NAG FL Interface
s11abf (arcsinh)

Settings help

FL Name Style:

FL Specification Language:

1 Purpose

s11abf returns the value of the inverse hyperbolic sine, arcsinhx, via the function name.

2 Specification

Fortran Interface
Function s11abf ( x, ifail)
Real (Kind=nag_wp) :: s11abf
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: x
C Header Interface
#include <nag.h>
double  s11abf_ (const double *x, Integer *ifail)
The routine may be called by the names s11abf or nagf_specfun_arcsinh.

3 Description

s11abf calculates an approximate value for the inverse hyperbolic sine of its argument, arcsinhx.
For |x|1 it is based on the Chebyshev expansion
arcsinhx=x×y(t)=xr=0crTr(t),   where ​t=2x2-1.  
For |x|>1 it uses the fact that
arcsinhx=signx×ln(|x|+x2+1) .  
This form is used directly for 1<|x|<10k, where k=n/2+1, and the machine uses approximately n decimal place arithmetic.
For |x|10k, x2+1 is equal to |x| to within the accuracy of the machine and hence we can guard against premature overflow and, without loss of accuracy, calculate

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.
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


7 Accuracy

If δ and ε are the relative errors in the argument and the result, respectively, then in principle
|ε| | x 1+x2 arcsinhx δ| .  
That is, the relative error in the argument, x, is amplified by a factor at least x1+x2arcsinhx , in the result.
The equality should hold if δ is greater than the machine precision (δ due to data errors etc.) but if δ is simply due to round-off in the machine representation it is possible that an extra figure may be lost in internal calculation round-off.
The behaviour of the amplification factor is shown in the following graph:
GnuplotProduced by GNUPLOT 5.4 patchlevel 6 10−1 100 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 ε/δ x x/(sqrt(1 + x**2) * asinh(x))
Figure 1
It should be noted that this factor is always less than or equal to one. For large x we have the absolute error in the result, E, in principle, given by
This means that eventually accuracy is limited by machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s11abf 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 (s11abfe.f90)

10.2 Program Data

Program Data (s11abfe.d)

10.3 Program Results

Program Results (s11abfe.r)