NAG FL Interface
s17aef (bessel_​j0_​real)

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1 Purpose

s17aef returns the value of the Bessel function J0(x), via the function name.

2 Specification

Fortran Interface
Function s17aef ( x, ifail)
Real (Kind=nag_wp) :: s17aef
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x
C Header Interface
#include <nag.h>
double  s17aef_ (const double *x, Integer *ifail)
The routine may be called by the names s17aef or nagf_specfun_bessel_j0_real.

3 Description

s17aef evaluates an approximation to the Bessel function of the first kind J0(x).
Note:  J0(-x)=J0(x), so the approximation need only consider x0.
The routine is based on three Chebyshev expansions:
For 0<x8,
J0(x)=r=0arTr(t),   with ​t=2 ( x8) 2 -1.  
For x>8,
J0(x)= 2πx {P0(x)cos(x-π4)-Q0(x)sin(x- π4)} ,  
where P0(x)=r=0brTr(t),
and Q0(x)= 8xr=0crTr(t),
with t=2 (8x) 2-1.
For x near zero, J0(x)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 J0(x); only the amplitude, 2π|x| , 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 |x|1/machine precision (see the Users' Note for your implementation for details).

4 References

NIST Digital Library of Mathematical Functions
Clenshaw C W (1962) Chebyshev Series for Mathematical Functions Mathematical tables HMSO

5 Arguments

1: x Real (Kind=nag_wp) Input
On entry: the argument x of the function.
2: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry 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:
On entry, x=value.
Constraint: |x|value.
|x| is too large, the function returns the amplitude of the J0 oscillation, 2/(π|x|).
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Let δ be the relative error in the argument and E be the absolute error in the result. (Since J0(x) oscillates about zero, absolute error and not relative error is significant.)
If δ is somewhat larger than the machine precision (e.g., if δ is due to data errors etc.), then E and δ are approximately related by:
(provided E is also within machine bounds). Figure 1 displays the behaviour of the amplification factor |xJ1(x)|.
However, if δ 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, J0(x) 2π|x| cos(x- π4). The amplitude 2π|x| can be calculated with reasonable accuracy for all x, but cos(x- π4) cannot. If x- π4 is written as 2Nπ+θ where N is an integer and 0θ<2π, then cos(x- π4) is determined by θ only. If xδ-1, θ 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 J0(x) and the routine must fail.
Figure 1
Figure 1

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s17aef is not threaded in any implementation.

9 Further Comments


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)
GnuplotProduced by GNUPLOT 5.4 patchlevel 6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −20 −10 0 10 20 J0(x) x "s17aefe.r" Example Program Returned Values for the Bessel Function J0(x)