# NAG Library Function Document

## 1Purpose

nag_arccosh (s11acc) returns the value of the inverse hyperbolic cosine, $\mathrm{arccosh}x$. The result is in the principal positive branch.

## 2Specification

 #include #include
 double nag_arccosh (double x, NagError *fail)

## 3Description

nag_arccosh (s11acc) calculates an approximate value for the inverse hyperbolic cosine, $\mathrm{arccosh}x$. It is based on the relation
 $arccosh⁡x=lnx+x2-1.$
This form is used directly for $1, 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
 $arccosh⁡x=ln⁡2+ln⁡x.$

## 4References

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

## 5Arguments

1:    $\mathbf{x}$doubleInput
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}\ge 1.0$.
2:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL_ARG_LT
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\ge 1.0$.
The function has been called with an argument less than $1.0$, for which $\mathrm{arccosh}x$ is not defined.

## 7Accuracy

If $\delta$ and $\epsilon$ are the relative errors in the argument and result respectively, then in principle
 $ε≃ x x2-1 arccosh⁡x ×δ .$
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:
Figure 1
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
 $E∼δ.$
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
 $E∼δ.$
That is, there will be approximately half as many decimal places correct in the result as there were correct figures in the argument.

## 8Parallelism and Performance

nag_arccosh (s11acc) is not threaded in any implementation.

None.

## 10Example

This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.

### 10.1Program Text

Program Text (s11acce.c)

### 10.2Program Data

Program Data (s11acce.d)

### 10.3Program Results

Program Results (s11acce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017