NAG Library Routine Document
s18acf (bessel_k0_real)
1
Purpose
s18acf returns the value of the modified Bessel function ${K}_{0}\left(x\right)$, via the function name.
2
Specification
Fortran Interface
Real (Kind=nag_wp)  ::  s18acf  Integer, Intent (Inout)  ::  ifail  Real (Kind=nag_wp), Intent (In)  ::  x 

C Header Interface
#include nagmk26.h
double 
s18acf_ (const double *x, Integer *ifail) 

3
Description
s18acf evaluates an approximation to the modified Bessel function of the second kind ${K}_{0}\left(x\right)$.
Note: ${K}_{0}\left(x\right)$ is undefined for $x\le 0$ and the routine will fail for such arguments.
The routine is based on five Chebyshev expansions:
For
$0<x\le 1$,
For
$1<x\le 2$,
For
$2<x\le 4$,
For
$x>4$,
For
$x$ near zero,
${K}_{0}\left(x\right)\simeq \gamma \mathrm{ln}\left(\frac{x}{2}\right)$, where
$\gamma $ denotes Euler's constant. This approximation is used when
$x$ is sufficiently small for the result to be correct to
machine precision.
For large $x$, where there is a danger of underflow due to the smallness of ${K}_{0}$, the result is set exactly to zero.
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:
 ${\mathbf{ifail}}=1$

${\mathbf{x}}\le 0.0$, ${K}_{0}$ 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 $\delta $ and $\epsilon $ be the relative errors in the argument and result respectively.
If
$\delta $ is somewhat larger than the
machine precision (i.e., if
$\delta $ is due to data errors etc.), then
$\epsilon $ and
$\delta $ are approximately related by:
Figure 1 shows the behaviour of the error amplification factor
However, if $\delta $ is of the same order as machine precision, then rounding errors could make $\epsilon $ slightly larger than the above relation predicts.
For small $x$, the amplification factor is approximately
$\left\frac{1}{\mathrm{ln}x}\right$, which implies strong attenuation of the error, but in general $\epsilon $ can never be less than the machine precision.
For large $x$, $\epsilon \simeq x\delta $ and we have strong amplification of the relative error. Eventually ${K}_{0}$, which is asymptotically given by $\frac{{e}^{x}}{\sqrt{x}}$, becomes so small that it cannot be calculated without underflow and hence the routine will return zero. Note that for large $x$ the errors will be dominated by those of the standard function exp.
8
Parallelism and Performance
s18acf is not threaded in any implementation.
None.
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 (s18acfe.f90)
10.2
Program Data
Program Data (s18acfe.d)
10.3
Program Results
Program Results (s18acfe.r)