public static double s17ak(
	double x,
	out int ifail
)

Public Shared Function s17ak ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double s17ak(
	double x, 
	[OutAttribute] int% ifail
)

static member s17ak : 
        x : float * 
        ifail : int byref -> float 

s17ak returns a value for the derivative of the Airy function $\mathrm{Bi}\left(x\right)$.

s17ak calculates an approximate value for the derivative of the Airy function $\mathrm{Bi}\left(x\right)$. It is based on a number of Chebyshev expansions.

For $x<-5$, 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 the variable $t=-2{\left(\frac{5}{x}\right)}^3-1$.

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

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

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

For $x\ge 9$, where $z=\frac{2}{3}\sqrt{x^3}$ and $v\left(t\right)$ 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 ${\mathrm{Bi}}^{\prime }\left(0\right)$. This saves time and avoids possible underflows in calculation.

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

For large positive arguments, where ${\mathrm{Bi}}^{\prime }$ grows in an essentially exponential manner, there is a danger of overflow 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 appropriate. In the positive region the function has essentially exponential behaviour and hence relative error is needed. 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 $x$, positive or negative, errors are strongly attenuated by the function and hence will effectively be bounded by the machine precision.

For moderate to large negative $x$, the error is, like the function, oscillatory. However, the amplitude of the absolute error grows like $\frac{{\left|x\right|}^{7/4}}{\sqrt{\pi }}$. Therefore it becomes impossible to calculate the function with any accuracy if ${\left|x\right|}^{7/4}>\frac{\sqrt{\pi }}{\delta }$.

For large positive $x$, the relative error amplification is considerable: $\frac{\epsilon }{\delta }\sim \sqrt{x^3}$. However, very large arguments are not possible due to the danger of overflow. Thus in practice the actual amplification that occurs is limited.

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.