nag_arctanh (s11aac) (PDF version)
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NAG Library Manual

NAG Library Function Document

nag_arctanh (s11aac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_arctanh (s11aac) returns the value of the inverse hyperbolic tangent, arctanhx.

2  Specification

#include <nag.h>
#include <nags.h>
double  nag_arctanh (double x, NagError *fail)

3  Description

nag_arctanh (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=12ln1+x 1-x .
For x1, the function fails as arctanhx is undefined.

4  References

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

5  Arguments

1:     xdoubleInput
On entry: the argument x of the function.
Constraint: x<1.0.
2:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

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.
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 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.

8  Parallelism and Performance

Not applicable.

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)

nag_arctanh (s11aac) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014