s09ab returns the value of the inverse circular cosine, arccosx; the result is in the principal range 0,π.

Syntax

C#
public static double s09ab(
	double x,
	out int ifail
)
Visual Basic
Public Shared Function s09ab ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
public:
static double s09ab(
	double x, 
	[OutAttribute] int% ifail
)
F#
static member s09ab : 
        x : float * 
        ifail : int byref -> float 

Parameters

x
Type: System..::..Double
On entry: the argument x of the function.
Constraint: x1.0.
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

s09ab returns the value of the inverse circular cosine, arccosx; the result is in the principal range 0,π.

Description

s09ab calculates an approximate value for the inverse circular cosine, arccosx. It is based on the Chebyshev expansion
arcsinx=x×yt=xr=0arTrt
where -12x12,   and  t=4x2-1.
For x212,  arccosx=π2-arcsinx.
For -1x<-12,  arccosx=π-arcsin1-x2.
For 12<x1,  arccosx=arcsin1-x2.
For x>1,  arccosx is undefined and the method fails.

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
s09ab has been called with x>1.0, for which arccos is undefined. A zero result is returned.
ifail=-9000
An error occured, see message report.

Accuracy

If δ and ε are the relative errors in the argument and the result, respectively, then in principle
εxarccosx1-x2×δ.
The equality should hold if δ is greater than the machine precision (δ is due to data errors etc.), but if δ is due simply to round-off in the machine it is possible that rounding etc. in internal calculations may lose one extra figure.
The behaviour of the amplification factor xarccosx1-x2 is shown in the graph below.
In the region of x=0 this factor tends to zero and the accuracy will be limited by the machine precision. For x close to one, 1-xδ, the above analysis is not applicable owing to the fact that both the argument and the result are bounded x1, 0arccosxπ.
In the region of x-1 we have εδ, that is the result will have approximately half as many correct significant figures as the argument.
In the region x+1, we have that the absolute error in the result, E, is given by Eδ, that is the result will have approximately half as many decimal places correct as there are correct figures in the argument.
Figure 1
Figure 1

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#): s09abe.cs

Example program data: s09abe.d

Example program results: s09abe.r

See Also