s17ah returns a value of the Airy function, Bix.


public static double s17ah(
	double x,
	out int ifail
Visual Basic
Public Shared Function s17ah ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
static double s17ah(
	double x, 
	[OutAttribute] int% ifail
static member s17ah : 
        x : float * 
        ifail : int byref -> float 


Type: System..::..Double
On entry: the argument x of the function.
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

s17ah returns a value of the Airy function, Bix.


s17ah evaluates an approximation to the Airy function Bix. It is based on a number of Chebyshev expansions.
For x<-5,
where z=π4+23-x3 and at and bt are expansions in the variable t=-25x3-1.
For -5x0,
where f and g are expansions in t=-2x53-1.
For 0<x<4.5,
where y is an expansion in t=4x/9-1.
For 4.5x<9,
where v is an expansion in t=4x/9-3.
For x9,
where z=23x3 and u is an expansion in t=218z-1.
For x<machine precision, the result is set directly to Bi0. This both saves time and avoids possible intermediate underflows.
For large negative arguments, it becomes impossible to calculate the phase of the oscillating function with any accuracy so the method must fail. This occurs if x<-32ε2/3, where ε is the machine precision.
For large positive arguments, there is a danger of causing overflow since Bi grows in an essentially exponential manner, so the method must fail.


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:
x is too large and positive. On failure, the method returns zero. (see the Users' Note for your implementation for details)
x is too large and negative. On failure, the method returns zero. See also the Users' Note for your implementation.
An error occured, see message report.


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, ε, are related in principle to the relative error in the argument, δ, by
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, errors are strongly damped and hence will be bounded essentially by the machine precision.
For moderate to large negative x, the error behaviour is clearly oscillatory but the amplitude of the error grows like amplitude Eδx5/4π.
However the phase error will be growing roughly as 23x3 and hence all accuracy will be lost for large negative arguments. This is due to the impossibility of calculating sin and cos to any accuracy if 23x3>1δ.
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 causing overflow and errors are therefore limited in practice.

Parallelism and Performance


Further Comments



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#): s17ahe.cs

Example program data: s17ahe.d

Example program results: s17ahe.r

See Also