NAG CL Interface
s19acc (kelvin_ker)
1
Purpose
s19acc returns a value for the Kelvin function .
2
Specification
| double |
s19acc (double x,
NagError *fail) |
|
The function may be called by the names: s19acc, nag_specfun_kelvin_ker or nag_kelvin_ker.
3
Description
s19acc evaluates an approximation to the Kelvin function .
Note: for the function is undefined and at it is infinite so we need only consider .
The function 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
.
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 function fails.
4
References
5
Arguments
-
1:
– double
Input
-
On entry: the argument of the function.
Constraint:
.
-
2:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_INTERNAL_ERROR
-
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
- NE_REAL_ARG_GT
-
On entry,
. The function returns zero.
Constraint:
.
x is too large, the result underflows and the function returns zero.
- NE_REAL_ARG_LE
-
On entry, .
Constraint: .
The function is undefined and returns zero.
7
Accuracy
Let
be the absolute error in the result,
be the relative error in the result and
be the relative error in the argument. If
is somewhat larger than the
machine precision, then we have:
For very small
, the relative error amplification factor is approximately given by
, which implies a strong attenuation of relative error. However,
in general cannot be less than the
machine precision.
For small , errors are damped 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 the absolute accuracy for 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 asymptotically behaves like , becomes so small that it cannot be calculated without causing underflow, and the function returns zero. Note that for large the errors are dominated by those of the standard math library function exp.
8
Parallelism and Performance
s19acc 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
NE_REAL_ARG_GT.
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