NAG Library Routine Document
s19acf (kelvin_ker)
1
Purpose
s19acf returns a value for the Kelvin function $\mathrm{ker}x$, via the function name.
2
Specification
Fortran Interface
Real (Kind=nag_wp)  ::  s19acf  Integer, Intent (Inout)  ::  ifail  Real (Kind=nag_wp), Intent (In)  ::  x 

C Header Interface
#include nagmk26.h
double 
s19acf_ (const double *x, Integer *ifail) 

3
Description
s19acf evaluates an approximation to the Kelvin function $\mathrm{ker}x$.
Note: for $x<0$ the function is undefined and at $x=0$ it is infinite so we need only consider $x>0$.
The routine is based on several Chebyshev expansions:
For
$0<x\le 1$,
where
$f\left(t\right)$,
$g\left(t\right)$ and
$y\left(t\right)$ are expansions in the variable
$t=2{x}^{4}1$.
For
$1<x\le 3$,
where
$q\left(t\right)$ is an expansion in the variable
$t=x2$.
For
$x>3$,
where
$\beta =\frac{x}{\sqrt{2}}+\frac{\pi}{8}$, and
$c\left(t\right)$ and
$d\left(t\right)$ are expansions in the variable
$t=\frac{6}{x}1$.
When
$x$ is sufficiently close to zero, the result is computed as
and when
$x$ is even closer to zero, simply as
$\mathrm{ker}x=\gamma \mathrm{log}\left(\frac{x}{2}\right)$.
For large $x$, $\mathrm{ker}x$ is asymptotically given by $\sqrt{\frac{\pi}{2x}}{e}^{x/\sqrt{2}}$ and this becomes so small that it cannot be computed without underflow and the routine fails.
4
References
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
5
Arguments
 1: $\mathbf{x}$ – Real (Kind=nag_wp)Input

On entry: the argument $x$ of the function.
Constraint:
${\mathbf{x}}>0.0$.
 2: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit:
${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
 ${\mathbf{ifail}}=1$

On entry,
x is too large: the result underflows. On soft failure, the routine returns zero. See also the
Users' Note for your implementation.
 ${\mathbf{ifail}}=2$

On entry, ${\mathbf{x}}\le 0.0$: the function is undefined. On soft failure the routine returns zero.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
Let
$E$ be the absolute error in the result,
$\epsilon $ be the relative error in the result and
$\delta $ be the relative error in the argument. If
$\delta $ is somewhat larger than the
machine precision, then we have:
For very small
$x$, the relative error amplification factor is approximately given by
$\frac{1}{\left\mathrm{log}\left(x\right)\right}$, which implies a strong attenuation of relative error. However,
$\epsilon $ in general cannot be less than the
machine precision.
For small $x$, errors are damped by the function and hence are limited by the machine precision.
For medium and large $x$, 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 $x$, the amplitude of the absolute error decays like $\sqrt{\frac{\pi x}{2}}{e}^{x/\sqrt{2}}$ which implies a strong attenuation of error. Eventually, $\mathrm{ker}x$, which asymptotically behaves like $\sqrt{\frac{\pi}{2x}}{e}^{x/\sqrt{2}}$, becomes so small that it cannot be calculated without causing underflow, and the routine returns zero. Note that for large $x$ the errors are dominated by those of the standard function exp.
8
Parallelism and Performance
s19acf is not threaded in any implementation.
Underflow may occur for a few values of $x$ close to the zeros of $\mathrm{ker}x$, below the limit which causes a failure with ${\mathbf{ifail}}={\mathbf{1}}$.
10
Example
This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.
10.1
Program Text
Program Text (s19acfe.f90)
10.2
Program Data
Program Data (s19acfe.d)
10.3
Program Results
Program Results (s19acfe.r)