# NAG Library Routine Document

## 1Purpose

s09aaf returns the value of the inverse circular sine, $\mathrm{arcsin}x$, via the function name. The value is in the principal range $\left(-\pi /2,\pi /2\right)$.

## 2Specification

Fortran Interface
 Function s09aaf ( x,
 Real (Kind=nag_wp) :: s09aaf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include nagmk26.h
 double s09aaf_ (const double *x, Integer *ifail)

## 3Description

s09aaf calculates an approximate value for the inverse circular sine, $\mathrm{arcsin}x$. It is based on the Chebyshev expansion
 $arcsin⁡x=x×yx=x∑′r=0arTrt$
where $-\frac{1}{\sqrt{2}}\le x\le \frac{1}{\sqrt{2}}$ and $t=4{x}^{2}-1$.
For ${x}^{2}\le \frac{1}{2}\text{, }\mathrm{arcsin}x=x×y\left(x\right)$.
For $\frac{1}{2}<{x}^{2}\le 1\text{, }\mathrm{arcsin}x=\mathrm{sign}x\left\{\frac{\pi }{2}-\mathrm{arcsin}\sqrt{1-{x}^{2}}\right\}$.
For ${x}^{2}>1\text{, }\mathrm{arcsin}x$ is undefined and the routine fails.
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

## 5Arguments

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the argument $x$ of the function.
Constraint: $\left|{\mathbf{x}}\right|\le 1.0$.
2:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
The routine has been called with an argument greater than $1.0$ in absolute value; $\mathrm{arcsin}x$ is undefined and the routine returns zero.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

If $\delta$ and $\epsilon$ are the relative errors in the argument and result, respectively, then in principle
 $ε≃ x arcsin⁡x 1-x2 ×δ .$
That is, a relative error in the argument $x$ is amplified by at least a factor $\frac{x}{\mathrm{arcsin}x\sqrt{1-{x}^{2}}}$ in the result.
The equality should hold if $\delta$ is greater than the machine precision ($\delta$ is a result of data errors etc.) but if $\delta$ is produced simply by round-off error in the machine it is possible that rounding in internal calculations may lose an extra figure in the result.
This factor stays close to one except near $\left|x\right|=1$ where its behaviour is shown in the following graph. Figure 1
For $\left|x\right|$ close to unity, $1-\left|x\right|\sim \delta$, the above analysis is no longer applicable owing to the fact that both argument and result are subject to finite bounds, ($\left|x\right|\le 1$ and $\left|\mathrm{arcsin}x\right|\le \frac{1}{2}\pi$). In this region $\epsilon \sim \sqrt{\delta }$; that is the result will have approximately half as many correct significant figures as the argument.
For $\left|x\right|=1$ the result will be correct to full machine precision.

## 8Parallelism and Performance

s09aaf 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 (s09aafe.f90)

### 10.2Program Data

Program Data (s09aafe.d)

### 10.3Program Results

Program Results (s09aafe.r)

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