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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_bessel_j0_real_vector (s17as)

## Purpose

nag_specfun_bessel_j0_real_vector (s17as) returns an array of values of the Bessel function J0(x)${J}_{0}\left(x\right)$.

## Syntax

[f, ivalid, ifail] = s17as(x, 'n', n)
[f, ivalid, ifail] = nag_specfun_bessel_j0_real_vector(x, 'n', n)

## Description

nag_specfun_bessel_j0_real_vector (s17as) evaluates an approximation to the Bessel function of the first kind J0(xi)${J}_{0}\left({x}_{i}\right)$ for an array of arguments xi${x}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$.
Note:  J0(x) = J0(x)${J}_{0}\left(-x\right)={J}_{0}\left(x\right)$, so the approximation need only consider x0$x\ge 0$.
The function is based on three Chebyshev expansions:
For 0 < x8$0,
 J0(x) = ∑′ arTr(t),   with ​t = 2(x/8)2 − 1. r = 0
$J0(x)=∑′r=0arTr(t), with ​t=2 ( x8) 2 -1.$
For x > 8$x>8$,
 J0(x) = sqrt(2/(πx)) {P0(x)cos(x − π/4) − Q0(x)sin(x − π/4)} , $J0(x)= 2πx {P0(x)cos(x-π4)-Q0(x)sin(x- π4)} ,$
where P0(x) = r = 0 brTr(t)${P}_{0}\left(x\right)=\underset{r=0}{{\sum }^{\prime }}\phantom{\rule{0.25em}{0ex}}{b}_{r}{T}_{r}\left(t\right)$,
and Q0(x) = 8/xr = 0 crTr(t)${Q}_{0}\left(x\right)=\frac{8}{x}\underset{r=0}{{\sum }^{\prime }}\phantom{\rule{0.25em}{0ex}}{c}_{r}{T}_{r}\left(t\right)$,
with t = 2 (8/x)21$t=2{\left(\frac{8}{x}\right)}^{2}-1$.
For x$x$ near zero, J0(x)1${J}_{0}\left(x\right)\simeq 1$. This approximation is used when x$x$ is sufficiently small for the result to be correct to machine precision.
For very large x$x$, it becomes impossible to provide results with any reasonable accuracy (see Section [Accuracy]), hence the function fails. Such arguments contain insufficient information to determine the phase of oscillation of J0(x)${J}_{0}\left(x\right)$; only the amplitude, sqrt(2/(π|x|))$\sqrt{\frac{2}{\pi |x|}}$, can be determined and this is returned on soft failure. The range for which this occurs is roughly related to machine precision; the function will fail if |x|1 / machine precision.

## 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

## Parameters

### Compulsory Input Parameters

1:     x(n) – double array
n, the dimension of the array, must satisfy the constraint n0${\mathbf{n}}\ge 0$.
The argument xi${x}_{\mathit{i}}$ of the function, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array x.
n$n$, the number of points.
Constraint: n0${\mathbf{n}}\ge 0$.

None.

### Output Parameters

1:     f(n) – double array
J0(xi)${J}_{0}\left({x}_{i}\right)$, the function values.
2:     ivalid(n) – int64int32nag_int array
ivalid(i)${\mathbf{ivalid}}\left(\mathit{i}\right)$ contains the error code for xi${x}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
ivalid(i) = 0${\mathbf{ivalid}}\left(i\right)=0$
No error.
ivalid(i) = 1${\mathbf{ivalid}}\left(i\right)=1$
 On entry, xi${x}_{i}$ is too large. f(i)${\mathbf{f}}\left(\mathit{i}\right)$ contains the amplitude of the J0${J}_{0}$ oscillation, sqrt(2/(π|xi|))$\sqrt{\frac{2}{\pi |{x}_{i}|}}$.
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W ifail = 1${\mathbf{ifail}}=1$
On entry, at least one value of x was invalid.
ifail = 2${\mathbf{ifail}}=2$
Constraint: n0${\mathbf{n}}\ge 0$.

## Accuracy

Let δ$\delta$ be the relative error in the argument and E$E$ be the absolute error in the result. (Since J0(x)${J}_{0}\left(x\right)$ oscillates about zero, absolute error and not relative error is significant.)
If δ$\delta$ is somewhat larger than the machine precision (e.g., if δ$\delta$ is due to data errors etc.), then E$E$ and δ$\delta$ are approximately related by:
 E ≃ |xJ1(x)|δ $E≃|xJ1(x)|δ$
(provided E$E$ is also within machine bounds). Figure 1 displays the behaviour of the amplification factor |xJ1(x)|$|x{J}_{1}\left(x\right)|$.
However, if δ$\delta$ is of the same order as machine precision, then rounding errors could make E$E$ slightly larger than the above relation predicts.
For very large x$x$, the above relation ceases to apply. In this region, J0(x)sqrt(2/(π|x|))cos(xπ/4)${J}_{0}\left(x\right)\simeq \sqrt{\frac{2}{\pi |x|}}\mathrm{cos}\left(x-\frac{\pi }{4}\right)$. The amplitude sqrt(2/(π|x|))$\sqrt{\frac{2}{\pi |x|}}$ can be calculated with reasonable accuracy for all x$x$, but cos(xπ/4)$\mathrm{cos}\left(x-\frac{\pi }{4}\right)$ cannot. If xπ/4 $x-\frac{\pi }{4}$ is written as 2Nπ + θ$2N\pi +\theta$ where N$N$ is an integer and 0θ < 2π$0\le \theta <2\pi$, then cos(xπ/4) $\mathrm{cos}\left(x-\frac{\pi }{4}\right)$ is determined by θ$\theta$ only. If xδ1$x\gtrsim {\delta }^{-1}$, θ$\theta$ cannot be determined with any accuracy at all. Thus if x$x$ is greater than, or of the order of, the inverse of the machine precision, it is impossible to calculate the phase of J0(x)${J}_{0}\left(x\right)$ and the function must fail.
Figure 1

None.

## Example

```function nag_specfun_bessel_j0_real_vector_example
x = [0; 0.5; 1; 3; 6; 8; 10; -1; 1000];
[f, ivalid, ifail] = nag_specfun_bessel_j0_real_vector(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end
```
```

X           Y
0.000e+00   1.000e+00    0
5.000e-01   9.385e-01    0
1.000e+00   7.652e-01    0
3.000e+00  -2.601e-01    0
6.000e+00   1.506e-01    0
8.000e+00   1.717e-01    0
1.000e+01  -2.459e-01    0
-1.000e+00   7.652e-01    0
1.000e+03   2.479e-02    0

```
```function s17as_example
x = [0; 0.5; 1; 3; 6; 8; 10; -1; 1000];
[f, ivalid, ifail] = s17as(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
fprintf('%12.3e%12.3e%5d\n', x(i), f(i), ivalid(i));
end
```
```

X           Y
0.000e+00   1.000e+00    0
5.000e-01   9.385e-01    0
1.000e+00   7.652e-01    0
3.000e+00  -2.601e-01    0
6.000e+00   1.506e-01    0
8.000e+00   1.717e-01    0
1.000e+01  -2.459e-01    0
-1.000e+00   7.652e-01    0
1.000e+03   2.479e-02    0

```