naginterfaces.library.specfun.bessel_​k1_​real_​vector

naginterfaces.library.specfun.bessel_k1_real_vector(x)

bessel_k1_real_vector returns an array of values of the modified Bessel function $K_1\left(x\right)$.

For full information please refer to the NAG Library document for s18ar

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s18arf.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 $\left(n\right)$

$K_1\left(x_i\right)$, 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.

$ivalid[i-1]=1$

$x_i\le 0.0$, $K_1\left(x_i\right)$ is undefined. $f[\text{i}-1]$ contains $0.0$.

$ivalid[i-1]=2$

$x_i$ is too small, there is a danger of overflow. $f[\text{i}-1]$ contains zero. The threshold value is the same as for $errno$ = 2 in bessel_k1_real().

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

bessel_k1_real_vector evaluates an approximation to the modified Bessel function of the second kind $K_1\left(x_i\right)$ for an array of arguments $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

Note: $K_1\left(x\right)$ is undefined for $x\le 0$ and the function will fail for such arguments.

The function is based on five Chebyshev expansions:

For $0<x\le 1$,

$$K_1\left(x\right)=\frac{1}{x}+xln\left(x\right){\sum ^{\prime }}_{r=0}{a}_r{T}_r\left(t\right)-x{\sum ^{\prime }}_{r=0}{b}_r{T}_r\left(t\right)\text{, where}t=2x^2-1\text{.}$$

For $1<x\le 2$,

$$K_1\left(x\right)=e^{-x}{\sum ^{\prime }}_{r=0}{c}_r{T}_r\left(t\right)\text{, where}t=2x-3\text{.}$$

For $2<x\le 4$,

$$K_1\left(x\right)=e^{-x}{\sum ^{\prime }}_{r=0}{d}_r{T}_r\left(t\right)\text{, where}t=x-3\text{.}$$

For $x>4$,

$$K_1\left(x\right)=\frac{e^{-x}}{\sqrt{x}}{\sum ^{\prime }}_{r=0}{e}_r{T}_r\left(t\right)\text{, where}t=\frac{9-x}{1+x}\text{.}$$

For $x$ near zero, $K_1\left(x\right)\simeq \frac{1}{x}$. This approximation is used when $x$ is sufficiently small for the result to be correct to machine precision. For very small $x$ it is impossible to calculate $\frac{1}{x}$ without overflow and the function must fail.

For large $x$, where there is a danger of underflow due to the smallness of $K_1$, the result is set exactly to zero.

References

NIST Digital Library of Mathematical Functions