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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_stat_pdf_vavilov (g01mu)

## Purpose

nag_stat_pdf_vavilov (g01mu) returns the value of the Vavilov density function φV(λ;κ,β2)${\varphi }_{V}\left(\lambda \text{;}\kappa ,{\beta }^{2}\right)$.
It is intended to be used after a call to nag_stat_init_vavilov (g01zu).

## Syntax

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

## Description

nag_stat_pdf_vavilov (g01mu) evaluates an approximation to the Vavilov density function φV(λ;κ,β2)${\varphi }_{V}\left(\lambda \text{;}\kappa ,{\beta }^{2}\right)$ given by
 c + i∞ φV(λ;κ,β2) = 1/(2πi) ∫ eλsf(s;κ,β2)ds, c − i∞
$ϕV(λ;κ,β2)=12πi ∫c-i∞ c+i∞eλsf(s;κ,β2)ds,$
where κ > 0$\kappa >0$ and 0β21$0\le {\beta }^{2}\le 1$, c$c$ is an arbitrary real constant and
 f(s;κ,β2) = C(κ,β2)exp{slnκ + (s + κβ2)[ln(s/κ) + E1(s/κ)] − κexp( − s/κ)} . $f(s;κ,β2)=C(κ,β2)exp{sln⁡κ+(s+κβ2) [ln(sκ)+E1 (sκ) ]-κexp(-sκ) } .$
E1(x) = 0xt1(1et)dt${E}_{1}\left(x\right)=\underset{0}{\overset{x}{\int }}{t}^{-1}\left(1-{e}^{-t}\right)dt$ is the exponential integral, C(κ,β2) = exp{κ(1 + γβ2)}$C\left(\kappa ,{\beta }^{2}\right)=\mathrm{exp}\left\{\kappa \left(1+\gamma {\beta }^{2}\right)\right\}$ and γ$\gamma$ 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$\kappa \le 0.01$, the Vavilov distribution can be replaced by the Landau distribution since λV = (λLlnκ) / κ${\lambda }_{V}=\left({\lambda }_{L}-\mathrm{ln}\kappa \right)/\kappa$. For values of κ10$\kappa \ge 10$, the Vavilov distribution can be replaced by a Gaussian distribution with mean μ = γ1β2lnκ$\mu =\gamma -1-{\beta }^{2}-\mathrm{ln}\kappa$ and variance σ2 = (2β2) / 2κ${\sigma }^{2}=\left(2-{\beta }^{2}\right)/2\kappa$.

## References

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

## Parameters

### Compulsory Input Parameters

1:     x – double scalar
The argument λ$\lambda$ of the function.
2:     rcomm(322$322$) – double array
This must be the same parameter rcomm as returned by a previous call to nag_stat_init_vavilov (g01zu).

None.

None.

### Output Parameters

1:     result – double scalar
The result of the function.
2:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
Either the initialization function has not been called prior to the first call of this function or a communication array has become corrupted.

## Accuracy

At least five significant digits are usually correct.

nag_stat_pdf_vavilov (g01mu) can be called repeatedly with different values of λ$\lambda$ provided that the values of κ$\kappa$ and β2${\beta }^{2}$ remain unchanged between calls. Otherwise, nag_stat_init_vavilov (g01zu) must be called again.

## Example

```function nag_stat_pdf_vavilov_example
x = 2.5;
rkappa = 0.4;
beta2 = 0.1;
mode = int64(0);
[xl, xu, work, ifail] = nag_stat_init_vavilov(rkappa, beta2, mode);
[result, ifail] = nag_stat_pdf_vavilov(x, work)
```
```

result =

0.0837

ifail =

0

```
```function g01mu_example
x = 2.5;
rkappa = 0.4;
beta2 = 0.1;
mode = int64(0);
[xl, xu, work, ifail] = g01zu(rkappa, beta2, mode);
[result, ifail] = g01mu(x, work)
```
```

result =

0.0837

ifail =

0

```