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
arctanhx=x×y(t)=xr=0arTr(t)  
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:
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

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
ε1/lnδ  
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

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.

10.1 Program Text

Program Text (s11aafe.f90)

10.2 Program Data

Program Data (s11aafe.d)

10.3 Program Results

Program Results (s11aafe.r)