kappa = `1`,
		double	beta2 = `1`,
		double	epsilonPM = `5E-4`,
		double	epsilon = `1E-5`
	)

Initialize an object to calculate the Vavilov distribution.

Parameters

kappa	The parameter $\kappa$, which must be in the range $\kappa \ge 0.001 $
beta2	The parameter $\beta^2$, which must be in the range $0 \le \beta^2 \le 1 $
epsilonPM	$\epsilon^+ = \epsilon^-$ in Eqs. (2.1) and (2.2) of Schorr's paper; gives an estimate on the integral of the cumulative distribution function outside the range $\lambda_{min} \le \lambda \le \lambda_{max}$ where the approximation is valid.
epsilon	$\epsilon$ in Eq. (4.10) of Schorr's paper; determines the accuracy of the series expansion.

Definition at line 52 of file VavilovAccurate.cxx.

◆ ~VavilovAccurate()

ROOT::Math::VavilovAccurate::~VavilovAccurate ( )

virtual

Destructor.

Definition at line 58 of file VavilovAccurate.cxx.

Member Function Documentation

◆ Cdf() [1/2]

double ROOT::Math::VavilovAccurate::Cdf ( double x ) const

virtual

Evaluate the Vavilov cumulative probability density function.

Parameters

x	The Landau parameter $x = \lambda_L$

Implements ROOT::Math::Vavilov.

Definition at line 257 of file VavilovAccurate.cxx.

◆ Cdf() [2/2]

double ROOT::Math::VavilovAccurate::Cdf	(	double	x,
		double	kappa,
		double	beta2
	)

virtual

Evaluate the Vavilov cumulative probability density function, and set kappa and beta2, if necessary.

Parameters

x	The Landau parameter $x = \lambda_L$
kappa	The parameter $\kappa$, which must be in the range $\kappa \ge 0.001 $
beta2	The parameter $\beta^2$, which must be in the range $0 \le \beta^2 \le 1 $

Implements ROOT::Math::Vavilov.

Definition at line 292 of file VavilovAccurate.cxx.

◆ Cdf_c() [1/2]

double ROOT::Math::VavilovAccurate::Cdf_c ( double x ) const

virtual

Evaluate the Vavilov complementary cumulative probability density function.

Parameters

x	The Landau parameter $x = \lambda_L$

Implements ROOT::Math::Vavilov.

Definition at line 297 of file VavilovAccurate.cxx.

◆ Cdf_c() [2/2]

double ROOT::Math::VavilovAccurate::Cdf_c	(	double	x,
		double	kappa,
		double	beta2
	)

virtual

Evaluate the Vavilov complementary cumulative probability density function, and set kappa and beta2, if necessary.

Parameters

x	The Landau parameter $x = \lambda_L$
kappa	The parameter $\kappa$, which must be in the range $\kappa \ge 0.001 $
beta2	The parameter $\beta^2$, which must be in the range $0 \le \beta^2 \le 1 $

Implements ROOT::Math::Vavilov.

Definition at line 332 of file VavilovAccurate.cxx.

◆ E1plLog()

double ROOT::Math::VavilovAccurate::E1plLog ( double x )

staticprivate

Definition at line 645 of file VavilovAccurate.cxx.

◆ G116f1()

double ROOT::Math::VavilovAccurate::G116f1 ( double x ) const

private

Definition at line 487 of file VavilovAccurate.cxx.

◆ G116f2()

double ROOT::Math::VavilovAccurate::G116f2 ( double x ) const

private

Definition at line 496 of file VavilovAccurate.cxx.

◆ GetBeta2()

double ROOT::Math::VavilovAccurate::GetBeta2 ( ) const

virtual

Return the current value of $\beta^2$.

Implements ROOT::Math::Vavilov.

Definition at line 672 of file VavilovAccurate.cxx.

◆ GetEpsilon()

double ROOT::Math::VavilovAccurate::GetEpsilon ( ) const

Return the current value of $\epsilon$.

Definition at line 704 of file VavilovAccurate.cxx.

◆ GetEpsilonPM()

double ROOT::Math::VavilovAccurate::GetEpsilonPM ( ) const

Return the current value of $\epsilon^+ = \epsilon^-$.

Definition at line 700 of file VavilovAccurate.cxx.

◆ GetInstance() [1/2]

VavilovAccurate * ROOT::Math::VavilovAccurate::GetInstance ( )

static

Returns a static instance of class VavilovFast.

Definition at line 451 of file VavilovAccurate.cxx.

◆ GetInstance() [2/2]

VavilovAccurate * ROOT::Math::VavilovAccurate::GetInstance	(	double	kappa,
		double	beta2
	)

static

Returns a static instance of class VavilovFast, and sets the values of kappa and beta2.

Parameters

kappa	The parameter $\kappa$, which must be in the range $\kappa \ge 0.001 $
beta2	The parameter $\beta^2$, which must be in the range $0 \le \beta^2 \le 1 $

Definition at line 456 of file VavilovAccurate.cxx.

◆ GetKappa()

double ROOT::Math::VavilovAccurate::GetKappa ( ) const

virtual

Return the current value of $\kappa$.

Implements ROOT::Math::Vavilov.

Definition at line 668 of file VavilovAccurate.cxx.

◆ GetLambdaMax()

double ROOT::Math::VavilovAccurate::GetLambdaMax ( ) const

virtual

Return the maximum value of $\lambda$ for which $p(\lambda; \kappa, \beta^2)$ is nonzero in the current approximation.

Implements ROOT::Math::Vavilov.

Definition at line 664 of file VavilovAccurate.cxx.

◆ GetLambdaMin()

double ROOT::Math::VavilovAccurate::GetLambdaMin ( ) const

virtual

Return the minimum value of $\lambda$ for which $p(\lambda; \kappa, \beta^2)$ is nonzero in the current approximation.

Implements ROOT::Math::Vavilov.

Definition at line 660 of file VavilovAccurate.cxx.

◆ GetNTerms()

double ROOT::Math::VavilovAccurate::GetNTerms ( ) const

Return the number of terms used in the series expansion.

Definition at line 708 of file VavilovAccurate.cxx.

◆ InitQuantile()

void ROOT::Math::VavilovAccurate::InitQuantile ( ) const

private

Definition at line 186 of file VavilovAccurate.cxx.

◆ Mode() [1/2]

double ROOT::Math::VavilovAccurate::Mode ( ) const

virtual

Return the value of $\lambda$ where the pdf is maximal.

Reimplemented from ROOT::Math::Vavilov.

Definition at line 676 of file VavilovAccurate.cxx.

◆ Mode() [2/2]

double ROOT::Math::VavilovAccurate::Mode	(	double	kappa,
		double	beta2
	)

virtual

Return the value of $\lambda$ where the pdf is maximal function, and set kappa and beta2, if necessary.

Parameters

kappa	The parameter $\kappa$, which must be in the range $\kappa \ge 0.001 $
beta2	The parameter $\beta^2$, which must be in the range $0 \le \beta^2 \le 1 $

Reimplemented from ROOT::Math::Vavilov.

Definition at line 695 of file VavilovAccurate.cxx.

◆ Pdf() [1/2]

double ROOT::Math::VavilovAccurate::Pdf ( double x ) const

virtual

Evaluate the Vavilov probability density function.

Parameters

x	The Landau parameter $x = \lambda_L$

Implements ROOT::Math::Vavilov.

Definition at line 218 of file VavilovAccurate.cxx.

◆ Pdf() [2/2]

double ROOT::Math::VavilovAccurate::Pdf	(	double	x,
		double	kappa,
		double	beta2
	)

virtual

Evaluate the Vavilov probability density function, and set kappa and beta2, if necessary.

Parameters

x	The Landau parameter $x = \lambda_L$
kappa	The parameter $\kappa$, which must be in the range $\kappa \ge 0.001 $
beta2	The parameter $\beta^2$, which must be in the range $0 \le \beta^2 \le 1 $

Implements ROOT::Math::Vavilov.

Definition at line 252 of file VavilovAccurate.cxx.

◆ Quantile() [1/2]

double ROOT::Math::VavilovAccurate::Quantile ( double z ) const

virtual

Evaluate the inverse of the Vavilov cumulative probability density function.

Parameters

z	The argument $z$, which must be in the range $0 \le z \le 1$

Implements ROOT::Math::Vavilov.

Definition at line 337 of file VavilovAccurate.cxx.

◆ Quantile() [2/2]

double ROOT::Math::VavilovAccurate::Quantile	(	double	z,
		double	kappa,
		double	beta2
	)

virtual

Evaluate the inverse of the Vavilov cumulative probability density function, and set kappa and beta2, if necessary.

Parameters

z	The argument $z$, which must be in the range $0 \le z \le 1$
kappa	The parameter $\kappa$, which must be in the range $\kappa \ge 0.001 $
beta2	The parameter $\beta^2$, which must be in the range $0 \le \beta^2 \le 1 $

Implements ROOT::Math::Vavilov.

Definition at line 384 of file VavilovAccurate.cxx.

◆ Quantile_c() [1/2]

double ROOT::Math::VavilovAccurate::Quantile_c ( double z ) const

virtual

Evaluate the inverse of the complementary Vavilov cumulative probability density function.

Parameters

z	The argument $z$, which must be in the range $0 \le z \le 1$

Implements ROOT::Math::Vavilov.

Definition at line 389 of file VavilovAccurate.cxx.

◆ Quantile_c() [2/2]

double ROOT::Math::VavilovAccurate::Quantile_c	(	double	z,
		double	kappa,
		double	beta2
	)

virtual

Evaluate the inverse of the complementary Vavilov cumulative probability density function, and set kappa and beta2, if necessary.

Parameters

z	The argument $z$, which must be in the range $0 \le z \le 1$
kappa	The parameter $\kappa$, which must be in the range $\kappa \ge 0.001 $
beta2	The parameter $\beta^2$, which must be in the range $0 \le \beta^2 \le 1 $

Implements ROOT::Math::Vavilov.

Definition at line 446 of file VavilovAccurate.cxx.

◆ Rzero()

int ROOT::Math::VavilovAccurate::Rzero	(	double	a,
		double	b,
		double &	x0,
		double	eps,
		int	mxf,
		double(VavilovAccurate::*)(double) const	f
	)		const

private

Definition at line 505 of file VavilovAccurate.cxx.

◆ Set()

void ROOT::Math::VavilovAccurate::Set	(	double	kappa,
		double	beta2,
		double	epsilonPM = `5E-4`,
		double	epsilon = `1E-5`
	)

(Re)Initialize the object

Parameters

kappa	The parameter $\kappa$, which must be in the range $\kappa \ge 0.001 $
beta2	The parameter $\beta^2$, which must be in the range $0 \le \beta^2 \le 1 $
epsilonPM	$\epsilon^+ = \epsilon^-$ in Eqs. (2.1) and (2.2) of Schorr's paper; gives an estimate on the integral of the cumulative distribution function outside the range $\lambda_{min} \le \lambda \le \lambda_{max}$ where the approximation is valid.
epsilon	$\epsilon$ in Eq. (4.10) of Schorr's paper; determines the accuracy of the series expansion.

Definition at line 67 of file VavilovAccurate.cxx.

◆ SetKappaBeta2()

void ROOT::Math::VavilovAccurate::SetKappaBeta2	(	double	kappa,
		double	beta2
	)

virtual

Change $\kappa$ and $\beta^2$ and recalculate coefficients if necessary.

Parameters

kappa	The parameter $\kappa$, which must be in the range $\kappa \ge 0.001 $
beta2	The parameter $\beta^2$, which must be in the range $0 \le \beta^2 \le 1 $

Implements ROOT::Math::Vavilov.

Definition at line 63 of file VavilovAccurate.cxx.

Member Data Documentation

◆ fA_cdf

double ROOT::Math::VavilovAccurate::fA_cdf[MAXTERMS+1]

private

Definition at line 333 of file VavilovAccurate.h.

◆ fA_pdf

double ROOT::Math::VavilovAccurate::fA_pdf[MAXTERMS+1]

private

Definition at line 333 of file VavilovAccurate.h.

◆ fB_cdf

double ROOT::Math::VavilovAccurate::fB_cdf[MAXTERMS+1]

private

Definition at line 333 of file VavilovAccurate.h.

◆ fB_pdf

double ROOT::Math::VavilovAccurate::fB_pdf[MAXTERMS+1]

private

Definition at line 333 of file VavilovAccurate.h.

◆ fBeta2

double ROOT::Math::VavilovAccurate::fBeta2

private

Definition at line 334 of file VavilovAccurate.h.

◆ fEpsilon

double ROOT::Math::VavilovAccurate::fEpsilon

private

Definition at line 335 of file VavilovAccurate.h.

◆ fEpsilonPM

double ROOT::Math::VavilovAccurate::fEpsilonPM

private

Definition at line 335 of file VavilovAccurate.h.

◆ fgInstance

VavilovAccurate * ROOT::Math::VavilovAccurate::fgInstance = 0

staticprivate

Definition at line 345 of file VavilovAccurate.h.

◆ fH

double ROOT::Math::VavilovAccurate::fH[8]

private

Definition at line 333 of file VavilovAccurate.h.

◆ fKappa

double ROOT::Math::VavilovAccurate::fKappa

private

Definition at line 334 of file VavilovAccurate.h.

◆ fLambda

double ROOT::Math::VavilovAccurate::fLambda[kNquantMax]

mutableprivate

Definition at line 341 of file VavilovAccurate.h.

◆ fNQuant

int ROOT::Math::VavilovAccurate::fNQuant

mutableprivate

Definition at line 338 of file VavilovAccurate.h.

◆ fOmega

double ROOT::Math::VavilovAccurate::fOmega

private

Definition at line 333 of file VavilovAccurate.h.

◆ fQuant

double ROOT::Math::VavilovAccurate::fQuant[kNquantMax]

mutableprivate

Definition at line 340 of file VavilovAccurate.h.

◆ fQuantileInit

bool ROOT::Math::VavilovAccurate::fQuantileInit

mutableprivate

Definition at line 337 of file VavilovAccurate.h.

◆ fT

double ROOT::Math::VavilovAccurate::fT

private

Definition at line 333 of file VavilovAccurate.h.

◆ fT0

double ROOT::Math::VavilovAccurate::fT0

private

Definition at line 333 of file VavilovAccurate.h.

◆ fT1

double ROOT::Math::VavilovAccurate::fT1

private

Definition at line 333 of file VavilovAccurate.h.

◆ fX0

double ROOT::Math::VavilovAccurate::fX0

private

Definition at line 333 of file VavilovAccurate.h.

The documentation for this class was generated from the following files:

math/mathmore/inc/Math/VavilovAccurate.h
math/mathmore/src/VavilovAccurate.cxx

x	The Landau parameter \(x = \lambda_L\)
kappa	The parameter \(\kappa\), which must be in the range \(\kappa \ge 0.001 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

x	The Landau parameter \(x = \lambda_L\)
kappa	The parameter \(\kappa\), which must be in the range \(\kappa \ge 0.001 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

x	The Landau parameter \(x = \lambda_L\)
kappa	The parameter \(\kappa\), which must be in the range \(\kappa \ge 0.001 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

z	The argument \(z\), which must be in the range \(0 \le z \le 1\)
kappa	The parameter \(\kappa\), which must be in the range \(\kappa \ge 0.001 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

z	The argument \(z\), which must be in the range \(0 \le z \le 1\)
kappa	The parameter \(\kappa\), which must be in the range \(\kappa \ge 0.001 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

Public Member Functions

Static Public Member Functions

Private Types

Private Member Functions

Static Private Member Functions

Private Attributes

Static Private Attributes

Member Enumeration Documentation

◆ anonymous enum

◆ anonymous enum

Constructor & Destructor Documentation

◆ VavilovAccurate()

◆ ~VavilovAccurate()

Member Function Documentation

◆ Cdf() [1/2]

◆ Cdf() [2/2]

◆ Cdf_c() [1/2]

◆ Cdf_c() [2/2]

◆ E1plLog()

◆ G116f1()

◆ G116f2()

◆ GetBeta2()

◆ GetEpsilon()

◆ GetEpsilonPM()

◆ GetInstance() [1/2]

◆ GetInstance() [2/2]

◆ GetKappa()

◆ GetLambdaMax()

◆ GetLambdaMin()

◆ GetNTerms()

◆ InitQuantile()

◆ Mode() [1/2]

◆ Mode() [2/2]

◆ Pdf() [1/2]

◆ Pdf() [2/2]

◆ Quantile() [1/2]

◆ Quantile() [2/2]

◆ Quantile_c() [1/2]

◆ Quantile_c() [2/2]

◆ Rzero()

◆ Set()

◆ SetKappaBeta2()

Member Data Documentation

◆ fA_cdf

◆ fA_pdf

◆ fB_cdf

◆ fB_pdf

◆ fBeta2

◆ fEpsilon

◆ fEpsilonPM

◆ fgInstance

◆ fH

◆ fKappa

◆ fLambda

◆ fNQuant

◆ fOmega

◆ fQuant

◆ fQuantileInit

◆ fT

◆ fT0

◆ fT1

◆ fX0