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

naginterfaces.library.specfun.airy_ai_real_vector(x)

airy_ai_real_vector returns an array of values for the Airy function, $Ai\left(x\right)$.

For full information please refer to the NAG Library document for s17au

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/s/s17auf.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 $\left(n\right)$

$Ai\left(x_i\right)$, 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_ai_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_ai_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_ai_real_vector evaluates an approximation to the Airy function, $Ai\left(x_i\right)$ 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 $x<-5$,

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

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 $-5\le x\le 0$,

$$Ai\left(x\right)=f\left(t\right)-xg\left(t\right)\text{,}$$

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

For $0<x<4.5$,

$$Ai\left(x\right)=e^{-3x/2}y\left(t\right)\text{,}$$

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

For $4.5\le x<9$,

$$Ai\left(x\right)=e^{-5x/2}u\left(t\right)\text{,}$$

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

For $x\ge 9$,

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

where $z=\frac{2}{3}\sqrt{x^3}$ and $v$ 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 $Ai\left(0\right)$. This both saves time and guards against underflow in intermediate calculations.

For large negative arguments, it becomes impossible to calculate the phase of the oscillatory function with any precision and 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, where $\text{Ai}$ decays in an essentially exponential manner, there is a danger of underflow so the function must fail.

References

NIST Digital Library of Mathematical Functions