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:

$ifail = 1$: nag_specfun_arccos (s09ab) has been called with $|x| > 1.0$ , for which arccos is undefined. A zero result is returned.

$ifail = - 99$: 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 the result, respectively, then in principle

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

The equality should hold if

δ

is greater than the machine precision (

δ

is due to data errors etc.), but if

δ

is due simply to round-off 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}{\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

|x|

close to one,

1 - |x| \sim δ

, the above analysis is not applicable owing to the fact that both the argument and the result are bounded

|x| \leq 1

0 \leq \arccos x \leq π

In the region of

x \sim - 1

we have

ε \sim \sqrt{δ}

, 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{δ}

, that is the result will have approximately half as many decimal places correct as there are correct figures in the argument.

Figure 1

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: s09ab_example

function s09ab_example


fprintf('s09ab example results\n\n');

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

for j=1:n
  [result(j), ifail] = s09ab(x(j));
end

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

s09ab example results

      x        arccos(x)
  -5.000e-01   2.094e+00
   1.000e-01   1.471e+00
   9.000e-01   4.510e-01

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

Chapter Contents

Chapter Introduction

NAG Toolbox