NAG Library Routine Document

g01erf (prob_vonmises)


    1  Purpose
    7  Accuracy


g01erf returns the probability associated with the lower tail of the von Mises distribution between -π and π through the function name.


Fortran Interface
Function g01erf ( t, vk, ifail)
Real (Kind=nag_wp):: g01erf
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: t, vk
C Header Interface
#include nagmk26.h
double  g01erf_ (const double *t, const double *vk, Integer *ifail)


The von Mises distribution is a symmetric distribution used in the analysis of circular data. The lower tail area of this distribution on the circle with mean direction μ0=0 and concentration argument kappa, κ, can be written as
PrΘθ:κ=12πI0κ -πθeκcosΘdΘ,  
where θ is reduced modulo 2π so that -πθ<π and κ0. Note that if θ=π then g01erf returns a probability of 1. For very small κ the distribution is almost the uniform distribution, whereas for κ all the probability is concentrated at one point.
The method of calculation for small κ involves backwards recursion through a series expansion in terms of modified Bessel functions, while for large κ an asymptotic Normal approximation is used.
In the case of small κ the series expansion of Pr(Θθ: κ) can be expressed as
PrΘθ:κ=12+θ 2π +1πI0κ n=1n-1Inκsinnθ,  
where Inκ is the modified Bessel function. This series expansion can be represented as a nested expression of terms involving the modified Bessel function ratio Rn,
Rnκ=Inκ In-1κ ,  n=1,2,3,,  
which is calculated using backwards recursion.
For large values of κ (see Section 7) an asymptotic Normal approximation is used. The angle Θ is transformed to the nearly Normally distributed variate Z,
bκ=2π eκ I0κ  
and bκ is computed from a continued fraction approximation. An approximation to order κ-4 of the asymptotic normalizing series for z is then used. Finally the Normal probability integral is evaluated.
For a more detailed analysis of the methods used see Hill (1977).


Hill G W (1977) Algorithm 518: Incomplete Bessel function I0: The Von Mises distribution ACM Trans. Math. Software 3 279–284
Mardia K V (1972) Statistics of Directional Data Academic Press


1:     t – Real (Kind=nag_wp)Input
On entry: θ, the observed von Mises statistic measured in radians.
2:     vk – Real (Kind=nag_wp)Input
On entry: the concentration parameter κ, of the von Mises distribution.
Constraint: vk0.0.
3:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1​ 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​ 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 -1​ or ​1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

Error Indicators and Warnings

If on entry 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:
On entry, vk=value.
Constraint: vk0.0.
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.
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.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


g01erf uses one of two sets of constants depending on the value of machine precision. One set gives an accuracy of six digits and uses the Normal approximation when vk6.5, the other gives an accuracy of 12 digits and uses the Normal approximation when vk50.0.

Parallelism and Performance

g01erf is not threaded in any implementation.

Further Comments

Using the series expansion for small κ the time taken by g01erf increases linearly with κ; for larger κ, for which the asymptotic Normal approximation is used, the time taken is much less.
If angles outside the region -πθ<π are used care has to be taken in evaluating the probability of being in a region θ1θθ2 if the region contains an odd multiple of π, 2n+1π. The value of Fθ2;κ-Fθ1;κ will be negative and the correct probability should then be obtained by adding one to the value.


This example inputs four values from the von Mises distribution along with the values of the argument κ. The probabilities are computed and printed.

Program Text

Program Text (g01erfe.f90)

Program Data

Program Data (g01erfe.d)

Program Results

Program Results (g01erfe.r)

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