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NAG Toolbox

NAG Toolbox: nag_specfun_arcsinh (s11ab)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_arcsinh (s11ab) returns the value of the inverse hyperbolic sine, arcsinhx, via the function name.


[result, ifail] = s11ab(x)
[result, ifail] = nag_specfun_arcsinh(x)


nag_specfun_arcsinh (s11ab) calculates an approximate value for the inverse hyperbolic sine of its argument, arcsinhx.
For x1 it is based on the Chebyshev expansion
arcsinhx=x×yt=xr=0crTrt,   where ​t=2x2-1.  
For x>1 it uses the fact that
arcsinhx=signx×lnx+x2+1 .  
This form is used directly for 1<x<10k, where k=n/2+1, and the machine uses approximately n decimal place arithmetic.
For x10k, x2+1 is equal to x to within the accuracy of the machine and hence we can guard against premature overflow and, without loss of accuracy, calculate


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.

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



If δ and ε are the relative errors in the argument and the result, respectively, then in principle
ε x 1+x2 arcsinhx δ .  
That is, the relative error in the argument, x, is amplified by a factor at least x1+x2arcsinhx , in the result.
The equality should hold if δ is greater than the machine precision (δ due to data errors etc.) but if δ is simply due to round-off in the machine representation it is possible that an extra figure may be lost in internal calculation round-off.
The behaviour of the amplification factor is shown in the following graph:
Figure 1
Figure 1
It should be noted that this factor is always less than or equal to one. For large x we have the absolute error in the result, E, in principle, given by
This means that eventually accuracy is limited by 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 s11ab_example

fprintf('s11ab example results\n\n');

x = [-2    -0.5    1     6];
n = size(x,2);
result = x;

for j=1:n
  [result(j), ifail] = s11ab(x(j));

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

s11ab example results

      x        arcsinh(x)
  -2.000e+00  -1.444e+00
  -5.000e-01  -4.812e-01
   1.000e+00   8.814e-01
   6.000e+00   2.492e+00

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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