NAG FL Interface
s09aaf (arcsin)

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1 Purpose

s09aaf returns the value of the inverse circular sine, arcsinx, via the function name. The value is in the principal range (-π/2,π/2).

2 Specification

Fortran Interface
Function s09aaf ( x, ifail)
Real (Kind=nag_wp) :: s09aaf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x
C Header Interface
#include <nag.h>
double  s09aaf_ (const double *x, Integer *ifail)
The routine may be called by the names s09aaf or nagf_specfun_arcsin.

3 Description

s09aaf calculates an approximate value for the inverse circular sine, arcsinx. It is based on the Chebyshev expansion
where - 12x 12 and t=4x2-1.
For x2 12,  arcsinx=x×y(x).
For 12<x21,  arcsinx=signx { π2-arcsin1-x2} .
For x2>1,  arcsinx 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 result, respectively, then in principle
|ε| | x arcsinx 1-x2 ×δ| .  
That is, a relative error in the argument x is amplified by at least a factor xarcsinx1-x2 in the result.
The equality should hold if δ is greater than the machine precision (δ is a result of data errors etc.) but if δ is produced simply by round-off error in the machine it is possible that rounding in internal calculations may lose an extra figure in the result.
This factor stays close to one except near |x|=1 where its behaviour is shown in the following graph.
Figure 1
Figure 1
For |x| close to unity, 1-|x|δ, the above analysis is no longer applicable owing to the fact that both argument and result are subject to finite bounds, (|x|1 and |arcsinx|12π). In this region εδ; that is the result will have approximately half as many correct significant figures as the argument.
For |x|=1 the result will be correct to full machine precision.

8 Parallelism and Performance

s09aaf 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 (s09aafe.f90)

10.2 Program Data

Program Data (s09aafe.d)

10.3 Program Results

Program Results (s09aafe.r)