public static double s11ac(
	double x,
	out int ifail
)

Public Shared Function s11ac ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double s11ac(
	double x, 
	[OutAttribute] int% ifail
)

static member s11ac : 
        x : float * 
        ifail : int byref -> float 

s11ac returns the value of the inverse hyperbolic cosine, $\mathrm{arccosh} x$. The result is in the principal positive branch.

s11ac calculates an approximate value for the inverse hyperbolic cosine, $\mathrm{arccosh} x$. It is based on the relation This form is used directly for $1<x<{10}^k$, where $k=n/2+1$, and the machine uses approximately $n$ decimal place arithmetic.

For $x\ge {10}^k$, $\sqrt{x^2-1}$ is equal to $\sqrt{x}$ 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

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 is amplified by a factor at least $\frac{x}{\sqrt{x^2-1}\mathrm{arccosh} x}$ in the result. The equality should apply if $\delta$ is greater than the machine precision ($\delta$ due to data errors etc.) but if $\delta$ is simply a result of round-off in the machine representation it is possible that an extra figure may be lost in internal calculation and round-off. The behaviour of the amplification factor is shown in the following graph:

It should be noted that for $x>2$ the factor is always less than $1.0$. For large $x$ we have the absolute error $E$ in the result, in principle, given by This means that eventually accuracy is limited by machine precision. More significantly for $x$ close to $1$, $x-1\sim \delta$, the above analysis becomes inapplicable due to the fact that both function and argument are bounded, $x\ge 1$, $\mathrm{arccosh} x\ge 0$. In this region we have That is, there will be approximately half as many decimal places correct in the result as there were correct figures in the argument.

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.