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## 1Purpose

s20adc returns a value for the Fresnel integral $C\left(x\right)$.

## 2Specification

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The function may be called by the names: s20adc, nag_specfun_fresnel_c or nag_fresnel_c.

## 3Description

s20adc evaluates an approximation to the Fresnel integral
 $C(x)=∫0xcos(π2t2)dt.$
Note:  $C\left(x\right)=-C\left(-x\right)$, so the approximation need only consider $x\ge 0.0$.
The function is based on three Chebyshev expansions:
For $0,
 $C(x)=x∑′r=0arTr(t), with ​ t=2 (x3) 4-1.$
For $x>3$,
 $C(x)=12+f(x)xsin(π2x2)-g(x)x3cos(π2x2) ,$
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 $x$, $C\left(x\right)\simeq x$. This approximation is used when $x$ is sufficiently small for the result to be correct to machine precision.
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, $C\left(x\right)\simeq \frac{1}{2}$. However, there will be considerable difficulties in calculating $\mathrm{sin}\left(\frac{\pi }{2}{x}^{2}\right)$ accurately before this final limiting value can be used. Since $\mathrm{sin}\left(\frac{\pi }{2}{x}^{2}\right)$ 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 $\mathrm{sin}\left(\frac{\pi }{2}{x}^{2}\right)$ depends on $\theta$ and on $N$ modulo $4$. By exploiting this fact, it is possible to retain some significance in the calculation of $\mathrm{sin}\left(\frac{\pi }{2}{x}^{2}\right)$ 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.

## 4References

NIST Digital Library of Mathematical Functions

## 5Arguments

1: $\mathbf{x}$double Input
On entry: the argument $x$ of the function.

None.

## 7Accuracy

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:
 $ε≃ | x cos( π2x2) C(x) |δ.$
Figure 1 shows the behaviour of the error amplification factor $|\frac{x\mathrm{cos}\left(\frac{\pi }{2}{x}^{2}\right)}{C\left(x\right)}|$.
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 $x$, $\epsilon \simeq \delta$ and there is no amplification of relative error.
For moderately large values of $x$,
 $|ε|≃ |2xcos(π2x2)||δ|$
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.

None.

## 10Example

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