# NAG Library Routine Document

## 1Purpose

s20aqf returns an array of values for the Fresnel integral $S\left(x\right)$.

## 2Specification

Fortran Interface
 Subroutine s20aqf ( n, x, f,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Out) :: f(n)
#include <nagmk26.h>
 void s20aqf_ (const Integer *n, const double x[], double f[], Integer *ifail)

## 3Description

s20aqf evaluates an approximation to the Fresnel integral
 $Sxi=∫0xisinπ2t2dt$
for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
Note:  $S\left(x\right)=-S\left(-x\right)$, so the approximation need only consider $x\ge 0.0$.
The routine is based on three Chebyshev expansions:
For $0,
 $Sx=x3∑′r=0arTrt, with ​ t=2 x3 4-1.$
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.

## 4References

NIST Digital Library of Mathematical Functions

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3:     $\mathbf{f}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: $S\left({x}_{i}\right)$, the function values.
4:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 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:
 $ε≃ x sin π2 x2 Sx δ.$
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 $x$,
 $ε ≃ 2x sin π2 x2 δ$
and the result will be subject to increasingly large amplification of errors. However, the above relation breaks down for large values of $x$ (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

## 8Parallelism and Performance

s20aqf is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (s20aqfe.f90)

### 10.2Program Data

Program Data (s20aqfe.d)

### 10.3Program Results

Program Results (s20aqfe.r)