NAG FL Interface
s09abf (arccos)

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1 Purpose

s09abf returns the value of the inverse circular cosine, arccosx, via the function name; the result is in the principal range (0,π).

2 Specification

Fortran Interface
Function s09abf ( x, ifail)
Real (Kind=nag_wp) :: s09abf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x
C Header Interface
#include <nag.h>
double  s09abf_ (const double *x, Integer *ifail)
The routine may be called by the names s09abf or nagf_specfun_arccos.

3 Description

s09abf calculates an approximate value for the inverse circular cosine, arccosx. It is based on the Chebyshev expansion
where -12x 12,   and  t=4x2-1.
For x2 12,  arccosx= π2-arcsinx.
For -1x< -12,  arccosx=π-arcsin1-x2.
For 12<x1,  arccosx=arcsin1-x2.
For |x|>1,  arccosx is undefined and the routine fails.

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: x Real (Kind=nag_wp) Input
On entry: the argument x of the function.
Constraint: |x|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 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, x=value.
Constraint: |x|1.
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.
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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

If δ and ε are the relative errors in the argument and the result, respectively, then in principle
|ε| | x arccosx 1-x2 ×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 xarccosx1-x2 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|δ, the above analysis is not applicable owing to the fact that both the argument and the result are bounded |x|1, 0arccosxπ.
In the region of x-1 we have εδ, that is the result will have approximately half as many correct significant figures as the argument.
In the region x+1, we have that the absolute error in the result, E, is given by Eδ, that is the result will have approximately half as many decimal places correct as there are correct figures in the argument.
Figure 1
Figure 1

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s09abf is not threaded in any implementation.

9 Further Comments


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.

10.1 Program Text

Program Text (s09abfe.f90)

10.2 Program Data

Program Data (s09abfe.d)

10.3 Program Results

Program Results (s09abfe.r)