naginterfaces.library.specfun.kelvin_​bei_​vector

naginterfaces.library.specfun.kelvin_bei_vector(x)

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

For full information please refer to the NAG Library document for s19ap

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/s/s19apf.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 $\left(n\right)$

$bei\left(x_i\right)$, 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.

$ivalid[i-1]=1$

$abs\left(x_i\right)$ is too large for an accurate result to be returned. $f[\text{i}-1]$ contains zero. The threshold value is the same as for $errno$ = 1 in kelvin_bei().

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

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

The function is based on several Chebyshev expansions:

For $0\le x\le 5$,

$$bei\left(x\right)=\frac{x^2}{4}{\sum ^{\prime }}_{r=0}{a}_r{T}_r\left(t\right)\text{, with}t=2{\left(\frac{x}{5}\right)}^4-1\text{;}$$

For $x>5$,

$$bei\left(x\right)=\frac{e^{x/\sqrt{2}}}{\sqrt{2\pi x}}[(1+\frac{1}{x}a\left(t\right))\sin \left(\alpha \right)-\frac{1}{x}b\left(t\right)\cos \left(\alpha \right)]$$

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

where $\alpha =\frac{x}{\sqrt{2}}-\frac{\pi }{8}$, $\beta =\frac{x}{\sqrt{2}}+\frac{\pi }{8}$,

and $a\left(t\right)$, $b\left(t\right)$, $c\left(t\right)$, and $d\left(t\right)$ are expansions in the variable $t=\frac{10}{x}-1$.

When $x$ is sufficiently close to zero, the result is computed as $bei\left(x\right)=\frac{x^2}{4}$. If this result would underflow, the result returned is $bei\left(x\right)=0.0$.

For large $x$, there is a danger of the result being totally inaccurate, as the error amplification factor grows in an essentially exponential manner;, therefore, the function must fail.

References

NIST Digital Library of Mathematical Functions