Base class describing a Vavilov distribution.

The Vavilov distribution is defined in P.V. Vavilov: Ionization losses of high-energy heavy particles, Sov. Phys. JETP 5 (1957) 749 [Zh. Eksp. Teor. Fiz. 32 (1957) 920].

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 Vavilov, 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.

Two subclasses are provided:

VavilovFast uses the algorithm by A. Rotondi and P. Montagna, Fast calculation of Vavilov distribution, Nucl. Instr. and Meth. B47 (1990) 215-224, which has been implemented in CERNLIB (G115).
VavilovAccurate uses the algorithm by 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).

Both subclasses store coefficients needed to calculate \(p(\lambda; \kappa, \beta^2)\) for fixed values of \(\kappa\) and \(\beta^2\). Changing these values is computationally expensive.

VavilovFast is about 5 times faster for the calculation of the Pdf than VavilovAccurate; initialization takes about 100 times longer than calculation of the Pdf value. For the quantile calculation, VavilovFast is 30 times faster for the initialization, and 6 times faster for subsequent calculations. Initialization for Quantile takes 27 (11) times longer than subsequent calls for VavilovFast (VavilovAccurate).

Definition at line 120 of file Vavilov.h.

Public Member Functions
	Vavilov ()
	Default constructor.

virtual	~Vavilov ()
	Destructor.

virtual double	Cdf (double x) const =0
	Evaluate the Vavilov cumulative probability density function.

virtual double	Cdf (double x, double kappa, double beta2)=0
	Evaluate the Vavilov cumulative probability density function, and set kappa and beta2, if necessary.

virtual double	Cdf_c (double x) const =0
	Evaluate the Vavilov complementary cumulative probability density function.

virtual double	Cdf_c (double x, double kappa, double beta2)=0
	Evaluate the Vavilov complementary cumulative probability density function, and set kappa and beta2, if necessary.

virtual double	GetBeta2 () const =0
	Return the current value of \(\beta^2\).

virtual double	GetKappa () const =0
	Return the current value of \(\kappa\).

virtual double	GetLambdaMax () const =0
	Return the maximum value of \(\lambda\) for which \(p(\lambda; \kappa, \beta^2)\) is nonzero in the current approximation.

virtual double	GetLambdaMin () const =0
	Return the minimum value of \(\lambda\) for which \(p(\lambda; \kappa, \beta^2)\) is nonzero in the current approximation.

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

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

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

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.

virtual double	Pdf (double x) const =0
	Evaluate the Vavilov probability density function.

virtual double	Pdf (double x, double kappa, double beta2)=0
	Evaluate the Vavilov probability density function, and set kappa and beta2, if necessary.

virtual double	Quantile (double z) const =0
	Evaluate the inverse of the Vavilov cumulative probability density function.

virtual double	Quantile (double z, double kappa, double beta2)=0
	Evaluate the inverse of the Vavilov cumulative probability density function, and set kappa and beta2, if necessary.

virtual double	Quantile_c (double z) const =0
	Evaluate the inverse of the complementary Vavilov cumulative probability density function.

virtual double	Quantile_c (double z, double kappa, double beta2)=0
	Evaluate the inverse of the complementary Vavilov cumulative probability density function, and set kappa and beta2, if necessary.

virtual void	SetKappaBeta2 (double kappa, double beta2)=0
	Change \(\kappa\) and \(\beta^2\) and recalculate coefficients if necessary.

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

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

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

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

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

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

#include <Math/Vavilov.h>

Inheritance diagram for ROOT::Math::Vavilov:

[legend]

Constructor & Destructor Documentation

◆ Vavilov()

ROOT::Math::Vavilov::Vavilov ( )

Default constructor.

Definition at line 46 of file Vavilov.cxx.

◆ ~Vavilov()

ROOT::Math::Vavilov::~Vavilov ( )

virtual

Destructor.

Definition at line 50 of file Vavilov.cxx.

Member Function Documentation

◆ Cdf() [1/2]

virtual double ROOT::Math::Vavilov::Cdf ( double x ) const

pure virtual

Evaluate the Vavilov cumulative probability density function.

Parameters

x	The Landau parameter \(x = \lambda_L\)

Implemented in ROOT::Math::VavilovAccurate, and ROOT::Math::VavilovFast.

◆ Cdf() [2/2]

virtual double ROOT::Math::Vavilov::Cdf	(	double	x,
		double	kappa,
		double	beta2
	)

pure 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 should be in the range \(0.01 \le \kappa \le 10 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

Implemented in ROOT::Math::VavilovAccurate, and ROOT::Math::VavilovFast.

◆ Cdf_c() [1/2]

virtual double ROOT::Math::Vavilov::Cdf_c ( double x ) const

pure virtual

Evaluate the Vavilov complementary cumulative probability density function.

Parameters

x	The Landau parameter \(x = \lambda_L\)

Implemented in ROOT::Math::VavilovAccurate, and ROOT::Math::VavilovFast.

◆ Cdf_c() [2/2]

virtual double ROOT::Math::Vavilov::Cdf_c	(	double	x,
		double	kappa,
		double	beta2
	)

pure 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 should be in the range \(0.01 \le \kappa \le 10 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

Implemented in ROOT::Math::VavilovAccurate, and ROOT::Math::VavilovFast.

◆ GetBeta2()

virtual double ROOT::Math::Vavilov::GetBeta2 ( ) const

pure virtual

Return the current value of \(\beta^2\).

Implemented in ROOT::Math::VavilovAccurate, and ROOT::Math::VavilovFast.

◆ GetKappa()

virtual double ROOT::Math::Vavilov::GetKappa ( ) const

pure virtual

Return the current value of \(\kappa\).

Implemented in ROOT::Math::VavilovAccurate, and ROOT::Math::VavilovFast.

◆ GetLambdaMax()

virtual double ROOT::Math::Vavilov::GetLambdaMax ( ) const

pure virtual

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

Implemented in ROOT::Math::VavilovAccurate, and ROOT::Math::VavilovFast.

◆ GetLambdaMin()

virtual double ROOT::Math::Vavilov::GetLambdaMin ( ) const

pure virtual

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

Implemented in ROOT::Math::VavilovAccurate, and ROOT::Math::VavilovFast.

◆ Kurtosis() [1/2]

double ROOT::Math::Vavilov::Kurtosis ( ) const

virtual

Return the theoretical kurtosis \(\gamma_2 = \frac{1/3 - \beta^2/4}{\kappa^3 \sigma^4}\).

Definition at line 105 of file Vavilov.cxx.

◆ Kurtosis() [2/2]

double ROOT::Math::Vavilov::Kurtosis	(	double	kappa,
		double	beta2
	)

static

Return the theoretical kurtosis \(\gamma_2 = \frac{1/3 - \beta^2/4}{\kappa^3 \sigma^4}\).

Parameters

kappa	The parameter \(\kappa\), which should be in the range \(0.01 \le \kappa \le 10 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

Definition at line 109 of file Vavilov.cxx.

◆ Mean() [1/2]

double ROOT::Math::Vavilov::Mean ( ) const

virtual

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

Definition at line 80 of file Vavilov.cxx.

◆ Mean() [2/2]

double ROOT::Math::Vavilov::Mean	(	double	kappa,
		double	beta2
	)

static

Return the theoretical Mean \(\mu = \gamma-1- \ln \kappa - \beta^2\).

Parameters

kappa	The parameter \(\kappa\), which should be in the range \(0.01 \le \kappa \le 10 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

Definition at line 84 of file Vavilov.cxx.

◆ Mode() [1/2]

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

virtual

Return the value of \(\lambda\) where the pdf is maximal.

Reimplemented in ROOT::Math::VavilovAccurate.

Definition at line 56 of file Vavilov.cxx.

◆ Mode() [2/2]

double ROOT::Math::Vavilov::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 should be in the range \(0.01 \le \kappa \le 10 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

Reimplemented in ROOT::Math::VavilovAccurate.

Definition at line 75 of file Vavilov.cxx.

◆ Pdf() [1/2]

virtual double ROOT::Math::Vavilov::Pdf ( double x ) const

pure virtual

Evaluate the Vavilov probability density function.

Parameters

x	The Landau parameter \(x = \lambda_L\)

Implemented in ROOT::Math::VavilovAccurate, and ROOT::Math::VavilovFast.

◆ Pdf() [2/2]

virtual double ROOT::Math::Vavilov::Pdf	(	double	x,
		double	kappa,
		double	beta2
	)

pure 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 should be in the range \(0.01 \le \kappa \le 10 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

Implemented in ROOT::Math::VavilovAccurate, and ROOT::Math::VavilovFast.

◆ Quantile() [1/2]

virtual double ROOT::Math::Vavilov::Quantile ( double z ) const

pure virtual

Evaluate the inverse of the Vavilov cumulative probability density function.

Parameters

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

Implemented in ROOT::Math::VavilovAccurate, and ROOT::Math::VavilovFast.

◆ Quantile() [2/2]

virtual double ROOT::Math::Vavilov::Quantile	(	double	z,
		double	kappa,
		double	beta2
	)

pure 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 should be in the range \(0.01 \le \kappa \le 10 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

Implemented in ROOT::Math::VavilovAccurate, and ROOT::Math::VavilovFast.

◆ Quantile_c() [1/2]

virtual double ROOT::Math::Vavilov::Quantile_c ( double z ) const

pure virtual

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

Parameters

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

Implemented in ROOT::Math::VavilovAccurate, and ROOT::Math::VavilovFast.

◆ Quantile_c() [2/2]

virtual double ROOT::Math::Vavilov::Quantile_c	(	double	z,
		double	kappa,
		double	beta2
	)

pure 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 should be in the range \(0.01 \le \kappa \le 10 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

Implemented in ROOT::Math::VavilovAccurate, and ROOT::Math::VavilovFast.

◆ SetKappaBeta2()

virtual void ROOT::Math::Vavilov::SetKappaBeta2	(	double	kappa,
		double	beta2
	)

pure virtual

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

Parameters

kappa	The parameter \(\kappa\), which should be in the range \(0.01 \le \kappa \le 10 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

Implemented in ROOT::Math::VavilovAccurate, and ROOT::Math::VavilovFast.

◆ Skewness() [1/2]

double ROOT::Math::Vavilov::Skewness ( ) const

virtual

Return the theoretical skewness \(\gamma_1 = \frac{1/2 - \beta^2/3}{\kappa^2 \sigma^3} \).

Definition at line 96 of file Vavilov.cxx.

◆ Skewness() [2/2]

double ROOT::Math::Vavilov::Skewness	(	double	kappa,
		double	beta2
	)

static

Return the theoretical skewness \(\gamma_1 = \frac{1/2 - \beta^2/3}{\kappa^2 \sigma^3} \).

Parameters

kappa	The parameter \(\kappa\), which should be in the range \(0.01 \le \kappa \le 10 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

Definition at line 100 of file Vavilov.cxx.

◆ Variance() [1/2]

double ROOT::Math::Vavilov::Variance ( ) const

virtual

Return the theoretical variance \(\sigma^2 = \frac{1 - \beta^2/2}{\kappa}\).

Definition at line 88 of file Vavilov.cxx.

◆ Variance() [2/2]

double ROOT::Math::Vavilov::Variance	(	double	kappa,
		double	beta2
	)

static

Return the theoretical Variance \(\sigma^2 = \frac{1 - \beta^2/2}{\kappa}\).

Parameters

kappa	The parameter \(\kappa\), which should be in the range \(0.01 \le \kappa \le 10 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

Definition at line 92 of file Vavilov.cxx.

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

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

Public Member Functions

Static Public Member Functions

Constructor & Destructor Documentation

◆ Vavilov()

◆ ~Vavilov()

Member Function Documentation

◆ Cdf() [1/2]

◆ Cdf() [2/2]

◆ Cdf_c() [1/2]

◆ Cdf_c() [2/2]

◆ GetBeta2()

◆ GetKappa()

◆ GetLambdaMax()

◆ GetLambdaMin()

◆ Kurtosis() [1/2]

◆ Kurtosis() [2/2]

◆ Mean() [1/2]

◆ Mean() [2/2]

◆ 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]

◆ SetKappaBeta2()

◆ Skewness() [1/2]

◆ Skewness() [2/2]

◆ Variance() [1/2]

◆ Variance() [2/2]