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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_arcsinh (s11ab)

## Purpose

nag_specfun_arcsinh (s11ab) returns the value of the inverse hyperbolic sine, $\mathrm{arcsinh}x$, via the function name.

## Syntax

[result, ifail] = s11ab(x)
[result, ifail] = nag_specfun_arcsinh(x)

## Description

nag_specfun_arcsinh (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
 $arcsinh⁡x=x×yt=x∑′r=0crTrt, where ​t=2x2-1.$
For $\left|x\right|>1$ it uses the fact that
 $arcsinh⁡x=sign⁡x×lnx+x2+1 .$
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
 $arcsinh⁡x=sign⁡x×ln⁡2+lnx.$

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{x}$ – double scalar
The argument $x$ of the function.

None.

### Output Parameters

1:     $\mathrm{result}$ – double scalar
The result of the function.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

None.

## Accuracy

If $\delta$ and $\epsilon$ are the relative errors in the argument and the result, respectively, then in principle
 $ε≃ x 1+x2 arcsinh⁡x δ .$
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: Figure 1
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
 $E∼δ.$
This means that eventually accuracy is limited by machine precision.

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.
```function s11ab_example

fprintf('s11ab example results\n\n');

x = [-2    -0.5    1     6];
n = size(x,2);
result = x;

for j=1:n
[result(j), ifail] = s11ab(x(j));
end

disp('      x        arcsinh(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

```
```s11ab example results

x        arcsinh(x)
-2.000e+00  -1.444e+00
-5.000e-01  -4.812e-01
1.000e+00   8.814e-01
6.000e+00   2.492e+00
```