NAG Library Function Document
nag_arctanh (s11aac) returns the value of the inverse hyperbolic tangent, .
||nag_arctanh (double x,
nag_arctanh (s11aac) calculates an approximate value for the inverse hyperbolic tangent of its argument, .
it is based on the Chebyshev expansion
, it uses
, the function fails as
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
x – doubleInput
On entry: the argument of the function.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG
On entry, .
The function has been called with an argument greater than or equal to in magnitude, for which is not defined.
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.
8 Parallelism and Performance
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.
10.1 Program Text
Program Text (s11aace.c)
10.2 Program Data
Program Data (s11aace.d)
10.3 Program Results
Program Results (s11aace.r)