NAG Library Routine Document
g05srf (dist_vonmises)
1
Purpose
g05srf generates a vector of pseudorandom numbers from a von Mises distribution with concentration parameter $\kappa $.
2
Specification
Fortran Interface
Integer, Intent (In)  ::  n  Integer, Intent (Inout)  ::  state(*), ifail  Real (Kind=nag_wp), Intent (In)  ::  vk  Real (Kind=nag_wp), Intent (Out)  ::  x(n) 

C Header Interface
#include nagmk26.h
void 
g05srf_ (const Integer *n, const double *vk, Integer state[], double x[], Integer *ifail) 

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:
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 nonrepeatable 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
Arguments
 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).
6
Error 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, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 0$.
 ${\mathbf{ifail}}=2$

On entry,
${\mathbf{vk}}\le 0.0$ or
vk too large:
${\mathbf{vk}}=\u2329\mathit{\text{value}}\u232a$.
 ${\mathbf{ifail}}=3$

On entry,
state vector has been corrupted or not initialized.
 ${\mathbf{ifail}}=99$
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.
7
Accuracy
Not applicable.
8
Parallelism 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 implementationspecific information.
For a given number of random variates the generation time increases slightly with increasing $\kappa $.
10
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.
10.1
Program Text
Program Text (g05srfe.f90)
10.2
Program Data
Program Data (g05srfe.d)
10.3
Program Results
Program Results (g05srfe.r)