VavilovFast.h

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// @(#)root/mathmore:$Id$
// Authors: B. List 29.4.2010
 
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// Header file for class VavilovFast
//
// Created by: blist  at Thu Apr 29 11:19:00 2010
//
// Last update: Thu Apr 29 11:19:00 2010
//
#ifndef ROOT_Math_VavilovFast
#define ROOT_Math_VavilovFast
 
 
/**
   @ingroup StatFunc
 */
 
 
#include "Math/Vavilov.h"
 
namespace ROOT {
namespace Math {
 
//____________________________________________________________________________
/**
   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="https://cern-tex.web.cern.ch/cern-tex/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
 */
 
 
class VavilovFast: public Vavilov {
 
public:
 
 
   /**
      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$0 \le \beta^2 \le 1 \f$
   */
 
  VavilovFast(double kappa=1, double beta2=1);
 
 
   /**
     Destructor
   */
   virtual ~VavilovFast();
 
 
public:
 
   /**
       Evaluate the Vavilov probability density function
 
       @param x The Landau parameter \f$x = \lambda_L\f$
   */
   double Pdf (double x) const;
 
   /**
       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 Pdf (double x, double kappa, double beta2);
 
   /**
       Evaluate the Vavilov cumulative probability density function
 
       @param x The Landau parameter \f$x = \lambda_L\f$
   */
   double Cdf (double x) const;
 
   /**
       Evaluate the Vavilov cumulative 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, double kappa, double beta2);
 
   /**
       Evaluate the Vavilov complementary cumulative probability density function
 
       @param x The Landau parameter \f$x = \lambda_L\f$
   */
   double Cdf_c (double x) const;
 
   /**
       Evaluate the Vavilov complementary cumulative 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, double kappa, double beta2);
 
   /**
       Evaluate the inverse of the Vavilov cumulative 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) const;
 
   /**
       Evaluate the inverse of the Vavilov cumulative 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 (double z, double kappa, double beta2);
 
   /**
       Evaluate the inverse of the complementary Vavilov cumulative 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) const;
 
   /**
       Evaluate the inverse of the complementary Vavilov cumulative 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, 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$0 \le \beta^2 \le 1 \f$
   */
   virtual void SetKappaBeta2 (double kappa, double beta2);
 
   /**
      Return the minimum value of \f$\lambda\f$ for which \f$p(\lambda; \kappa, \beta^2)\f$
      is nonzero in the current approximation
   */
   virtual double GetLambdaMin() const;
 
   /**
      Return the maximum value of \f$\lambda\f$ for which \f$p(\lambda; \kappa, \beta^2)\f$
      is nonzero in the current approximation
   */
   virtual double GetLambdaMax() const;
 
   /**
      Return the current value of \f$\kappa\f$
   */
   virtual double GetKappa()     const;
 
   /**
      Return the current value of \f$\beta^2\f$
   */
   virtual double GetBeta2()     const;
 
   /**
      Returns a static instance of class VavilovFast
   */
   static VavilovFast *GetInstance();
 
   /**
      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$0 \le \beta^2 \le 1 \f$
   */
   static VavilovFast *GetInstance(double kappa, double beta2);
 
 
private:
   double fKappa;
   double fBeta2;
 
   double fAC[14];
   double fHC[9];
   double fWCM[201];
   int    fItype;
   int    fNpt;
 
   static VavilovFast *fgInstance;
 
};
 
   /**
       The Vavilov probability density function
 
       @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$0 \le \beta^2 \le 1 \f$
 
       @ingroup PdfFunc
   */
double vavilov_fast_pdf (double x, double kappa, double beta2);
 
   /**
       The Vavilov cumulative probability density function
 
       @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$0 \le \beta^2 \le 1 \f$
 
       @ingroup ProbFunc
   */
double vavilov_fast_cdf (double x, double kappa, double beta2);
 
   /**
       The Vavilov complementary cumulative probability density function
 
       @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$0 \le \beta^2 \le 1 \f$
 
       @ingroup ProbFunc
   */
double vavilov_fast_cdf_c (double x, double kappa, double beta2);
 
   /**
       The inverse of the Vavilov cumulative probability density function
 
       @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$0 \le \beta^2 \le 1 \f$
 
      @ingroup QuantFunc
   */
double vavilov_fast_quantile (double z, double kappa, double beta2);
 
   /**
       The inverse of the complementary Vavilov cumulative probability density function
 
       @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$0 \le \beta^2 \le 1 \f$
 
      @ingroup QuantFunc
   */
double vavilov_fast_quantile_c (double z, double kappa, double beta2);
 
} // namespace Math
} // namespace ROOT
 
#endif /* ROOT_Math_VavilovFast */