NAG Library Function Document

nag_arccosh (s11acc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_arccosh (s11acc) returns the value of the inverse hyperbolic cosine,

arccosh x

. The result is in the principal positive branch.

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_arccosh (double x, NagError *fail)

3 Description

nag_arccosh (s11acc) calculates an approximate value for the inverse hyperbolic cosine,

arccosh x

. It is based on the relation

arccosh x = \ln (x + \sqrt{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 \geq 10^{k}, \sqrt{x^{2} - 1}

is equal to

\sqrt{x}

to within the accuracy of the machine and hence we can guard against premature overflow and, without loss of accuracy, calculate

arccosh x = \ln 2 + \ln x .

4 References

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

5 Arguments

1: x – doubleInput: On entry: the argument $x$ of the function.
Constraint: $x \geq 1.0$ .
2: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_REAL_ARG_LT: On entry, x must not be less than 1.0: $x = 〈value〉$ .
$arccosh x$ is not defined and the result returned is zero.

7 Accuracy

δ

and

ε

are the relative errors in the argument and result respectively, then in principle

|ε| ≃ |\frac{x}{\sqrt{x^{2} - 1} arccosh x} δ| .

That is the relative error in the argument is amplified by a factor at least

\frac{x}{\sqrt{x^{2} - 1} arccosh x}

in the result. The equality should apply if

δ

is greater than the machine precision (

δ

due to data error etc.), but if

δ

is simply a result of round-off in the machine representation, it is possible that an extra figure may be lost in internal calculation and round-off.

It should be noted that for

x > 2

the factor is always less than 1.0. For large

x

we have the absolute error

E

in the result, in principle, given by

E \sim δ .

This means that eventually accuracy is limited by machine precision. More significantly for

x

close to 1,

x - 1 \sim δ

, the above analysis becomes inapplicable due to the fact that both function and argument are bounded,

x \geq 1

arccosh x \geq 0

. In this region we have

E \sim \sqrt{δ} .

That is, there will be approximately half as many decimal places correct in the result as there were correct figures in the argument.

8 Further Comments

None.

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.

NAG Library Function Documentnag_arccosh (s11acc)