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g05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_rngs_von_mises (g05lpc)

## 1  Purpose

nag_rngs_von_mises (g05lpc) generates a vector of pseudorandom numbers from a von Mises distribution with concentration parameter $\kappa$.

## 2  Specification

 #include #include
 void nag_rngs_von_mises (double vk, Integer n, double x[], Integer igen, Integer iseed[], NagError *fail)

## 3  Description

The von Mises distribution is a symmetric distribution used in the analysis of circular data. The probability density function of this distribution on the circle with mean direction ${\mu }_{0}=0$ and concentration parameter kappa, $\kappa$, 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 functions nag_rngs_init_repeatable (g05kbc) (for a repeatable sequence if computed sequentially) or nag_rngs_init_nonrepeatable (g05kcc) (for a non-repeatable sequence) must be called prior to the first call to nag_rngs_von_mises (g05lpc).

## 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:     vkdoubleInput
On entry: $\kappa$, the concentration parameter of the required von Mises distribution.
Constraint: ${\mathbf{vk}}>0.0$.
2:     nIntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
3:     x[n]doubleOutput
On exit: the $n$ pseudorandom numbers from the specified von Mises distribution.
4:     igenIntegerInput
On entry: must contain the identification number for the generator to be used to return a pseudorandom number and should remain unchanged following initialization by a prior call to nag_rngs_init_repeatable (g05kbc) or nag_rngs_init_nonrepeatable (g05kcc).
5:     iseed[$4$]IntegerCommunication Array
On entry: contains values which define the current state of the selected generator.
On exit: contains updated values defining the new state of the selected generator.
6:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL
On entry, ${\mathbf{vk}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{vk}}>0.0$.

Not applicable.

## 8  Further Comments

For a given number of random variates the generation time increases slightly with increasing $\kappa$.
If vk is supplied too large (i.e., ${\mathbf{vk}}>\sqrt{\left({\mathbf{nag_real_largest_number}}\left(\right)\right)}$) then floating point overflow will occur in internal calculation.

## 9  Example

This example prints the first five pseudorandom real numbers from a von Mises distribution with $\kappa =1.0$, generated by a single call to nag_rngs_von_mises (g05lpc), after initialization by nag_rngs_init_repeatable (g05kbc).

### 9.1  Program Text

Program Text (g05lpce.c)

None.

### 9.3  Program Results

Program Results (g05lpce.r)