NAG Library Routine Document
S11AAF returns the value of the inverse hyperbolic tangent, , via the function name.
|REAL (KIND=nag_wp) S11AAF
S11AAF calculates an approximate value for the inverse hyperbolic tangent of its argument, .
it is based on the Chebyshev expansion
, it uses
, the routine fails as
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 greater than or equal to in magnitude, for which is not defined. On soft failure, the result is returned as 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 at least a factor
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 then 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:
The factor is not significantly greater than one except for arguments close to
. However in the region where
is close to one,
, the above analysis is inapplicable since
is bounded by definition,
. In this region where arctanh is tending to infinity we have
which implies an obvious, unavoidable serious loss of accuracy near
, e.g., if
significant figures, the result for
would be correct to at most about one figure.
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 (s11aafe.f90)
9.2 Program Data
Program Data (s11aafe.d)
9.3 Program Results
Program Results (s11aafe.r)