naginterfaces.library.specfun.bessel_​i0_​real_​vector

naginterfaces.library.specfun.bessel_i0_real_vector(x)

bessel_i0_real_vector returns an array of values of the modified Bessel function $I_0\left(x\right)$.

For full information please refer to the NAG Library document for s18as

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/s/s18asf.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 $\left(n\right)$

$I_0\left(x_i\right)$, 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. $f[\text{i}-1]$ contains the approximate value of $I_0\left(x_i\right)$ at the nearest valid argument. The threshold value is the same as for $errno$ = 1 in bessel_i0_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_i0_real_vector evaluates an approximation to the modified Bessel function of the first kind $I_0\left(x_i\right)$ for an array of arguments $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

Note: $I_0(-x)=I_0\left(x\right)$, so the approximation need only consider $x\ge 0$.

The function is based on three Chebyshev expansions:

For $0<x\le 4$,

$$I_0\left(x\right)=e^x{\sum ^{\prime }}_{r=0}{a}_r{T}_r\left(t\right)\text{, where}t=2\left(\frac{x}{4}\right)-1\text{.}$$

For $4<x\le 12$,

$$I_0\left(x\right)=e^x{\sum ^{\prime }}_{r=0}{b}_r{T}_r\left(t\right)\text{, where}t=\frac{x-8}{4}\text{.}$$

For $x>12$,

$$I_0\left(x\right)=\frac{e^x}{\sqrt{x}}{\sum ^{\prime }}_{r=0}{c}_r{T}_r\left(t\right)\text{, where}t=2\left(\frac{12}{x}\right)-1\text{.}$$

For small $x$, $I_0\left(x\right)\simeq 1$. This approximation is used when $x$ is sufficiently small for the result to be correct to machine precision.

For large $x$, the function must fail because of the danger of overflow in calculating $e^x$.

References

NIST Digital Library of Mathematical Functions