NAG Library Function Document

nag_fresnel_s (s20acc)

nag_fresnel_s (s20acc) returns a value for the Fresnel integral $S\left(x\right)$.

nag_fresnel_s (s20acc) evaluates an approximation to the Fresnel integral Note:  $S\left(x\right)=-S\left(-x\right)$, so the approximation need only consider $x\ge 0.0$.

The function is based on three Chebyshev expansions:

For $0<x\le 3$, For $x>3$, where $f\left(x\right)={\displaystyle \underset{r=0}{{\sum }^{\prime }}}{b}_r{T}_r\left(t\right)$,

and $g\left(x\right)={\displaystyle \underset{r=0}{{\sum }^{\prime }}}{c}_r{T}_r\left(t\right)$,

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 $\cos \left(\frac{\pi }{2}{x}^2\right)$ accurately before this final limiting value can be used. Since $\cos \left(\frac{\pi }{2}{x}^2\right)$ 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 $\cos \left(\frac{\pi }{2}{x}^2\right)$ depends on $\theta$ and on $N$ modulo $4$. By exploiting this fact, it is possible to retain significance in the calculation of $\cos \left(\frac{\pi }{2}{x}^2\right)$ 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.

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

None.

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: Figure 1 shows the behaviour of the error amplification factor $\left|\frac{x\sin \left(\frac{\pi }{2}{x}^2\right)}{S\left(x\right)}\right|$.

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 $x$, 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.

nag_fresnel_s (s20acc) is not threaded in any implementation.

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