naginterfaces.library.specfun.airy_​bi_​real_​vector

naginterfaces.library.specfun.airy_bi_real_vector(x)

airy_bi_real_vector returns an array of values of the Airy function, $Bi\left(x\right)$.

For full information please refer to the NAG Library document for s17av

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/s/s17avf.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 $\left(n\right)$

$Bi\left(x_i\right)$, 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 and positive. $f[\text{i}-1]$ contains zero. The threshold value is the same as for $errno$ = 1 in airy_bi_real().

$ivalid[i-1]=2$

$x_i$ is too large and negative. $f[\text{i}-1]$ contains zero. The threshold value is the same as for $errno$ = 2 in airy_bi_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

airy_bi_real_vector evaluates an approximation to the Airy function $Bi\left(x_i\right)$ for an array of arguments $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$. It is based on a number of Chebyshev expansions.

For $x<-5$,

$$Bi\left(x\right)=\frac{a\left(t\right)\cos \left(z\right)+b\left(t\right)\sin \left(z\right)}{{(-x)}^{1/4}}\text{,}$$

where $z=\frac{\pi }{4}+\frac{2}{3}\sqrt{-x^3}$ and $a\left(t\right)$ and $b\left(t\right)$ are expansions in the variable $t=-2{\left(\frac{5}{x}\right)}^3-1$.

For $-5\le x\le 0$,

$$Bi\left(x\right)=\sqrt{3}(f\left(t\right)+xg\left(t\right))\text{,}$$

where $f$ and $g$ are expansions in $t=-2{\left(\frac{x}{5}\right)}^3-1$.

For $0<x<4.5$,

$$Bi\left(x\right)=e^{11x/8}y\left(t\right)\text{,}$$

where $y$ is an expansion in $t=4x/9-1$.

For $4.5\le x<9$,

$$Bi\left(x\right)=e^{5x/2}v\left(t\right)\text{,}$$

where $v$ is an expansion in $t=4x/9-3$.

For $x\ge 9$,

$$Bi\left(x\right)=\frac{e^{z}u\left(t\right)}{x^{1/4}}\text{,}$$

where $z=\frac{2}{3}\sqrt{x^3}$ and $u$ is an expansion in $t=2\left(\frac{18}{z}\right)-1$.

For $\left|x\right|<\text{machine precision}$, the result is set directly to $Bi\left(0\right)$. This both saves time and avoids possible intermediate underflows.

For large negative arguments, it becomes impossible to calculate the phase of the oscillating function with any accuracy so the function must fail. This occurs if $x<-{\left(\frac{3}{2ϵ}\right)}^{2/3}$, where $ϵ$ is the machine precision.

For large positive arguments, there is a danger of causing overflow since Bi grows in an essentially exponential manner, so the function must fail.

References

NIST Digital Library of Mathematical Functions