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 VavilovFast, 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 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).

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 \(\kappa\) must be in the range \(0.01 \le \kappa \le 12\).

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

Average times on a Pentium Core2 Duo P8400 2.26GHz:

9.9us per call to SetKappaBeta2 or constructor
0.095us per call to Pdf, Cdf
3.7us per first call to Quantile after SetKappaBeta2 or constructor
0.137us per subsequent call to Quantile

Benno List, June 2010

Definition at line 116 of file VavilovFast.h.

Public Member Functions
	VavilovFast (double kappa=1, double beta2=1)
	Initialize an object to calculate the Vavilov distribution.

virtual	~VavilovFast ()
	Destructor.

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

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

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

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

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

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

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

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

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

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

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

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.

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

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.

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

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

virtual	~Vavilov ()
	Destructor.

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	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 VavilovFast *	GetInstance ()
	Returns a static instance of class VavilovFast.

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

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

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

Private Attributes
double	fAC [14]

double	fBeta2

double	fHC [9]

int	fItype

double	fKappa

int	fNpt

double	fWCM [201]

Static Private Attributes
static VavilovFast *	fgInstance = 0

#include <Math/VavilovFast.h>

Inheritance diagram for ROOT::Math::VavilovFast:

[legend]

Constructor & Destructor Documentation

◆ VavilovFast()

ROOT::Math::VavilovFast::VavilovFast	(	double	kappa = `1`,
		double	beta2 = `1`
	)

Initialize an object to calculate the Vavilov distribution.

Parameters

kappa	The parameter \(\kappa\), which must be in the range \(0.01 \le \kappa \le 12 \)
beta2	The parameter \(\beta^2\), which must be in the range \(0 \le \beta^2 \le 1 \)

Definition at line 51 of file VavilovFast.cxx.

◆ ~VavilovFast()

ROOT::Math::VavilovFast::~VavilovFast ( )

virtual

Destructor.

Definition at line 57 of file VavilovFast.cxx.

Member Function Documentation

◆ Cdf() [1/2]

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

virtual

Evaluate the Vavilov cumulative probability density function.

Parameters

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

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 \(0.01 \le \kappa \le 12 \)
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 Attributes

Static Private Attributes

Constructor & Destructor Documentation

◆ VavilovFast()

◆ ~VavilovFast()

Member Function Documentation

◆ Cdf() [1/2]

◆ Cdf() [2/2]

◆ Cdf_c() [1/2]

◆ Cdf_c() [2/2]

◆ GetBeta2()

◆ GetInstance() [1/2]

◆ GetInstance() [2/2]

◆ GetKappa()

◆ GetLambdaMax()

◆ GetLambdaMin()

◆ Pdf() [1/2]

◆ Pdf() [2/2]

◆ Quantile() [1/2]

◆ Quantile() [2/2]

◆ Quantile_c() [1/2]

◆ Quantile_c() [2/2]

◆ SetKappaBeta2()

Member Data Documentation

◆ fAC

◆ fBeta2

◆ fgInstance

◆ fHC

◆ fItype

◆ fKappa

◆ fNpt

◆ fWCM