NAG Library Routine Document
s09abf (arccos)
1
Purpose
s09abf returns the value of the inverse circular cosine, $\mathrm{arccos}x$, via the function name; the result is in the principal range $\left(0,\pi \right)$.
2
Specification
Fortran Interface
Real (Kind=nag_wp)  ::  s09abf  Integer, Intent (Inout)  ::  ifail  Real (Kind=nag_wp), Intent (In)  ::  x 

C Header Interface
#include nagmk26.h
double 
s09abf_ (const double *x, Integer *ifail) 

3
Description
s09abf calculates an approximate value for the inverse circular cosine,
$\mathrm{arccos}x$. It is based on the Chebyshev expansion
where
$\frac{1}{\sqrt{2}}\le x\le \frac{1}{\sqrt{2}}\text{, \hspace{1em} and \hspace{1em}}t=4{x}^{2}1$.
For ${x}^{2}\le \frac{1}{2}\text{, \hspace{1em}}\mathrm{arccos}x=\frac{\pi}{2}\mathrm{arcsin}x$.
For $1\le x<\frac{1}{\sqrt{2}}\text{, \hspace{1em}}\mathrm{arccos}x=\pi \mathrm{arcsin}\sqrt{1{x}^{2}}$.
For $\frac{1}{\sqrt{2}}<x\le 1\text{, \hspace{1em}}\mathrm{arccos}x=\mathrm{arcsin}\sqrt{1{x}^{2}}$.
For $\leftx\right>1\text{, \hspace{1em}}\mathrm{arccos}x$ is undefined and the routine fails.
4
References
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
5
Arguments
 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).
6
Error 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:
 ${\mathbf{ifail}}=1$

s09abf has been called with $\left{\mathbf{x}}\right>1.0$, for which arccos is undefined. A zero result is returned.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
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.
7
Accuracy
If
$\delta $ and
$\epsilon $ are the relative errors in the argument and the result, respectively, then in principle
The equality should hold if
$\delta $ is greater than the
machine precision (
$\delta $ is due to data errors etc.), but if
$\delta $ is due simply to roundoff in the machine it is possible that rounding etc. in internal calculations may lose one extra figure.
The behaviour of the amplification factor $\frac{x}{\mathrm{arccos}x\sqrt{1{x}^{2}}}$ is shown in the graph below.
In the region of $x=0$ this factor tends to zero and the accuracy will be limited by the machine precision. For $\leftx\right$ close to one, $1\leftx\right\sim \delta $, the above analysis is not applicable owing to the fact that both the argument and the result are bounded $\leftx\right\le 1$, $0\le \mathrm{arccos}x\le \pi $.
In the region of $x\sim 1$ we have $\epsilon \sim \sqrt{\delta}$, that is the result will have approximately half as many correct significant figures as the argument.
In the region $x\sim +1$, we have that the absolute error in the result, $E$, is given by $E\sim \sqrt{\delta}$, that is the result will have approximately half as many decimal places correct as there are correct figures in the argument.
8
Parallelism and Performance
s09abf is not threaded in any implementation.
None.
10
Example
This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.
10.1
Program Text
Program Text (s09abfe.f90)
10.2
Program Data
Program Data (s09abfe.d)
10.3
Program Results
Program Results (s09abfe.r)