The routine may be called by the names s18adf or nagf_specfun_bessel_k1_real.
s18adf evaluates an approximation to the modified Bessel function of the second kind .
Note: is undefined for and the routine will fail for such arguments.
The routine is based on five Chebyshev expansions:
For near zero, . This approximation is used when is sufficiently small for the result to be correct to machine precision. For very small on some machines, it is impossible to calculate without overflow and the routine must fail.
For large , where there is a danger of underflow due to the smallness of , the result is set exactly to zero.
On entry: ifail must be set to , . If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value is recommended. If the output of error messages is undesirable, then the value is recommended. Otherwise, if you are not familiar with this argument, the recommended value is . When the value is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, .
is undefined and the function returns zero.
On entry, . Constraint: . x is too small, there is a danger of overflow and the function returns approximately the largest representable value.
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
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 the machine precision, then rounding errors could make slightly larger than the above relation predicts.
For small , and there is no amplification of errors.
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 routine will return zero. Note that for large the errors will be dominated by those of the standard function exp.
8Parallelism and Performance
s18adf 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.