# NAG C Library Function Document

## 1Purpose

nag_fresnel_c_vector (s20arc) returns an array of values for the Fresnel integral $C\left(x\right)$.

## 2Specification

 #include #include
 void nag_fresnel_c_vector (Integer n, const double x[], double f[], NagError *fail)

## 3Description

nag_fresnel_c_vector (s20arc) evaluates an approximation to the Fresnel integral
 $Cxi=∫0xicosπ2t2dt$
for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
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,
 $Cx=x∑′r=0arTrt, with ​ t=2 x3 4-1.$
For $x>3$,
 $Cx=12+fxxsinπ2x2-gxx3cosπ2x2 ,$
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:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2:    $\mathbf{x}\left[{\mathbf{n}}\right]$const doubleInput
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3:    $\mathbf{f}\left[{\mathbf{n}}\right]$doubleOutput
On exit: $C\left({x}_{i}\right)$, the function values.
4:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 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:
 $ε≃ x cos π2 x2 Cx δ.$
Figure 1 shows the behaviour of the error amplification factor $\left|\frac{x\mathrm{cos}\left(\frac{\pi }{2}{x}^{2}\right)}{C\left(x\right)}\right|$.
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 $x$, $\epsilon \simeq \delta$ and there is no amplification of relative error.
For moderately large values of $x$,
 $ε ≃ 2x cos π2 x2 δ$
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

nag_fresnel_c_vector (s20arc) 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 (s20arce.c)

### 10.2Program Data

Program Data (s20arce.d)

### 10.3Program Results

Program Results (s20arce.r)