NAG FL Interface
s11aaf (arctanh)

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1 Purpose

s11aaf returns the value of the inverse hyperbolic tangent, arctanhx, 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 <nag.h>
double  s11aaf_ (const double *x, Integer *ifail)
The routine may be called by the names s11aaf or nagf_specfun_arctanh.

3 Description

s11aaf 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 routine fails as arctanhx is undefined.

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

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 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
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.
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.
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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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.
s11aaf 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 (s11aafe.f90)

10.2 Program Data

Program Data (s11aafe.d)

10.3 Program Results

Program Results (s11aafe.r)