nag_bessel_j1_vector (s17atc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_bessel_j1_vector (s17atc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_bessel_j1_vector (s17atc) returns an array of values of the Bessel function J1x.

2  Specification

#include <nag.h>
#include <nags.h>
void  nag_bessel_j1_vector (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)

3  Description

nag_bessel_j1_vector (s17atc) evaluates an approximation to the Bessel function of the first kind J1xi for an array of arguments xi, for i=1,2,,n.
Note:  J1-x=-J1x, so the approximation need only consider x0.
The function is based on three Chebyshev expansions:
For 0<x8,
J1x=x8r=0arTrt,   with ​t=2 x8 2-1.
For x>8,
J1x=2πx P1xcosx-3π4-Q1xsinx-3π4
where P1x=r=0brTrt,
and Q1x= 8xr=0crTrt,
with t=2 8x 2-1.
For x near zero, J1x 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 J1x; only the amplitude, 2πx , 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 x1/machine precision (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:     nIntegerInput
On entry: n, the number of points.
Constraint: n0.
2:     x[n]const doubleInput
On entry: the argument xi of the function, for i=1,2,,n.
3:     f[n]doubleOutput
On exit: J1xi, the function values.
4:     ivalid[n]IntegerOutput
On exit: ivalid[i-1] contains the error code for xi, for i=1,2,,n.
ivalid[i-1]=0
No error.
ivalid[i-1]=1
On entry,xi is too large. f[i-1] contains the amplitude of the J1 oscillation, 2πxi .
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
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.
NW_IVALID
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 J1x 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:
ExJ0x-J1xδ
(provided E is also within machine bounds). Figure 1 displays the behaviour of the amplification factor xJ0x-J1x.
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, J1x 2πx cosx- 3π4. The amplitude 2πx  can be calculated with reasonable accuracy for all x, but cosx- 3π4 cannot. If x- 3π4  is written as 2Nπ+θ where N is an integer and 0θ<2π, then cosx- 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 J1x and the function must fail.
Figure 1
Figure 1

8  Further Comments

None.

9  Example

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

9.1  Program Text

Program Text (s17atce.c)

9.2  Program Data

Program Data (s17atce.d)

9.3  Program Results

Program Results (s17atce.r)


nag_bessel_j1_vector (s17atc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012