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

Constraint: $| x | \leq 1.0$ .
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: $| x | \leq 1$ .

$ifail = - 99$: 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 the result, respectively, then in principle

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

The equality should hold if

δ

is greater than the machine precision (

δ

is due to data errors etc.), but if

δ

is due simply to round-off in the machine it is possible that rounding etc. in internal calculations may lose one extra figure.

The behaviour of the amplification factor

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

is shown in the graph below.

In the region of

x = 0

this factor tends to zero and the accuracy will be limited by the machine precision. For

| x |

close to one,

1 - | x | \sim δ

, the above analysis is not applicable owing to the fact that both the argument and the result are bounded

| x | \leq 1

0 \leq \arccos x \leq π

In the region of

x \sim - 1

we have

ε \sim \sqrt{δ}

, that is the result will have approximately half as many correct significant figures as the argument.

In the region

x \sim + 1

, we have that the absolute error in the result,

E

, is given by

E \sim \sqrt{δ}

, that is the result will have approximately half as many decimal places correct as there are correct figures in the argument.

Figure 1

8 Parallelism and Performance

s09abf 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.

s09ab: FL CL CPP AD

NAG FL Interfaces09abf (arccos)

▸▿ 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
s09abf (arccos)