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, .
For the function is based on a Chebyshev expansion.
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
On entry, must not be greater than or equal to 1.0: .
The function has been called with an argument greater than or equal to 1.0 in magnitude, for which arctanh is not defined. 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 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
and 1 agree to 6 significant figures, the result for
would be correct to at most about one figure.
The following program 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 (s11aace.c)
9.2 Program Data
Program Data (s11aace.d)
9.3 Program Results
Program Results (s11aace.r)