NAG Library Routine Document

s10aaf  (tanh)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

s10aaf returns a value for the hyperbolic tangent, tanhx, via the function name.

2
Specification

Fortran Interface
Function s10aaf ( x, ifail)
Real (Kind=nag_wp):: s10aaf
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: x
C Header Interface
#include nagmk26.h
double  s10aaf_ ( const double *x, Integer *ifail)

3
Description

s10aaf calculates an approximate value for the hyperbolic tangent of its argument, tanhx.
For x1 it is based on the Chebyshev expansion
tanhx=x×yt=xr=0arTrt  
where -1x1,  -1t1,   and  t=2x2-1.
For 1<x<E1 (see the Users' Note for your implementation for value of E1)
tanhx=e2x-1 e2x+1 .  
For xE1, tanhx=signx to within the representation accuracy of the machine and so this approximation is used.

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.
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

None.

7
Accuracy

If δ and ε are the relative errors in the argument and the result respectively, then in principle,
ε 2x sinh2x δ .  
That is, a relative error in the argument, x, is amplified by a factor approximately 2x sinh2x , 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 due simply to the round-off in the machine representation 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
Figure 1
It should be noted that this factor is always less than or equal to 1.0 and away from x=0 the accuracy will eventually be limited entirely by the precision of machine representation.

8
Parallelism and Performance

s10aaf 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 (s10aafe.f90)

10.2
Program Data

Program Data (s10aafe.d)

10.3
Program Results

Program Results (s10aafe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017