# NAG C Library Function Document

## 1Purpose

nag_prob_density_vavilov (g01muc) returns the value of the Vavilov density function ${\varphi }_{V}\left(\lambda \text{;}\kappa ,{\beta }^{2}\right)$.
It is intended to be used after a call to nag_init_vavilov (g01zuc).

## 2Specification

 #include #include
 double nag_prob_density_vavilov (double x, const double comm_arr[])

## 3Description

nag_prob_density_vavilov (g01muc) evaluates an approximation to the Vavilov density function ${\varphi }_{V}\left(\lambda \text{;}\kappa ,{\beta }^{2}\right)$ given by
 $ϕVλ;κ,β2=12πi ∫c-i∞ c+i∞eλsfs;κ,β2ds,$
where $\kappa >0$ and $0\le {\beta }^{2}\le 1$, $c$ is an arbitrary real constant and
 $fs;κ,β2=Cκ,β2expsln⁡κ+s+κβ2 lnsκ+E1 sκ -κexp-sκ .$
${E}_{1}\left(x\right)=\underset{0}{\overset{x}{\int }}{t}^{-1}\left(1-{e}^{-t}\right)dt$ is the exponential integral, $C\left(\kappa ,{\beta }^{2}\right)=\mathrm{exp}\left\{\kappa \left(1+\gamma {\beta }^{2}\right)\right\}$ and $\gamma$ is Euler's constant.
The method used is based on Fourier expansions. Further details can be found in Schorr (1974).
For values of $\kappa \le 0.01$, the Vavilov distribution can be replaced by the Landau distribution since ${\lambda }_{V}=\left({\lambda }_{L}-\mathrm{ln}\kappa \right)/\kappa$. For values of $\kappa \ge 10$, the Vavilov distribution can be replaced by a Gaussian distribution with mean $\mu =\gamma -1-{\beta }^{2}-\mathrm{ln}\kappa$ and variance ${\sigma }^{2}=\left(2-{\beta }^{2}\right)/2\kappa$.

## 4References

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

## 5Arguments

1:    $\mathbf{x}$doubleInput
On entry: the argument $\lambda$ of the function.
2:    $\mathbf{comm_arr}\left[322\right]$const doubleCommunication Array
On entry: this must be the same argument comm_arr as returned by a previous call to nag_init_vavilov (g01zuc).

None.

## 7Accuracy

At least five significant digits are usually correct.

## 8Parallelism and Performance

nag_prob_density_vavilov (g01muc) is not threaded in any implementation.

nag_prob_density_vavilov (g01muc) can be called repeatedly with different values of $\lambda$ provided that the values of $\kappa$ and ${\beta }^{2}$ remain unchanged between calls. Otherwise, nag_init_vavilov (g01zuc) must be called again. This is illustrated in Section 10.

## 10Example

This example evaluates ${\varphi }_{V}\left(\lambda \text{;}\kappa ,{\beta }^{2}\right)$ at $\lambda =2.5$, $\kappa =0.4$ and ${\beta }^{2}=0.1$, and prints the results.

### 10.1Program Text

Program Text (g01muce.c)

### 10.2Program Data

Program Data (g01muce.d)

### 10.3Program Results

Program Results (g01muce.r)