s17ak returns a value for the derivative of the Airy function Bix.

Syntax

C#
public static double s17ak(
	double x,
	out int ifail
)
Visual Basic
Public Shared Function s17ak ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
public:
static double s17ak(
	double x, 
	[OutAttribute] int% ifail
)
F#
static member s17ak : 
        x : float * 
        ifail : int byref -> float 

Parameters

x
Type: System..::..Double
On entry: the argument x of the function.
ifail
Type: System..::..Int32%
On exit: ifail=0 unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

s17ak returns a value for the derivative of the Airy function Bix.

Description

s17ak calculates an approximate value for the derivative of the Airy function Bix. It is based on a number of Chebyshev expansions.
For x<-5,
Bix=-x4-atsinz+btζcosz,
where z=π4+ζ, ζ=23-x3 and at and bt are expansions in the variable t=-25x3-1.
For -5x0,
Bix=3x2ft+gt,
where f and g are expansions in t=-2x53-1.
For 0<x<4.5,
Bix=e3x/2yt,
where yt is an expansion in t=4x/9-1.
For 4.5x<9,
Bix=e21x/8ut,
where ut is an expansion in t=4x/9-3.
For x9,
Bix=x4ezvt,
where z=23x3 and vt is an expansion in t=218z-1.
For x< the square of the machine precision, the result is set directly to Bi0. 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<-πε4/7, where ε is the machine precision.
For large positive arguments, where Bi grows in an essentially exponential manner, there is a danger of overflow so the method must fail.

References

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

Error Indicators and Warnings

Errors or warnings detected by the method:
ifail=1
x is too large and positive. On failure, the method returns zero. (see the Users' Note for your implementation for details)
ifail=2
x is too large and negative. On failure the method returns zero. (see the Users' Note for your implementation for details)
ifail=-9000
An error occured, see message report.

Accuracy

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 ε, are related in principle to the relative error in the argument δ, by
Ex2Bixδεx2BixBixδ.
In practice, approximate equality is the best that can be expected. When δ, ε 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 x7/4π. Therefore it becomes impossible to calculate the function with any accuracy if x7/4>πδ.
For large positive x, the relative error amplification is considerable: εδx3. However, very large arguments are not possible due to the danger of overflow. Thus in practice the actual amplification that occurs is limited.

Parallelism and Performance

None.

Further Comments

None.

Example

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

Example program (C#): s17ake.cs

Example program data: s17ake.d

Example program results: s17ake.r

See Also