G05 Chapter Contents
G05 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG05SRF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G05SRF generates a vector of pseudorandom numbers from a von Mises distribution with concentration parameter $\kappa$.

## 2  Specification

 SUBROUTINE G05SRF ( N, VK, STATE, X, IFAIL)
 INTEGER N, STATE(*), IFAIL REAL (KIND=nag_wp) VK, X(N)

## 3  Description

The von Mises distribution is a symmetric distribution used in the analysis of circular data. The PDF (probability density function) of this distribution on the circle with mean direction ${\mu }_{0}=0$ and concentration parameter $\kappa$, can be written as:
 $fθ= eκcos⁡θ 2πI0κ ,$
where $\theta$ is reduced modulo $2\pi$ so that $-\pi \le \theta <\pi$ and $\kappa \ge 0$. For very small $\kappa$ the distribution is almost the uniform distribution, whereas for $\kappa \to \infty$ all the probability is concentrated at one point.
The $n$ variates, ${\theta }_{1},{\theta }_{2},\dots ,{\theta }_{n}$, are generated using an envelope rejection method with a wrapped Cauchy target distribution as proposed by Best and Fisher (1979) and described by Dagpunar (1988).
One of the initialization routines G05KFF (for a repeatable sequence if computed sequentially) or G05KGF (for a non-repeatable sequence) must be called prior to the first call to G05SRF.

## 4  References

Best D J and Fisher N I (1979) Efficient simulation of the von Mises distribution Appl. Statist. 28 152–157
Dagpunar J (1988) Principles of Random Variate Generation Oxford University Press
Mardia K V (1972) Statistics of Directional Data Academic Press

## 5  Parameters

1:     N – INTEGERInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{N}}\ge 0$.
2:     VK – REAL (KIND=nag_wp)Input
On entry: $\kappa$, the concentration parameter of the required von Mises distribution.
Constraint: $0.0<{\mathbf{VK}}\le \sqrt{{\mathbf{X02ALF}}}/2.0$.
3:     STATE($*$) – INTEGER arrayCommunication Array
Note: the actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.
4:     X(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the $n$ pseudorandom numbers from the specified von Mises distribution.
5:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
On entry, ${\mathbf{N}}<0$.
${\mathbf{IFAIL}}=2$
 On entry, ${\mathbf{VK}}\le 0.0$, or ${\mathbf{VK}}>\sqrt{{\mathbf{X02ALF}}\left(\right)}/2.0$.
${\mathbf{IFAIL}}=3$
 On entry, STATE vector was not initialized or has been corrupted.

## 7  Accuracy

Not applicable.

For a given number of random variates the generation time increases slightly with increasing $\kappa$.

## 9  Example

This example prints the first five pseudorandom numbers from a von Mises distribution with $\kappa =1.0$, generated by a single call to G05SRF, after initialization by G05KFF.

### 9.1  Program Text

Program Text (g05srfe.f90)

### 9.2  Program Data

Program Data (g05srfe.d)

### 9.3  Program Results

Program Results (g05srfe.r)