# NAG Library Routine Document

## 1Purpose

g05srf generates a vector of pseudorandom numbers from a von Mises distribution with concentration parameter $\kappa$.

## 2Specification

Fortran Interface
 Subroutine g05srf ( n, vk, x,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: state(*), ifail Real (Kind=nag_wp), Intent (In) :: vk Real (Kind=nag_wp), Intent (Out) :: x(n)
#include nagmk26.h
 void g05srf_ (const Integer *n, const double *vk, Integer state[], double x[], Integer *ifail)

## 3Description

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.

## 4References

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

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{vk}$ – 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:     $\mathbf{state}\left(*\right)$ – 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:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the $n$ pseudorandom numbers from the specified von Mises distribution.
5:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, 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).

## 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:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{vk}}\le 0.0$ or vk too large: ${\mathbf{vk}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=3$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=-99$
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.

Not applicable.

## 8Parallelism and Performance

g05srf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

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

## 10Example

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.

### 10.1Program Text

Program Text (g05srfe.f90)

### 10.2Program Data

Program Data (g05srfe.d)

### 10.3Program Results

Program Results (g05srfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017