# NAG FL Interfaceg01muf (pdf_​vavilov)

## ▸▿ Contents

Settings help

FL Name Style:

FL Specification Language:

## 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)
#include <nag.h>
 double g01muf_ (const double *x, const double rcomm[], Integer *ifail)
The routine may be called by the names g01muf or nagf_stat_pdf_vavilov.

## 3Description

g01muf evaluates an approximation to the Vavilov density function ${\varphi }_{V}\left(\lambda \text{;}\kappa ,{\beta }^{2}\right)$ given by
 $ϕV(λ;κ,β2)=12πi ∫c-i∞ c+i∞eλsf(s;κ,β2)ds,$
where $\kappa >0$ and $0\le {\beta }^{2}\le 1$, $c$ is an arbitrary real constant and
 $f(s;κ,β2)=C(κ,β2)exp{sln⁡κ+(s+κβ2)[ln(sκ)+E1(sκ)]-κexp(-sκ)} .$
${E}_{1}\left(x\right)=\underset{0}{\overset{x}{\int }}{t}^{-1}\left(1-{e}^{-t}\right)dt$ 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) array Communication Array
On entry: this must be the same argument rcomm as returned by a previous call to g01zuf.
3: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ 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.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

At least five significant digits are usually correct.

## 8Parallelism and Performance

g01muf is not threaded in any implementation.

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.

## 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)