NAG Library Routine Document
S11ABF returns the value of the inverse hyperbolic sine, , via the function name.
|REAL (KIND=nag_wp) S11ABF
S11ABF calculates an approximate value for the inverse hyperbolic sine of its argument, .
it is based on the Chebyshev expansion
it uses the fact that
This form is used directly for
, and the machine uses approximately
decimal place arithmetic.
is equal to
to within the accuracy of the machine and hence we can guard against premature overflow and, without loss of accuracy, calculate
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
- 1: X – REAL (KIND=nag_wp)Input
On entry: the argument of the function.
- 2: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
are the relative errors in the argument and the result, respectively, then in principle
That is, the relative error in the argument,
, is amplified by a factor at least
, in the result.
The equality should hold if is greater than the machine precision ( due to data errors etc.) but if 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:
It should be noted that this factor is always less than or equal to one. For large
we have the absolute error in the result,
, in principle, given by
This means that eventually accuracy is limited by machine precision
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.
9.1 Program Text
Program Text (s11abfe.f90)
9.2 Program Data
Program Data (s11abfe.d)
9.3 Program Results
Program Results (s11abfe.r)