NAG CL Interface
s11aac (arctanh)

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1 Purpose

s11aac returns the value of the inverse hyperbolic tangent, arctanhx.

2 Specification

#include <nag.h>
double  s11aac (double x, NagError *fail)
The function may be called by the names: s11aac, nag_specfun_arctanh or nag_arctanh.

3 Description

s11aac 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=12ln(1+x 1-x ) .  
For |x|1, the function fails as arctanhx is undefined.

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: x double Input
On entry: the argument x of the function.
Constraint: |x|<1.0.
2: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
On entry, x=value.
Constraint: |x|<1.
The function has been called with an argument greater than or equal to 1.0 in magnitude, for which arctanh is not defined.

7 Accuracy

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 x(1-x2)arctanhx 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:
GnuplotProduced by GNUPLOT 5.4 patchlevel 6 100 101 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ε/δ x x/((1 - x**2) * atanh(x))
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-|x|δ, 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 |x|1, e.g., if x and 1 agree to 6 significant figures, the result for arctanhx would be correct to at most about one figure.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s11aac is not threaded in any implementation.

9 Further Comments


10 Example

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

10.1 Program Text

Program Text (s11aace.c)

10.2 Program Data

Program Data (s11aace.d)

10.3 Program Results

Program Results (s11aace.r)