naginterfaces.library.specfun.kelvin_​ker_​vector

naginterfaces.library.specfun.kelvin_ker_vector(x)

kelvin_ker_vector returns an array of values for the Kelvin function $ker\left(x\right)$.

For full information please refer to the NAG Library document for s19aq

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s19aqf.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 $\left(n\right)$

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

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

$ivalid[\text{i}-1]$ contains the error code for $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

$ivalid[i-1]=0$

No error.

$ivalid[i-1]=1$

$x_i$ is too large, the result underflows. $f[\text{i}-1]$ contains zero. The threshold value is the same as for $errno$ = 1 in kelvin_ker().

$ivalid[i-1]=2$

$x_i\le 0.0$, the function is undefined. $f[\text{i}-1]$ contains $0.0$.

Raises

NagValueError

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, at least one value of $x$ was invalid.

Check $ivalid$ for more information.

Notes

kelvin_ker_vector evaluates an approximation to the Kelvin function $ker\left(x_i\right)$ for an array of arguments $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

Note: for $x<0$ the function is undefined and at $x=0$ it is infinite so we need only consider $x>0$.

The function is based on several Chebyshev expansions:

For $0<x\le 1$,

$$ker\left(x\right)=-f\left(t\right)\log \left(x\right)+\frac{\pi }{16}{x}^{2}g\left(t\right)+y\left(t\right)$$

where $f\left(t\right)$, $g\left(t\right)$ and $y\left(t\right)$ are expansions in the variable $t=2x^4-1$.

For $1<x\le 3$,

$$ker\left(x\right)=exp(-\frac{11}{16}x)q\left(t\right)$$

where $q\left(t\right)$ is an expansion in the variable $t=x-2$.

For $x>3$,

$$ker\left(x\right)=\sqrt{\frac{\pi }{2x}}{e}^{-x/\sqrt{2}}[(1+\frac{1}{x}c\left(t\right))\cos \left(\beta \right)-\frac{1}{x}d\left(t\right)\sin \left(\beta \right)]$$

where $\beta =\frac{x}{\sqrt{2}}+\frac{\pi }{8}$, and $c\left(t\right)$ and $d\left(t\right)$ are expansions in the variable $t=\frac{6}{x}-1$.

When $x$ is sufficiently close to zero, the result is computed as

$$ker\left(x\right)=-\gamma -\log \left(\frac{x}{2}\right)+(\pi -\frac{3}{8}{x}^2)\frac{x^2}{16}$$

and when $x$ is even closer to zero, simply as $ker\left(x\right)=-\gamma -\log \left(\frac{x}{2}\right)$.

For large $x$, $ker\left(x\right)$ is asymptotically given by $\sqrt{\frac{\pi }{2x}}{e}^{-x/\sqrt{2}}$ and this becomes so small that it cannot be computed without underflow and the function fails.

References

NIST Digital Library of Mathematical Functions