NAG FL Interface
s18adf (bessel_k1_real)
1
Purpose
s18adf returns the value of the modified Bessel function , via the function name.
2
Specification
Fortran Interface
| Real (Kind=nag_wp) |
:: |
s18adf |
| Integer, Intent (Inout) |
:: |
ifail |
| Real (Kind=nag_wp), Intent (In) |
:: |
x |
|
C Header Interface
|
#include <nag.h>
| double |
s18adf_ (const double *x, Integer *ifail) |
|
C++ Header Interface
|
#include <nag.h> extern "C" {
| double |
s18adf_ (const double &x, Integer &ifail) |
}
|
The routine may be called by the names s18adf or nagf_specfun_bessel_k1_real.
3
Description
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
,
For
,
For
,
For
,
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.
4
References
5
Arguments
-
1:
– Real (Kind=nag_wp)
Input
-
On entry: the argument of the function.
Constraint:
.
-
2:
– Integer
Input/Output
-
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).
6
Error 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, .
Constraint: .
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
contact
NAG.
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.
7
Accuracy
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.
8
Parallelism and Performance
s18adf is not threaded in any implementation.
None.
10
Example
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.
10.1
Program Text
10.2
Program Data
10.3
Program Results
