NAG CL Interface
s17atc (bessel_​j1_​real_​vector)

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1 Purpose

s17atc returns an array of values of the Bessel function J1(x).

2 Specification

#include <nag.h>
void  s17atc (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)
The function may be called by the names: s17atc, nag_specfun_bessel_j1_real_vector or nag_bessel_j1_vector.

3 Description

s17atc evaluates an approximation to the Bessel function of the first kind J1(xi) for an array of arguments xi, for i=1,2,,n.
Note:  J1(-x)=-J1(x), so the approximation need only consider x0.
The function is based on three Chebyshev expansions:
For 0<x8,
J1(x)=x8r=0arTr(t),   with ​t=2 (x8) 2-1.  
For x>8,
J1(x)=2πx {P1(x)cos(x-3π4)-Q1(x)sin(x-3π4)}  
where P1(x)=r=0brTr(t),
and Q1(x)= 8xr=0crTr(t),
with t=2 ( 8x) 2-1.
For x near zero, J1(x) x2 . 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 function fails. Such arguments contain insufficient information to determine the phase of oscillation of J1(x); only the amplitude, 2π|x| , 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 |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: n Integer Input
On entry: n, the number of points.
Constraint: n0.
2: x[n] const double Input
On entry: the argument xi of the function, for i=1,2,,n.
3: f[n] double Output
On exit: J1(xi), the function values.
4: ivalid[n] Integer Output
On exit: ivalid[i-1] contains the error code for xi, for i=1,2,,n.
No error.
On entry, xi is too large. f[i-1] contains the amplitude of the J1 oscillation, 2π|xi| .
5: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
On entry, n=value.
Constraint: n0.
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 7.5 in the Introduction to the NAG Library CL 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 CL Interface for further information.
On entry, at least one value of x was invalid.
Check ivalid for more information.

7 Accuracy

Let δ be the relative error in the argument and E be the absolute error in the result. (Since J1(x) oscillates about zero, absolute error and not relative error is significant.)
If δ is somewhat larger than 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 |xJ0(x)-J1(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, J1(x) 2π|x| cos(x- 3π4). The amplitude 2π|x| can be calculated with reasonable accuracy for all x, but cos(x- 3π4) cannot. If x- 3π4 is written as 2Nπ+θ where N is an integer and 0θ<2π, then cos(x- 3π4) 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 reciprocal of machine precision, it is impossible to calculate the phase of J1(x) and the function must fail.
Figure 1
Figure 1

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s17atc is not threaded in any implementation.

9 Further Comments


10 Example

This example reads values of x from a file, evaluates the function at each value of xi and prints the results.

10.1 Program Text

Program Text (s17atce.c)

10.2 Program Data

Program Data (s17atce.d)

10.3 Program Results

Program Results (s17atce.r)