NAG Library Function Document
nag_arcsinh (s11abc) returns the value of the inverse hyperbolic sine, .
||nag_arcsinh (double x)
nag_arcsinh (s11abc) 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
On entry: the argument of the function.
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
Parallelism and Performance
nag_arcsinh (s11abc) is not threaded in any implementation.
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.
Program Text (s11abce.c)
Program Data (s11abce.d)
Program Results (s11abce.r)