naginterfaces.library.specfun.bessel_​i1_​real_​vector

naginterfaces.library.specfun.bessel_i1_real_vector(x)

bessel_i1_real_vector returns an array of values for the modified Bessel function $I_1\left(x\right)$.

For full information please refer to the NAG Library document for s18at

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/s/s18atf.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 $\left(n\right)$

$I_1\left(x_i\right)$, 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.

$ivalid[i-1]=1$

$x_i$ is too large. $f[\text{i}-1]$ contains the approximate value of $I_1\left(x_i\right)$ at the nearest valid argument. The threshold value is the same as for $errno$ = 1 in bessel_i1_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_i1_real_vector evaluates an approximation to the modified Bessel function of the first kind $I_1\left(x_i\right)$ for an array of arguments $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

Note: $I_1(-x)=-I_1\left(x\right)$, so the approximation need only consider $x\ge 0$.

The function is based on three Chebyshev expansions:

For $0<x\le 4$,

$$I_1\left(x\right)=x{\sum ^{\prime }}_{r=0}{a}_r{T}_r\left(t\right)\text{, where}t=2{\left(\frac{x}{4}\right)}^2-1\text{;}$$

For $4<x\le 12$,

$$I_1\left(x\right)=e^x{\sum ^{\prime }}_{r=0}{b}_r{T}_r\left(t\right)\text{, where}t=\frac{x-8}{4}\text{;}$$

For $x>12$,

$$I_1\left(x\right)=\frac{e^x}{\sqrt{x}}{\sum ^{\prime }}_{r=0}{c}_r{T}_r\left(t\right)\text{, where}t=2\left(\frac{12}{x}\right)-1\text{.}$$

For small $x$, $I_1\left(x\right)\simeq x$. 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 $I_1\left(x\right)$ cannot be represented without overflow.

References

NIST Digital Library of Mathematical Functions