s11aa returns the value of the inverse hyperbolic tangent, arctanhx.


public static double s11aa(
	double x,
	out int ifail
Visual Basic
Public Shared Function s11aa ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
static double s11aa(
	double x, 
	[OutAttribute] int% ifail
static member s11aa : 
        x : float * 
        ifail : int byref -> float 


Type: System..::..Double
On entry: the argument x of the function.
Constraint: x<1.0.
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

s11aa returns the value of the inverse hyperbolic tangent, arctanhx.


s11aa calculates an approximate value for the inverse hyperbolic tangent of its argument, arctanhx.
For x212 it is based on the Chebyshev expansion
where -12x12, -1t1,   and  t=4x2-1.
For 12<x2<1, it uses
For x1, the method fails as arctanhx is undefined.


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:
The method has been called with an argument greater than or equal to 1.0 in magnitude, for which arctanh is not defined. On failure, the result is returned as zero.
An error occured, see message report.


If δ and ε are the relative errors in the argument and result, respectively, then in principle
That is, the relative error in the argument, x, is amplified by at least a factor x1-x2arctanhx in the result. The equality should hold if δ is greater than the machine precision (δ due to data errors etc.) but if δ is simply due to round-off in the machine representation then it is possible that an extra figure may be lost in internal calculation round-off.
The behaviour of the amplification factor is shown in the following graph:
Figure 1
Figure 1
The factor is not significantly greater than one except for arguments close to x=1. However in the region where x is close to one, 1-xδ, the above analysis is inapplicable since x is bounded by definition, x<1. In this region where arctanh is tending to infinity we have
which implies an obvious, unavoidable serious loss of accuracy near x1, e.g., if x and 1 agree to 6 significant figures, the result for arctanhx would be correct to at most about one figure.

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#): s11aae.cs

Example program data: s11aae.d

Example program results: s11aae.r

See Also