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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_specfun_arctanh (s11aa)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_specfun_arctanh (s11aa) returns the value of the inverse hyperbolic tangent, arctanhx, via the function name.


[result, ifail] = s11aa(x)
[result, ifail] = nag_specfun_arctanh(x)


nag_specfun_arctanh (s11aa) calculates an approximate value for the inverse hyperbolic tangent of its argument, arctanhx.
For x212  it is based on the Chebyshev expansion
where - 12x 12, -1t1,   and  t=4x2-1.
For 12<x2<1, it uses
arctanhx=12ln1+x 1-x .  
For x1, the function fails as arctanhx is undefined.


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: x<1.0.

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 or equal to 1.0 in magnitude, for which arctanh is not defined. On soft failure, the result is returned as 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 1-x2 arctanhx ×δ .  
That is, the relative error in the argument, x, is amplified by at least a factor x1-x2arctanhx  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 then 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
The factor is not significantly greater than one except for arguments close to x=1. However in the region where x is close to one, 1-xδ, the above analysis is inapplicable since x is bounded by definition, x<1. In this region where arctanh is tending to infinity we have
which implies an obvious, unavoidable serious loss of accuracy near x1, e.g., if x and 1 agree to 6 significant figures, the result for arctanhx would be correct to at most about one figure.

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 s11aa_example

fprintf('s11aa example results\n\n');

x = [-0.5     0    0.5     -0.9999];
n = size(x,2);
result = x;

for j=1:n
  [result(j), ifail] = s11aa(x(j));

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

s11aa example results

      x        arctanh(x)
  -5.000e-01  -5.493e-01
   0.000e+00   0.000e+00
   5.000e-01   5.493e-01
  -9.999e-01  -4.952e+00

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

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