ROOT   Reference Guide
ROOT::Math::VavilovAccurate Class Reference

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 VavilovAccurateGetInstance ()
Returns a static instance of class VavilovFast. More...

static VavilovAccurateGetInstance (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
More...

double G116f2 (double x) const
More...

void InitQuantile () const
More...

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

## Static Private Member Functions

static double E1plLog (double x)
More...

## Private Attributes

double fA_cdf [MAXTERMS+1]
More...

double fA_pdf [MAXTERMS+1]
More...

double fB_cdf [MAXTERMS+1]
More...

double fB_pdf [MAXTERMS+1]
More...

double fBeta2
More...

double fEpsilon
More...

double fEpsilonPM
More...

double fH [8]
More...

double fKappa
More...

double fLambda [kNquantMax]
More...

int fNQuant
More...

double fOmega
More...

double fQuant [kNquantMax]
More...

bool fQuantileInit
More...

double fT
More...

double fT0
More...

double fT1
More...

double fX0
More...

## Static Private Attributes

static VavilovAccuratefgInstance = nullptr
More...

#include <Math/VavilovAccurate.h>

Inheritance diagram for ROOT::Math::VavilovAccurate:
[legend]

## ◆ 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.

## ◆ 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 51 of file VavilovAccurate.cxx.

## ◆ ~VavilovAccurate()

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

Destructor.

Definition at line 57 of file VavilovAccurate.cxx.

## ◆ 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 256 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 291 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 296 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 331 of file VavilovAccurate.cxx.

## ◆ E1plLog()

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

Definition at line 644 of file VavilovAccurate.cxx.

## ◆ G116f1()

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

Definition at line 486 of file VavilovAccurate.cxx.

## ◆ G116f2()

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

Definition at line 495 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 671 of file VavilovAccurate.cxx.

## ◆ GetEpsilon()

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

Return the current value of $$\epsilon$$.

Definition at line 703 of file VavilovAccurate.cxx.

## ◆ GetEpsilonPM()

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

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

Definition at line 699 of file VavilovAccurate.cxx.

## ◆ GetInstance() [1/2]

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

Returns a static instance of class VavilovFast.

Definition at line 450 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 455 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 667 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 663 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 659 of file VavilovAccurate.cxx.

## ◆ GetNTerms()

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

Return the number of terms used in the series expansion.

Definition at line 707 of file VavilovAccurate.cxx.

## ◆ InitQuantile()

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

Definition at line 185 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 675 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 694 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 217 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 251 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 336 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 383 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 388 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 445 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 504 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 66 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 62 of file VavilovAccurate.cxx.

## ◆ 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 = nullptr
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.

Libraries for ROOT::Math::VavilovAccurate:
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The documentation for this class was generated from the following files: