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

Constraint: $| x | < 1.0$ .
2: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. 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:

$ifail = 1$: On entry, $x = ⟨ value ⟩$ .
Constraint: $| x | < 1$ .
The routine has been called with an argument greater than or equal to $1.0$ in magnitude, for which $arctanh$ is not defined.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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

Background information to multithreading can be found in the Multithreading documentation.

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.

s11aa: FL CL CPP AD PY MB

NAG FL Interfaces11aaf (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

NAG FL Interface
s11aaf (arctanh)