NAG FL Interface
s19adf (kelvin_kei)
1
Purpose
s19adf returns a value for the Kelvin function via the function name.
2
Specification
Fortran Interface
| Real (Kind=nag_wp) |
:: |
s19adf |
| Integer, Intent (Inout) |
:: |
ifail |
| Real (Kind=nag_wp), Intent (In) |
:: |
x |
|
C Header Interface
|
#include <nag.h>
| double |
s19adf_ (const double *x, Integer *ifail) |
|
C++ Header Interface
|
#include <nag.h> extern "C" {
| double |
s19adf_ (const double &x, Integer &ifail) |
}
|
The routine may be called by the names s19adf or nagf_specfun_kelvin_kei.
3
Description
s19adf evaluates an approximation to the Kelvin function .
Note: for the function is undefined, so we need only consider .
The routine is based on several Chebyshev expansions:
For
,
where
,
and
are expansions in the variable
;
For
,
where
is an expansion in the variable
;
For
,
where
, and
and
are expansions in the variable
.
For , the function is undefined, and hence the routine fails and returns zero.
When
is sufficiently close to zero, the result is computed as
and when
is even closer to zero simply as
For large
,
is asymptotically given by
and this becomes so small that it cannot be computed without underflow and the routine fails.
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,
. The function returns zero.
Constraint:
.
x is too large, the result underflows and the function returns zero.
-
On entry, .
Constraint: .
The function is undefined and returns zero.
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
be the absolute error in the result, and
be the relative error in the argument. If
is somewhat larger than the machine representation error, then we have:
For small
, errors are attenuated by the function and hence are limited by the
machine precision.
For medium and large , the error behaviour, like the function itself, is oscillatory and hence only absolute accuracy of the function can be maintained. For this range of , the amplitude of the absolute error decays like , which implies a strong attenuation of error. Eventually, , which is asymptotically given by ,becomes so small that it cannot be calculated without causing underflow and therefore the routine returns zero. Note that for large , the errors are dominated by those of the standard function exp.
8
Parallelism and Performance
s19adf is not threaded in any implementation.
Underflow may occur for a few values of close to the zeros of , below the limit which causes a failure with .
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