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 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 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_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_REAL_ARG_LT: On entry, $x = 〈value〉$ .
Constraint: $x \geq 1.0$ .
The function has been called with an argument less than $1.0$ , for which $arccosh x$ is not defined.

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} \times 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 errors 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. The behaviour of the amplification factor is shown in the following graph:

Figure 1

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 Parallelism and Performance

Not applicable.

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.

NAG Library Function Documentnag_arccosh (s11acc)