public static double s11ab(
	double x
)

Public Shared Function s11ab ( _
	x As Double _
) As Double

public:
static double s11ab(
	double x
)

static member s11ab : 
        x : float -> float 

s11ab returns the value of the inverse hyperbolic sine, $\mathrm{arcsinh} x$.

s11ab calculates an approximate value for the inverse hyperbolic sine of its argument, $\mathrm{arcsinh} x$.

For $\left|x\right|\le 1$ it is based on the Chebyshev expansion For $\left|x\right|>1$ it uses the fact that This form is used directly for $1<\left|x\right|<{10}^k$, where $k=n/2+1$, and the machine uses approximately $n$ decimal place arithmetic.

For $\left|x\right|\ge {10}^k$, $\sqrt{x^2+1}$ is equal to $\left|x\right|$ to within the accuracy of the machine and hence we can guard against premature overflow and, without loss of accuracy, calculate

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

None.

If $\delta$ and $\epsilon$ are the relative errors in the argument and the result, respectively, then in principle That is, the relative error in the argument, $x$, is amplified by a factor at least $\frac{x}{\sqrt{1+x^2}\mathrm{arcsinh} x}$, 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 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:

It should be noted that this factor is always less than or equal to one. For large $x$ we have the absolute error in the result, $E$, in principle, given by This means that eventually accuracy is limited by machine precision.

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.