NAG Library Routine Document
S11ACF returns the value of the inverse hyperbolic cosine, , via the function name. The result is in the principal positive branch.
|REAL (KIND=nag_wp) S11ACF
S11ACF calculates an approximate value for the inverse hyperbolic cosine,
. It is based on the relation
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
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
The routine has been called with an argument less than , for which is not defined. The result returned is zero.
are the relative errors in the argument and 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 apply if
is greater than the machine precision
due to data errors etc.) but if
is simply a result of round-off in the machine representation it is possible that an extra figure may be lost in internal calculation and round-off. The behaviour of the amplification factor is shown in the following graph:
It should be noted that for
the factor is always less than
. For large
we have the absolute error
in the result, in principle, given by
This means that eventually accuracy is limited by machine precision
. More significantly for
, the above analysis becomes inapplicable due to the fact that both function and argument are bounded,
. In this region we have
That is, there will be approximately half as many decimal places correct in the result as there were correct figures in the argument.
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 (s11acfe.f90)
9.2 Program Data
Program Data (s11acfe.d)
9.3 Program Results
Program Results (s11acfe.r)