﻿ s20ac Method
s20ac returns a value for the Fresnel integral $S\left(x\right)$.

# Syntax

C#
```public static double s20ac(
double x
)```
Visual Basic
```Public Shared Function s20ac ( _
x As Double _
) As Double```
Visual C++
```public:
static double s20ac(
double x
)```
F#
```static member s20ac :
x : float -> float
```

#### Parameters

x
Type: System..::..Double
On entry: the argument $x$ of the function.

#### Return Value

s20ac returns a value for the Fresnel integral $S\left(x\right)$.

# Description

s20ac evaluates an approximation to the Fresnel integral
 $Sx=∫0xsinπ2t2dt.$
Note:  $S\left(x\right)=-S\left(-x\right)$, so the approximation need only consider $x\ge 0.0$.
The method is based on three Chebyshev expansions:
For $0,
 $Sx=x3∑′r=0arTrt, with ​t=2x34-1.$
For $x>3$,
 $Sx=12-fxxcosπ2x2-gxx3sinπ2x2,$
where $f\left(x\right)=\underset{r=0}{{\sum }^{\prime }}\phantom{\rule{0.25em}{0ex}}{b}_{r}{T}_{r}\left(t\right)$,
and $g\left(x\right)=\underset{r=0}{{\sum }^{\prime }}\phantom{\rule{0.25em}{0ex}}{c}_{r}{T}_{r}\left(t\right)$,
with $t=2{\left(\frac{3}{x}\right)}^{4}-1$.
For small $x$, $S\left(x\right)\simeq \frac{\pi }{6}{x}^{3}$. This approximation is used when $x$ is sufficiently small for the result to be correct to machine precision. For very small $x$, this approximation would underflow; the result is then set exactly to zero.
For large $x$, $f\left(x\right)\simeq \frac{1}{\pi }$ and $g\left(x\right)\simeq \frac{1}{{\pi }^{2}}$. Therefore for moderately large $x$, when $\frac{1}{{\pi }^{2}{x}^{3}}$ is negligible compared with $\frac{1}{2}$, the second term in the approximation for $x>3$ may be dropped. For very large $x$, when $\frac{1}{\pi x}$ becomes negligible, $S\left(x\right)\simeq \frac{1}{2}$. However there will be considerable difficulties in calculating $\mathrm{cos}\left(\frac{\pi }{2}{x}^{2}\right)$ accurately before this final limiting value can be used. Since $\mathrm{cos}\left(\frac{\pi }{2}{x}^{2}\right)$ is periodic, its value is essentially determined by the fractional part of ${x}^{2}$. If ${x}^{2}=N+\theta$ where $N$ is an integer and $0\le \theta <1$, then $\mathrm{cos}\left(\frac{\pi }{2}{x}^{2}\right)$ depends on $\theta$ and on $N$ modulo $4$. By exploiting this fact, it is possible to retain significance in the calculation of $\mathrm{cos}\left(\frac{\pi }{2}{x}^{2}\right)$ either all the way to the very large $x$ limit, or at least until the integer part of $\frac{x}{2}$ 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

# Error Indicators and Warnings

There are no failure exits from s20ac. The parameter _ifail has been included for consistency with other methods in this chapter.

# Accuracy

Let $\delta$ and $\epsilon$ be the relative errors in the argument and result respectively.
If $\delta$ is somewhat larger than the machine precision (i.e., if $\delta$ is due to data errors etc.), then $\epsilon$ and $\delta$ are approximately related by:
 $ε≃xsinπ2x2Sxδ.$
Figure 1 shows the behaviour of the error amplification factor $\left|\frac{x\mathrm{sin}\left(\frac{\pi }{2}{x}^{2}\right)}{S\left(x\right)}\right|$.
However if $\delta$ is of the same order as the machine precision, then rounding errors could make $\epsilon$ slightly larger than the above relation predicts.
For small $x$, $\epsilon \simeq 3\delta$ and hence there is only moderate amplification of relative error. Of course for very small $x$ where the correct result would underflow and exact zero is returned, relative error-control is lost.
For moderately large values of $x$,
 $ε≃2xsinπ2x2δ$
and the result will be subject to increasingly large amplification of errors. However the above relation breaks down for large values of $x$ (i.e., when $\frac{1}{{x}^{2}}$ is of the order of the machine precision); in this region the relative error in the result is essentially bounded by $\frac{2}{\pi x}$.
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.

None.

# Example

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

Example program (C#): s20ace.cs

Example program data: s20ace.d

Example program results: s20ace.r