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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_arccos (s09ab)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_arccos (s09ab) returns the value of the inverse circular cosine, arccosx, via the function name; the result is in the principal range 0,π.


[result, ifail] = s09ab(x)
[result, ifail] = nag_specfun_arccos(x)


nag_specfun_arccos (s09ab) calculates an approximate value for the inverse circular cosine, arccosx. It is based on the Chebyshev expansion
where -12x 12,   and  t=4x2-1.
For x2 12,  arccosx= π2-arcsinx.
For -1x< -12,  arccosx=π-arcsin1-x2.
For 12<x1,  arccosx=arcsin1-x2.
For x>1,  arccosx is undefined and the function fails.


Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications


Compulsory Input Parameters

1:     x – double scalar
The argument x of the function.
Constraint: x1.0.

Optional Input Parameters


Output Parameters

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:
nag_specfun_arccos (s09ab) has been called with x>1.0, for which arccos is undefined. A zero result is returned.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


If δ and ε are the relative errors in the argument and the result, respectively, then in principle
ε x arccosx 1-x2 ×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 xarccosx1-x2  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-xδ, the above analysis is not applicable owing to the fact that both the argument and the result are bounded x1, 0arccosxπ.
In the region of x-1 we have εδ, that is the result will have approximately half as many correct significant figures as the argument.
In the region x+1, we have that the absolute error in the result, E, is given by Eδ, that is the result will have approximately half as many decimal places correct as there are correct figures in the argument.
Figure 1
Figure 1

Further Comments



This example reads values of the argument x from a file, evaluates the function at each value of x and prints the results.
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));

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

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