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 less than $1.0$ , for which $arccosh x$ is not defined. The result returned is 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}{\sqrt{x^{2} - 1} arccosh x} \times δ| .

That is the relative error in the argument is amplified by a factor at least

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

in the result. The equality should apply if

δ

is greater than the machine precision (

δ

due to data errors etc.) but if

δ

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 \sim δ .

This means that eventually accuracy is limited by machine precision. More significantly for

x

close to

1

x - 1 \sim δ

, the above analysis becomes inapplicable due to the fact that both function and argument are bounded,

x \geq 1

arccosh x \geq 0

. In this region we have

E \sim \sqrt{δ} .

That is, there will be approximately half as many decimal places correct in the result as there were correct figures in the argument.

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: s11ac_example

function s11ac_example


fprintf('s11ac example results\n\n');

x = [1   2     5     10];
n = size(x,2);
result = x;

for j=1:n
  [result(j), ifail] = s11ac(x(j));
end

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

s11ac example results

      x        arccosh(x)
   1.000e+00   0.000e+00
   2.000e+00   1.317e+00
   5.000e+00   2.292e+00
   1.000e+01   2.993e+00

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

Chapter Contents

Chapter Introduction

NAG Toolbox