NAG CL Interface
s11abc (arcsinh)

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1 Purpose

s11abc returns the value of the inverse hyperbolic sine, arcsinhx.

2 Specification

#include <nag.h>
double  s11abc (double x)
The function may be called by the names: s11abc, nag_specfun_arcsinh or nag_arcsinh.

3 Description

s11abc calculates an approximate value for the inverse hyperbolic sine of its argument, arcsinhx.
For |x|1 it is based on the Chebyshev expansion
arcsinhx=x×y(t)=xr=0crTr(t),   where ​t=2x2-1.  
For |x|>1 it uses the fact that
arcsinhx=signx×ln(|x|+x2+1) .  
This form is used directly for 1<|x|<10k, where k=n/2+1, and the machine uses approximately n decimal place arithmetic.
For |x|10k, x2+1 is equal to |x| to within the accuracy of the machine and hence we can guard against premature overflow and, without loss of accuracy, calculate

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: x double Input
On entry: the argument x of the function.

6 Error Indicators and Warnings


7 Accuracy

If δ and ε are the relative errors in the argument and the result, respectively, then in principle
|ε| | x 1+x2 arcsinhx δ| .  
That is, the relative error in the argument, x, is amplified by a factor at least x1+x2arcsinhx , 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 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 10−1 100 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 ε/δ x x/(sqrt(1 + x**2) * asinh(x))
Figure 1
It should be noted that this factor is always less than or equal to one. For large x we have the absolute error in the result, E, in principle, given by
This means that eventually accuracy is limited by machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s11abc 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 (s11abce.c)

10.2 Program Data

Program Data (s11abce.d)

10.3 Program Results

Program Results (s11abce.r)