naginterfaces.library.specfun.fresnel_​s_​vector

naginterfaces.library.specfun.fresnel_s_vector(x)

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

For full information please refer to the NAG Library document for s20aq

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s20aqf.html

Parameters

xfloat, array-like, shape $\left(n\right)$

The argument $x_{\text{i}}$ of the function, for $\text{i}=1,2,\dots ,n$.

Returns

ffloat, ndarray, shape $\left(n\right)$

$S\left(x_i\right)$, the function values.

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

Notes

fresnel_s_vector evaluates an approximation to the Fresnel integral

$$S\left(x_i\right)={\int }_0^{x_i}\sin \left(\frac{\pi }{2}{t}^2\right)dt$$

for an array of arguments $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

Note: $S\left(x\right)=-S(-x)$, so the approximation need only consider $x\ge 0.0$.

The function is based on three Chebyshev expansions:

For $0<x\le 3$,

$$S\left(x\right)=x^3{\sum ^{\prime }}_{r=0}{a}_r{T}_r\left(t\right)\text{, with}t=2{\left(\frac{x}{3}\right)}^4-1\text{.}$$

For $x>3$,

$$S\left(x\right)=\frac{1}{2}-\frac{f\left(x\right)}{x}\cos \left(\frac{\pi }{2}{x}^2\right)-\frac{g\left(x\right)}{x^3}\sin \left(\frac{\pi }{2}{x}^2\right)\text{,}$$

where $f\left(x\right)={{\sum }^{\prime }}_{r=0}{b}_r{T}_r\left(t\right)$,

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

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.

References

NIST Digital Library of Mathematical Functions