NAG Library Function Document

nag_arctanh (s11aac)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

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

arctanh x

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,

arctanh x

For

x^{2} \leq \frac{1}{2}

it is based on the Chebyshev expansion

arctanh x = x \times y (t) = x \underset{r = 0}{\sum^{'}} a_{r} T_{r} (t)

where

- \frac{1}{\sqrt{2}} \leq x \leq \frac{1}{\sqrt{2}}

- 1 \leq t \leq 1

and t = 4 x^{2} - 1

For

\frac{1}{2} < x^{2} < 1

, it uses

arctanh x = \frac{1}{2} \ln (\frac{1 + x}{1 - x}) .

For

|x| \geq 1

, the function fails as

arctanh x

is undefined.

4 References

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

5 Arguments

1: x – doubleInput: On entry: the argument $x$ of the function.
Constraint: $|x| < 1.0$ .
2: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_INTERNAL_ERROR: 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.
NE_REAL_ARG_GE: 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

δ

and

ε

are the relative errors in the argument and result, respectively, then in principle

|ε| ≃ |\frac{x}{(1 - x^{2}) arctanh x} \times δ| .

That is, the relative error in the argument,

x

, is amplified by at least a factor

\frac{x}{(1 - x^{2}) arctanh x}

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

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| \sim 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

ε \sim 1 / \ln δ

which implies an obvious, unavoidable serious loss of accuracy near

|x| \sim 1

, e.g., if

x

and

1

agree to

6

significant figures, the result for

arctanh x

would be correct to at most about one figure.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

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.

NAG Library Function Documentnag_arctanh (s11aac)