nag_bessel_k0 (s18acc) evaluates an approximation to the modified Bessel function of the second kind .
Note: is undefined for and the function will fail for such arguments.
The function is based on five Chebyshev expansions:
For near zero, , where denotes Euler's constant. This approximation is used when is sufficiently small for the result to be correct to machine precision.
For large , where there is a danger of underflow due to the smallness of , the result is set exactly to zero.
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
On entry: the argument of the function.
– NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).
Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 18.104.22.168 in How to Use the NAG Library and its Documentation for further information.
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.
See Section 2.7.6 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 2.7.5 in How to Use the NAG Library and its Documentation for further information.
On entry, .
is undefined and the function returns zero.
Let and be the relative errors in the argument and result respectively.
If is somewhat larger than the machine precision (i.e., if is due to data errors etc.), then and are approximately related by:
Figure 1 shows the behaviour of the error amplification factor
However, if is of the same order as machine precision, then rounding errors could make slightly larger than the above relation predicts.
For small , the amplification factor is approximately
, which implies strong attenuation of the error, but in general can never be less than the machine precision.
For large , and we have strong amplification of the relative error. Eventually , which is asymptotically given by , becomes so small that it cannot be calculated without underflow and hence the function will return zero. Note that for large the errors will be dominated by those of the standard math library function exp.
Parallelism and Performance
nag_bessel_k0 (s18acc) is not threaded in any implementation.
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.