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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_fresnel_s_vector (s20aq)

## Purpose

nag_specfun_fresnel_s_vector (s20aq) returns an array of values for the Fresnel integral S(x)$S\left(x\right)$.

## Syntax

[f, ifail] = s20aq(x, 'n', n)
[f, ifail] = nag_specfun_fresnel_s_vector(x, 'n', n)

## Description

nag_specfun_fresnel_s_vector (s20aq) evaluates an approximation to the Fresnel integral
 xi S(xi) = ∫ sin(π/2t2)dt 0
$S(xi)=∫0xisin(π2t2)dt$
for an array of arguments xi${x}_{\mathit{i}}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$.
Note:  S(x) = S(x)$S\left(x\right)=-S\left(-x\right)$, so the approximation need only consider x0.0$x\ge 0.0$.
The function is based on three Chebyshev expansions:
For 0 < x3$0,
 S(x) = x3 ∑′ arTr(t),   with ​t = 2(x/3)4 − 1. r = 0
$S(x)=x3∑′r=0arTr(t), with ​ t=2 (x3) 4-1.$
For x > 3$x>3$,
 S(x) = (1/2) − (f(x))/xcos(π/2x2) − (g(x))/(x3)sin(π/2x2) , $S(x)=12-f(x)xcos(π2x2)-g(x)x3sin(π2x2) ,$
where f(x) = r = 0 brTr(t)$f\left(x\right)=\underset{r=0}{{\sum }^{\prime }}\phantom{\rule{0.25em}{0ex}}{b}_{r}{T}_{r}\left(t\right)$,
and g(x) = r = 0 crTr(t)$g\left(x\right)=\underset{r=0}{{\sum }^{\prime }}\phantom{\rule{0.25em}{0ex}}{c}_{r}{T}_{r}\left(t\right)$,
with t = 2 (3/x)41$t=2{\left(\frac{3}{x}\right)}^{4}-1$.
For small x$x$, S(x)π/6x3$S\left(x\right)\simeq \frac{\pi }{6}{x}^{3}$. This approximation is used when x$x$ is sufficiently small for the result to be correct to machine precision. For very small x$x$, this approximation would underflow; the result is then set exactly to zero.
For large x$x$, f(x)1/π $f\left(x\right)\simeq \frac{1}{\pi }$ and g(x)1/(π2) $g\left(x\right)\simeq \frac{1}{{\pi }^{2}}$. Therefore for moderately large x$x$, when 1/(π2x3) $\frac{1}{{\pi }^{2}{x}^{3}}$ is negligible compared with (1/2) $\frac{1}{2}$, the second term in the approximation for x > 3$x>3$ may be dropped. For very large x$x$, when 1/(πx) $\frac{1}{\pi x}$ becomes negligible, S(x)(1/2) $S\left(x\right)\simeq \frac{1}{2}$. However there will be considerable difficulties in calculating cos(π/2x2)$\mathrm{cos}\left(\frac{\pi }{2}{x}^{2}\right)$ accurately before this final limiting value can be used. Since cos(π/2x2)$\mathrm{cos}\left(\frac{\pi }{2}{x}^{2}\right)$ is periodic, its value is essentially determined by the fractional part of x2${x}^{2}$. If x2 = N + θ${x}^{2}=N+\theta$ where N$N$ is an integer and 0θ < 1$0\le \theta <1$, then cos(π/2x2)$\mathrm{cos}\left(\frac{\pi }{2}{x}^{2}\right)$ depends on θ$\theta$ and on N$N$ modulo 4$4$. By exploiting this fact, it is possible to retain significance in the calculation of cos(π/2x2)$\mathrm{cos}\left(\frac{\pi }{2}{x}^{2}\right)$ either all the way to the very large x$x$ limit, or at least until the integer part of x/2 $\frac{x}{2}$ is equal to the maximum integer allowed on the machine.

## References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

## 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
S(xi)$S\left({x}_{i}\right)$, the function values.
2:     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${\mathbf{ifail}}=1$
Constraint: n0${\mathbf{n}}\ge 0$.

## Accuracy

Let δ$\delta$ and ε$\epsilon$ be the relative errors in the argument and result respectively.
If δ$\delta$ is somewhat larger than the machine precision (i.e., if δ$\delta$ is due to data errors etc.), then ε$\epsilon$ and δ$\delta$ are approximately related by:
 ε ≃ |( x sin(π/2x2) )/(S(x))|δ. $ε≃ | x sin( π2 x2 ) S(x) |δ.$
Figure 1 shows the behaviour of the error amplification factor |( x sin(π/2x2) )/(S(x))| $|\frac{x\mathrm{sin}\left(\frac{\pi }{2}{x}^{2}\right)}{S\left(x\right)}|$.
However if δ$\delta$ is of the same order as the machine precision, then rounding errors could make ε$\epsilon$ slightly larger than the above relation predicts.
For small x$x$, ε3δ$\epsilon \simeq 3\delta$ and hence there is only moderate amplification of relative error. Of course for very small x$x$ where the correct result would underflow and exact zero is returned, relative error-control is lost.
For moderately large values of x$x$,
 ε ≃ |2xsin(π/2x2)| δ $ε ≃ | 2x sin( π2 x2 ) | δ$
and the result will be subject to increasingly large amplification of errors. However the above relation breaks down for large values of x$x$ (i.e., when 1/(x2) $\frac{1}{{x}^{2}}$ is of the order of the machine precision); in this region the relative error in the result is essentially bounded by 2/(πx) $\frac{2}{\pi x}$.
Hence the effects of error amplification are limited and at worst the relative error loss should not exceed half the possible number of significant figures.
Figure 1

None.

## Example

```function nag_specfun_fresnel_s_vector_example
x = [0; 0.5; 1; 2; 4; 5; 6; 8; 10; -1; 1000];
[f, ifail] = nag_specfun_fresnel_s_vector(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
fprintf('%12.3e%12.3e\n', x(i), f(i));
end
```
```

X           Y
0.000e+00   0.000e+00
5.000e-01   6.473e-02
1.000e+00   4.383e-01
2.000e+00   3.434e-01
4.000e+00   4.205e-01
5.000e+00   4.992e-01
6.000e+00   4.470e-01
8.000e+00   4.602e-01
1.000e+01   4.682e-01
-1.000e+00  -4.383e-01
1.000e+03   4.997e-01

```
```function s20aq_example
x = [0; 0.5; 1; 2; 4; 5; 6; 8; 10; -1; 1000];
[f, ifail] = s20aq(x);
fprintf('\n    X           Y\n');
for i=1:numel(x)
fprintf('%12.3e%12.3e\n', x(i), f(i));
end
```
```

X           Y
0.000e+00   0.000e+00
5.000e-01   6.473e-02
1.000e+00   4.383e-01
2.000e+00   3.434e-01
4.000e+00   4.205e-01
5.000e+00   4.992e-01
6.000e+00   4.470e-01
8.000e+00   4.602e-01
1.000e+01   4.682e-01
-1.000e+00  -4.383e-01
1.000e+03   4.997e-01

```