NAG Library Routine Document
S17AEF
1 Purpose
S17AEF returns the value of the Bessel function ${J}_{0}\left(x\right)$, via the function name.
2 Specification
REAL (KIND=nag_wp) S17AEF 
INTEGER 
IFAIL 
REAL (KIND=nag_wp) 
X 

3 Description
S17AEF evaluates an approximation to the Bessel function of the first kind ${J}_{0}\left(x\right)$.
Note: ${J}_{0}\left(x\right)={J}_{0}\left(x\right)$, so the approximation need only consider $x\ge 0$.
The routine is based on three Chebyshev expansions:
where ${P}_{0}\left(x\right)={\displaystyle \underset{r=0}{{\sum}^{\prime}}}\phantom{\rule{0.25em}{0ex}}{b}_{r}{T}_{r}\left(t\right)$,
and ${Q}_{0}\left(x\right)=\frac{8}{x}{\displaystyle \underset{r=0}{{\sum}^{\prime}}}\phantom{\rule{0.25em}{0ex}}{c}_{r}{T}_{r}\left(t\right)$,
with $t=2{\left(\frac{8}{x}\right)}^{2}1$.
For $x$ near zero, ${J}_{0}\left(x\right)\simeq 1$. This approximation is used when $x$ is sufficiently small for the result to be correct to machine precision.
For very large
$x$, it becomes impossible to provide results with any reasonable accuracy (see
Section 7), hence the routine fails. Such arguments contain insufficient information to determine the phase of oscillation of
${J}_{0}\left(x\right)$; only the amplitude,
$\sqrt{\frac{2}{\pi \leftx\right}}$, can be determined and this is returned on soft failure. The range for which this occurs is roughly related to
machine precision; the routine will fail if
$\leftx\right\gtrsim 1/\mathit{machineprecision}$ (see the
Users' Note for your implementation for details).
4 References
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Clenshaw C W (1962) Chebyshev Series for Mathematical Functions Mathematical tables HMSO
5 Arguments
 1: $\mathrm{X}$ – REAL (KIND=nag_wp)Input

On entry: the argument $x$ of the function.
 2: $\mathrm{IFAIL}$ – 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:
 ${\mathbf{IFAIL}}=1$

X is too large. On soft failure the routine returns the amplitude of the
${J}_{0}$ oscillation,
$\sqrt{\frac{2}{\pi \leftx\right}}$.
 ${\mathbf{IFAIL}}=99$
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 $ be the relative error in the argument and $E$ be the absolute error in the result. (Since ${J}_{0}\left(x\right)$ oscillates about zero, absolute error and not relative error is significant.)
If
$\delta $ is somewhat larger than the
machine precision (e.g., if
$\delta $ is due to data errors etc.), then
$E$ and
$\delta $ are approximately related by:
(provided
$E$ is also within machine bounds).
Figure 1 displays the behaviour of the amplification factor
$\leftx{J}_{1}\left(x\right)\right$.
However, if $\delta $ is of the same order as machine precision, then rounding errors could make $E$ slightly larger than the above relation predicts.
For very large $x$, the above relation ceases to apply. In this region, ${J}_{0}\left(x\right)\simeq \sqrt{\frac{2}{\pi \leftx\right}}\mathrm{cos}\left(x\frac{\pi}{4}\right)$. The amplitude $\sqrt{\frac{2}{\pi \leftx\right}}$ can be calculated with reasonable accuracy for all $x$, but $\mathrm{cos}\left(x\frac{\pi}{4}\right)$ cannot. If $x\frac{\pi}{4}$ is written as $2N\pi +\theta $ where $N$ is an integer and $0\le \theta <2\pi $, then $\mathrm{cos}\left(x\frac{\pi}{4}\right)$ is determined by $\theta $ only. If $x\gtrsim {\delta}^{1}$, $\theta $ cannot be determined with any accuracy at all. Thus if $x$ is greater than, or of the order of, the inverse of the machine precision, it is impossible to calculate the phase of ${J}_{0}\left(x\right)$ and the routine must fail.
8 Parallelism and Performance
S17AEF 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 (s17aefe.f90)
10.2 Program Data
Program Data (s17aefe.d)
10.3 Program Results
Program Results (s17aefe.r)