- Type: System..::..DoubleOn entry: the argument of the function.
s11ab calculates an approximate value for the inverse hyperbolic sine of its argument, .
For it is based on the Chebyshev expansion
For it uses the fact that
This form is used directly for , where , and the machine uses approximately decimal place arithmetic.
For , 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
If and 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.
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.
This example reads values of the argument from a file, evaluates the function at each value of and prints the results.