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

naginterfaces.library.specfun.airy_ai_deriv_vector(x)

airy_ai_deriv_vector returns an array of values of the derivative of the Airy function $Ai\left(x\right)$.

For full information please refer to the NAG Library document for s17aw

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s17awf.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 $\left(n\right)$

${\text{Ai}}^{\prime }\left(x_i\right)$, 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_deriv().

$ivalid[i-1]=2$

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

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_deriv_vector evaluates an approximation to the derivative of 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 $x<-5$,

$${\text{Ai}}^{\prime }\left(x\right)=\sqrt[4]{4-x}[a\left(t\right)\cos \left(z\right)+\frac{b\left(t\right)}{\zeta }\sin \left(z\right)]\text{,}$$

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

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

$${\text{Ai}}^{\prime }\left(x\right)=x^{2}f\left(t\right)-g\left(t\right)\text{,}$$

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

For $0<x<4.5$,

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

where $y\left(t\right)$ is an expansion in $t=4\left(\frac{x}{9}\right)-1$.

For $4.5\le x<9$,

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

where $v\left(t\right)$ is an expansion in $t=4\left(\frac{x}{9}\right)-3$.

For $x\ge 9$,

$${\text{Ai}}^{\prime }\left(x\right)=\sqrt[4]{4x}{e}^{-z}u\left(t\right)\text{,}$$

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

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

For large negative arguments, it becomes impossible to calculate a result for the oscillating function with any accuracy and so the function must fail. This occurs for $x<-{\left(\frac{\sqrt{\pi }}{ϵ}\right)}^{4/7}$, where $ϵ$ is the machine precision.

For large positive arguments, where ${\text{Ai}}^{\prime }$ decays in an essentially exponential manner, there is a danger of underflow so the function must fail.

References

NIST Digital Library of Mathematical Functions