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NAG Toolbox

NAG Toolbox: nag_specfun_arcsin (s09aa)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_arcsin (s09aa) returns the value of the inverse circular sine, arcsinx, via the function name. The value is in the principal range -π/2,π/2.


[result, ifail] = s09aa(x)
[result, ifail] = nag_specfun_arcsin(x)


nag_specfun_arcsin (s09aa) calculates an approximate value for the inverse circular sine, arcsinx. It is based on the Chebyshev expansion
where - 12x 12  and t=4x2-1.
For x2 12,  arcsinx=x×yx.
For 12<x21,  arcsinx=signx π2-arcsin1-x2 .
For x2>1,  arcsinx 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:
The function has been called with an argument greater than 1.0 in absolute value; arcsinx is undefined and the function returns zero.
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 result, respectively, then in principle
ε x arcsinx 1-x2 ×δ .  
That is, a relative error in the argument x is amplified by at least a factor xarcsinx1-x2  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
Figure 1
For x close to unity, 1-xδ, the above analysis is no longer applicable owing to the fact that both argument and result are subject to finite bounds, (x1 and arcsinx12π). In this region εδ; 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



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

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

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Chapter Introduction
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