1: $x$ – Real (Kind=nag_wp) Input: On entry: the argument $x$ of the function.

Constraint: $x \geq 1.0$ .
2: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. 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

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $x = ⟨ value ⟩$ .
Constraint: $x \geq 1.0$ .
The routine has been called with an argument less than $1.0$ , for which $arccosh x$ is not defined.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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

s11acf 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.

s11ac: FL CL CPP AD

NAG FL Interfaces11acf (arccosh)

▸▿ 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

NAG FL Interface
s11acf (arccosh)