Re: Another proposal for a "split" function to complement "mid"
Vladik
On 18 Mar 2012, at 20:40, Kreinovich, Vladik wrote:
> will it be a concern that asinh requires computing ln which is much slower than usual midpoint and most the alternatives propsed so far?
I am assuming that in B&B what one *does with* a box is significantly more expensive than *getting* the box. If it's B&B in several dimensions, one typically evaluates a multivariate function in interval mode over the box. Getting the box involves a split in just one dimension of a larger box, evaluating asinh twice and sinh once using my formula. Count half of that, as the split produces two boxes to go on the B&B queue.
> also, sounds somewhat arbitrary -- if someone can empirically check that this particular function works well (and better than other functions with a similar asymptotic behavior) that would make this proposal much more convincing -- and it will probably be possible to come up with a theoretical explanation
Earlier discussion suggested a good method to find a split point in [a,b] would behave like the arithmetic mean for smallish numbers, and like geometric mean for larger numbers of the same sign. asinh is the only elementary function that gives a mapping which implements that, being like log(x) for x >> 1, like x for |x| small, like -log(-x) for x << -1. That's why I chose it. But a function with similar behaviour would do, if it and its inverse are cheap to evaluate and guaranteed monotone.
John