public static double s11aa(
	double x,
	out int ifail
)

Public Shared Function s11aa ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double s11aa(
	double x, 
	[OutAttribute] int% ifail
)

static member s11aa : 
        x : float * 
        ifail : int byref -> float 

s11aa returns the value of the inverse hyperbolic tangent, $\mathrm{arctanh} x$.

s11aa calculates an approximate value for the inverse hyperbolic tangent of its argument, $\mathrm{arctanh} x$.

For $x^2\le \frac{1}{2}$ it is based on the Chebyshev expansion where $-\frac{1}{\sqrt{2}}\le x\le \frac{1}{\sqrt{2}}$, $-1\le t\le 1$, $\text{\hspace{1em} and \hspace{1em}}t=4x^2-1$.

For $\frac{1}{2}<x^2<1$, it uses For $\left|x\right|\ge 1$, the method fails as $\mathrm{arctanh} x$ is undefined.

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

Errors or warnings detected by the method:

If $\delta$ and $\epsilon$ 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 $\frac{x}{\left(1-x^2\right)\mathrm{arctanh} x}$ in the result. The equality should hold if $\delta$ is greater than the machine precision ($\delta$ due to data errors etc.) but if $\delta$ 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:

The factor is not significantly greater than one except for arguments close to $\left|x\right|=1$. However in the region where $\left|x\right|$ is close to one, $1-\left|x\right|\sim \delta$, the above analysis is inapplicable since $x$ is bounded by definition, $\left|x\right|<1$. In this region where arctanh is tending to infinity we have which implies an obvious, unavoidable serious loss of accuracy near $\left|x\right|\sim 1$, e.g., if $x$ and $1$ agree to $6$ significant figures, the result for $\mathrm{arctanh} x$ would be correct to at most about one figure.

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.