class ROOT::Math::VavilovFast: public ROOT::Math::Vavilov


   Class describing a Vavilov distribution.

   The probability density function of the Vavilov distribution
   as function of Landau's parameter is given by:
  \f[ 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\f]
   where \f$\phi(s) = e^{C} e^{\psi(s)}\f$
   with  \f$ C = \kappa (1+\beta^2 \gamma )\f$
   and \f$\psi(s)= s \ln \kappa + (s+\beta^2 \kappa)
               \cdot \left ( \int \limits_{0}^{1}
               \frac{1 - e^{\frac{-st}{\kappa}}}{t} \,d t- \gamma \right )
               - \kappa \, e^{\frac{-s}{\kappa}}\f$.
   \f$ \gamma = 0.5772156649\dots\f$ is Euler's constant.

   For the class VavilovFast,
   Pdf returns the Vavilov distribution as function of Landau's parameter
   \f$\lambda_L = \lambda_V/\kappa  - \ln \kappa\f$,
   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
   <A HREF="http://books.google.de/books?id=kmMrAAAAYAAJ&lpg=PP9&pg=PA187#v=onepage&q&f=false">Google books</A>

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

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

   For values \f$\kappa > 10\f$, the Gauss approximation should be used
   with \f$\mu\f$ and \f$\sigma\f$ 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,
   <A HREF="http://dx.doi.org/10.1016/0168-583X(90)90749-K">Nucl. Instr. and Meth. B47 (1990) 215-224</A>,
   which has been implemented in
   <A HREF="http://wwwasdoc.web.cern.ch/wwwasdoc/shortwrupsdir/g115/top.html">
   CERNLIB (G115)</A>.

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

   The parameter \f$\kappa\f$ must be in the range \f$0.01 \le \kappa \le 12\f$.

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

   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

   @ingroup StatFunc

Function Members (Methods)

public:

virtual	~VavilovFast()
virtual double	Cdf(double x) const
virtual double	Cdf(double x, double kappa, double beta2)
virtual double	Cdf_c(double x) const
virtual double	Cdf_c(double x, double kappa, double beta2)
virtual double	GetBeta2() const
static ROOT::Math::VavilovFast*	GetInstance()
static ROOT::Math::VavilovFast*	GetInstance(double kappa, double beta2)
virtual double	GetKappa() const
virtual double	GetLambdaMax() const
virtual double	GetLambdaMin() const
virtual double	ROOT::Math::Vavilov::Kurtosis() const
static double	ROOT::Math::Vavilov::Kurtosis(double kappa, double beta2)
virtual double	ROOT::Math::Vavilov::Mean() const
static double	ROOT::Math::Vavilov::Mean(double kappa, double beta2)
virtual double	ROOT::Math::Vavilov::Mode() const
virtual double	ROOT::Math::Vavilov::Mode(double kappa, double beta2)
ROOT::Math::VavilovFast&	operator=(const ROOT::Math::VavilovFast&)
virtual double	Pdf(double x) const
virtual double	Pdf(double x, double kappa, double beta2)
virtual double	Quantile(double z) const
virtual double	Quantile(double z, double kappa, double beta2)
virtual double	Quantile_c(double z) const
virtual double	Quantile_c(double z, double kappa, double beta2)
virtual void	SetKappaBeta2(double kappa, double beta2)
virtual double	ROOT::Math::Vavilov::Skewness() const
static double	ROOT::Math::Vavilov::Skewness(double kappa, double beta2)
virtual double	ROOT::Math::Vavilov::Variance() const
static double	ROOT::Math::Vavilov::Variance(double kappa, double beta2)
ROOT::Math::VavilovFast	VavilovFast(const ROOT::Math::VavilovFast&)
ROOT::Math::VavilovFast	VavilovFast(double kappa = 1, double beta2 = 1)

Data Members

private:

double	fAC[14]
double	fBeta2
double	fHC[9]
int	fItype
double	fKappa
int	fNpt
double	fWCM[201]
static ROOT::Math::VavilovFast*	fgInstance

Class Charts

Inheritance Inherited Members Includes Libraries

Function documentation

VavilovFast(double kappa = 1, double beta2 = 1)

      Initialize an object to calculate the Vavilov distribution

       @param kappa The parameter \f$\kappa\f$, which must be in the range \f$0.01 \le \kappa \le 12 \f$
       @param beta2 The parameter \f$\beta^2\f$, which must be in the range \f$0 \le \beta^2 \le 1 \f$

virtual ~VavilovFast()

     Destructor

double Pdf(double x) const

       Evaluate the Vavilov probability density function

       @param x The Landau parameter \f$x = \lambda_L\f$

double Pdf(double x, double kappa, double beta2)

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

       @param x The Landau parameter \f$x = \lambda_L\f$
       @param kappa The parameter \f$\kappa\f$, which must be in the range \f$0.01 \le \kappa \le 12 \f$
       @param beta2 The parameter \f$\beta^2\f$, which must be in the range \f$0 \le \beta^2 \le 1 \f$

double Cdf(double x) const

       Evaluate the Vavilov cummulative probability density function

       @param x The Landau parameter \f$x = \lambda_L\f$

double Cdf(double x, double kappa, double beta2)

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

       @param x The Landau parameter \f$x = \lambda_L\f$
       @param kappa The parameter \f$\kappa\f$, which must be in the range \f$0.01 \le \kappa \le 12 \f$
       @param beta2 The parameter \f$\beta^2\f$, which must be in the range \f$0 \le \beta^2 \le 1 \f$

double Cdf_c(double x) const

       Evaluate the Vavilov complementary cummulative probability density function

       @param x The Landau parameter \f$x = \lambda_L\f$

double Cdf_c(double x, double kappa, double beta2)

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

       @param x The Landau parameter \f$x = \lambda_L\f$
       @param kappa The parameter \f$\kappa\f$, which must be in the range \f$0.01 \le \kappa \le 12 \f$
       @param beta2 The parameter \f$\beta^2\f$, which must be in the range \f$0 \le \beta^2 \le 1 \f$

double Quantile(double z) const

       Evaluate the inverse of the Vavilov cummulative probability density function

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

double Quantile(double z, double kappa, double beta2)

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

       @param z The argument \f$z\f$, which must be in the range \f$0 \le z \le 1\f$
       @param kappa The parameter \f$\kappa\f$, which must be in the range \f$0.01 \le \kappa \le 12 \f$
       @param beta2 The parameter \f$\beta^2\f$, which must be in the range \f$0 \le \beta^2 \le 1 \f$

double Quantile_c(double z) const

       Evaluate the inverse of the complementary Vavilov cummulative probability density function

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

double Quantile_c(double z, double kappa, double beta2)

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

       @param z The argument \f$z\f$, which must be in the range \f$0 \le z \le 1\f$
       @param kappa The parameter \f$\kappa\f$, which must be in the range \f$0.01 \le \kappa \le 12 \f$
       @param beta2 The parameter \f$\beta^2\f$, which must be in the range \f$0 \le \beta^2 \le 1 \f$

void SetKappaBeta2(double kappa, double beta2)

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

       @param kappa The parameter \f$\kappa\f$, which must be in the range \f$0.01 \le \kappa \le 12 \f$
       @param beta2 The parameter \f$\beta^2\f$, which must be in the range \f$0 \le \beta^2 \le 1 \f$

double GetLambdaMin() const

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

double GetLambdaMax() const

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

double GetKappa() const

      Return the current value of \f$\kappa\f$

double GetBeta2() const

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

VavilovFast * GetInstance()

      Returns a static instance of class VavilovFast

VavilovFast * GetInstance(double kappa, double beta2)

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

       @param kappa The parameter \f$\kappa\f$, which must be in the range \f$0.01 \le \kappa \le 12 \f$
       @param beta2 The parameter \f$\beta^2\f$, which must be in the range \f$0 \le \beta^2 \le 1 \f$