NAG Library Function Document
nag_bessel_j1 (s17afc)
1
Purpose
nag_bessel_j1 (s17afc) returns the value of the Bessel function .
2
Specification
#include <nag.h> |
#include <nags.h> |
double |
nag_bessel_j1 (double x,
NagError *fail) |
|
3
Description
nag_bessel_j1 (s17afc) evaluates an approximation to the Bessel function of the first kind .
Note: , so the approximation need only consider .
The function is based on three Chebyshev expansions:
For
,
For
,
where
,
and ,
with .
For near zero, . This approximation is used when is sufficiently small for the result to be correct to machine precision.
For very large
, it becomes impossible to provide results with any reasonable accuracy (see
Section 7), hence the function fails. Such arguments contain insufficient information to determine the phase of oscillation of
; only the amplitude,
, can be determined and this is returned on failure. The range for which this occurs is roughly related to
machine precision; the function will fail if
(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:
– doubleInput
-
On entry: the argument of the function.
- 2:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_REAL_ARG_GT
-
On entry, .
Constraint: .
is too large, the function returns the amplitude of the oscillation, .
7
Accuracy
Let be the relative error in the argument and be the absolute error in the result. (Since oscillates about zero, absolute error and not relative error is significant.)
If
is somewhat larger than
machine precision (e.g., if
is due to data errors etc.), then
and
are approximately related by:
(provided
is also within machine bounds).
Figure 1 displays the behaviour of the amplification factor
.
However, if is of the same order as machine precision, then rounding errors could make slightly larger than the above relation predicts.
For very large , the above relation ceases to apply. In this region, . The amplitude can be calculated with reasonable accuracy for all , but cannot. If is written as where is an integer and , then is determined by only. If , cannot be determined with any accuracy at all. Thus if is greater than, or of the order of, the reciprocal of machine precision, it is impossible to calculate the phase of and the function must fail.
8
Parallelism and Performance
nag_bessel_j1 (s17afc) is not threaded in any implementation.
None.
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
Program Text (s17afce.c)
10.2
Program Data
Program Data (s17afce.d)
10.3
Program Results
Program Results (s17afce.r)