nag_arcsinh (s11abc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_arcsinh (s11abc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_arcsinh (s11abc) returns the value of the inverse hyperbolic sine, arcsinhx .

2  Specification

#include <nag.h>
#include <nags.h>
double  nag_arcsinh (double x)

3  Description

nag_arcsinh (s11abc) calculates an approximate value for the inverse hyperbolic sine of its argument, arcsinhx .
For x 1  the function is based on a Chebyshev expansion.
For x > 1  
arcsinhx = signx × ln x + x 2 + 1 .
This form is used directly for 1 < x < 10 k , where k = n / 2 + 1 , and the machine uses approximately n  decimal place arithmetic.
For x 10 k , x 2 + 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
arcsinhx = signx × ln2 + ln x

4  References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

5  Arguments

1:     xdoubleInput
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 + x 2 arcsinhx δ .
That is, the relative error in the argument, x , is amplified by a factor at least
x 1 + x 2 arcsinhx
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.
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
E δ .
This means that eventually accuracy is limited by machine precision.

8  Further Comments


9  Example

The following program reads values of the argument x  from a file, evaluates the function at each value of x  and prints the results.

9.1  Program Text

Program Text (s11abce.c)

9.2  Program Data

Program Data (s11abce.d)

9.3  Program Results

Program Results (s11abce.r)

nag_arcsinh (s11abc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012