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

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: The function has been called with an argument greater than $1.0$ in absolute value; $\arcsin x$ is undefined and the function returns zero.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

δ

and

ε

are the relative errors in the argument and result, respectively, then in principle

|ε| ≃ |\frac{x}{\arcsin x \sqrt{1 - x^{2}}} \times δ| .

That is, a relative error in the argument

x

is amplified by at least a factor

\frac{x}{\arcsin x \sqrt{1 - x^{2}}}

in the result.

The equality should hold if

δ

is greater than the machine precision (

δ

is a result of data errors etc.) but if

δ

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

|x| = 1

where its behaviour is shown in the following graph.

Figure 1

For

|x|

close to unity,

1 - |x| \sim δ

, the above analysis is no longer applicable owing to the fact that both argument and result are subject to finite bounds, (

|x| \leq 1

and

|\arcsin x| \leq \frac{1}{2} π

). In this region

ε \sim \sqrt{δ}

; that is the result will have approximately half as many correct significant figures as the argument.

For

|x| = 1

the result will be correct to full machine precision.

Further Comments

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.

Open in the MATLAB editor: s09aa_example

function s09aa_example


fprintf('s09aa example results\n\n');

x = [ -0.5    0.1     0.9];
n = size(x,2);
result = x;

for j=1:n
  [result(j), ifail] = s09aa(x(j));
end

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

s09aa example results

      x        arcsin(x)
  -5.000e-01  -5.236e-01
   1.000e-01   1.002e-01
   9.000e-01   1.120e+00

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox