NAG Library Routine Document
S18ADF returns the value of the modified Bessel function , via the function name.
|REAL (KIND=nag_wp) S18ADF
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:
. 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.
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
- 1: X – REAL (KIND=nag_wp)Input
On entry: the argument of the function.
- 2: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction 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 parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
, is undefined. On soft failure the routine returns zero.
is too small, there is a danger of overflow. On soft failure the routine returns approximately the largest representable value. (see the Users' Note
for your implementation for details)
Let and be the relative errors in the argument and result respectively.
is somewhat larger than the machine precision
is due to data errors etc.), then
are approximately related by:
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.
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.
9.1 Program Text
Program Text (s18adfe.f90)
9.2 Program Data
Program Data (s18adfe.d)
9.3 Program Results
Program Results (s18adfe.r)