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NAG Library Routine Document

s11aaf (arctanh)

▸▿ 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

s11aaf returns the value of the inverse hyperbolic tangent,

arctanh x

, via the function name.

2

Specification

Fortran Interface

Function s11aaf (

x, ifail)

Real (Kind=nag_wp)	::	s11aaf
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x

C Header Interface

#include nagmk26.h

double

s11aaf_ ( const double *x, Integer *ifail)

3

Description

s11aaf 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 routine 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: $x$ – Real (Kind=nag_wp)Input: On entry: the argument $x$ of the function.

Constraint: $|x| < 1.0$ .
2: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: The routine 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.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

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

s11aaf is not threaded in any implementation.

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 Routine Document

s11aaf (arctanh)

▸▿ 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