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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_stat_pdf_vavilov (g01mu)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_stat_pdf_vavilov (g01mu) returns the value of the Vavilov density function ϕVλ;κ,β2.
It is intended to be used after a call to nag_stat_init_vavilov (g01zu).


[result, ifail] = g01mu(x, rcomm)
[result, ifail] = nag_stat_pdf_vavilov(x, rcomm)


nag_stat_pdf_vavilov (g01mu) evaluates an approximation to the Vavilov density function ϕVλ;κ,β2 given by
ϕVλ;κ,β2=12πi c-i c+ieλsfs;κ,β2ds,  
where κ>0 and 0β21, c is an arbitrary real constant and
fs;κ,β2=Cκ,β2expslnκ+s+κβ2 lnsκ+E1 sκ -κexp-sκ .  
E1x=0xt-11-e-tdt is the exponential integral, Cκ,β2=expκ1+γβ2 and γ is Euler's constant.
The method used is based on Fourier expansions. Further details can be found in Schorr (1974).
For values of κ0.01, the Vavilov distribution can be replaced by the Landau distribution since λV=λL-lnκ/κ. For values of κ10, the Vavilov distribution can be replaced by a Gaussian distribution with mean μ=γ-1-β2-lnκ and variance σ2=2-β2/2κ.


Schorr B (1974) Programs for the Landau and the Vavilov distributions and the corresponding random numbers Comp. Phys. Comm. 7 215–224


Compulsory Input Parameters

1:     x – double scalar
The argument λ of the function.
2:     rcomm322 – double array
This must be the same argument rcomm as returned by a previous call to nag_stat_init_vavilov (g01zu).

Optional Input Parameters


Output Parameters

1:     result – double scalar
The result of the function.
2:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
Either the initialization function has not been called prior to the first call of this function or a communication array has become corrupted.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


At least five significant digits are usually correct.

Further Comments

nag_stat_pdf_vavilov (g01mu) can be called repeatedly with different values of λ provided that the values of κ and β2 remain unchanged between calls. Otherwise, nag_stat_init_vavilov (g01zu) must be called again.


This example evaluates ϕVλ;κ,β2 at λ=2.5, κ=0.4 and β2=0.1, and prints the results.
function g01mu_example

fprintf('g01mu example results\n\n');

x      = 2.5;
rkappa = 0.4;
beta2  = 0.1;

fprintf('  x     rkappa    beta2        y\n\n');

% Initialize rcomm array (new value of rkappa or beta2)
mode = int64(0);
[xl, xu, rcomm, ifail] = g01zu( ...
                               rkappa, beta2, mode);

% Compute value of Vavilov density function
[y, ifail] = g01mu( ...
                    x, rcomm);

% Display results
fprintf('%4.1f%9.1f%9.1f%15.4e\n', x, rkappa, beta2, y);

g01mu example results

  x     rkappa    beta2        y

 2.5      0.4      0.1     8.3675e-02

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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