# NAG Library Routine Document

## 1Purpose

g01muf 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 g01zuf.

## 2Specification

Fortran Interface
 Function g01muf ( x,
 Real (Kind=nag_wp) :: g01muf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x, rcomm(322)
C Header Interface
#include <nagmk26.h>
 double g01muf_ (const double *x, const double rcomm[], Integer *ifail)

## 3Description

g01muf 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:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the argument $\lambda$ of the function.
2:     $\mathbf{rcomm}\left(322\right)$ – Real (Kind=nag_wp) arrayCommunication Array
On entry: this must be the same argument rcomm as returned by a previous call to g01zuf.
3:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
Either the initialization routine has not been called prior to the first call of this routine or a communication array has become corrupted.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

At least five significant digits are usually correct.

## 8Parallelism and Performance

g01muf is not threaded in any implementation.

## 9Further Comments

g01muf can be called repeatedly with different values of $\lambda$ provided that the values of $\kappa$ and ${\beta }^{2}$ remain unchanged between calls. Otherwise, g01zuf 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 (g01mufe.f90)

### 10.2Program Data

Program Data (g01mufe.d)

### 10.3Program Results

Program Results (g01mufe.r)