public static double s17ag(
	double x,
	out int ifail
)

Public Shared Function s17ag ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double s17ag(
	double x, 
	[OutAttribute] int% ifail
)

static member s17ag : 
        x : float * 
        ifail : int byref -> float 

s17ag returns a value for the Airy function, $\mathrm{Ai}\left(x\right)$.

s17ag evaluates an approximation to the Airy function, $\mathrm{Ai}\left(x\right)$. It is based on a number of Chebyshev expansions:

For $x<-5$, 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 $-5\le x\le 0$, where $f$ and $g$ are expansions in $t=-2{\left(\frac{x}{5}\right)}^3-1\text{.}$

For $0<x<4.5$, where $y$ is an expansion in $t=4x/9-1$.

For $4.5\le x<9$, where $u$ is an expansion in $t=4x/9-3$.

For $x\ge 9$, where $z=\frac{2}{3}\sqrt{x^3}$ and $v$ is an expansion in $t=2\left(\frac{18}{z}\right)-1$.

For $\left|x\right|<\mathit{machine precision}$, the result is set directly to $\mathrm{Ai}\left(0\right)$. 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 method must fail. This occurs if $x<-{\left(\frac{3}{2\epsilon }\right)}^{2/3}$, where $\epsilon$ is the machine precision.

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

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

Errors or warnings detected by the method:

For negative arguments the function is oscillatory and hence absolute error is the appropriate measure. In the positive region the function is essentially exponential-like and here relative error is appropriate. The absolute error, $E$, and the relative error, $\epsilon$, are related in principle to the relative error in the argument, $\delta$, by In practice, approximate equality is the best that can be expected. When $\delta$, $\epsilon$ or $E$ is of the order of the machine precision, the errors in the result will be somewhat larger.

For small $x$, errors are strongly damped by the function and hence will be bounded by the machine precision.

For moderate negative $x$, the error behaviour is oscillatory but the amplitude of the error grows like However the phase error will be growing roughly like $\frac{2}{3}\sqrt{{\left|x\right|}^3}$ and hence all accuracy will be lost for large negative arguments due to the impossibility of calculating sin and cos to any accuracy if $\frac{2}{3}\sqrt{{\left|x\right|}^3}>\frac{1}{\delta }$.

For large positive arguments, the relative error amplification is considerable: This means a loss of roughly two decimal places accuracy for arguments in the region of $20$. However very large arguments are not possible due to the danger of setting underflow and so the errors are limited in practice.

None.

None.

This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.