I think you meant (1 > (|@- <.)) I also remember having problems with this issue, and using <.&.+. instead, but that was years ago, and I no longer remember the context.
As an aside, it is interesting to note that quaternions push the limits of this constraint, because of the distances between the corners of a 4 dimensional hypercube. (And it fails utterly on octonions. But mathematical properties of octonions are so divergent from those of regular numbers, it is difficult to show any practical consequences using octonions. It's hard to be suspicious of a mechanism only because the mechanism fails on numbers which are noncommutative and nonassociative...) -- Raul On Thu, Jan 26, 2012 at 1:29 PM, Marshall Lochbaum <mwlochb...@gmail.com> wrote: > Thanks for providing the link. > > The property that I had missed was "fractionality:" that (1 < (|@- <.)) . > This would ensure that while doing the Euclidian algorithm, the magnitudes > of numbers always decrease. However, this is not a necessary condition for > the Euclidian algorithm to work. Another sufficient condition is that > (((a|b)|a) <&| a) for all a and b. It seems that this is true for the floor > function |&.+. , although I haven't been able to prove it yet. > > Also note that J's builtin GCD does not use floor: it uses the nearest > integer. It seems that the complex remainder function was made the way it > was to facilitate complex GCD, but we can just as easily make it use a > nearest integer function derived from complex floor instead. > > Marshall > > On Thu, Jan 26, 2012 at 2:07 AM, Roger Hui <rogerhui.can...@gmail.com>wrote: > >> Eugene McDonnell's 1973 paper *Complex Floor* is now available at >> http://www.jsoftware.com/papers/eem/complexfloor.htm . >> ---------------------------------------------------------------------- >> For information about J forums see http://www.jsoftware.com/forums.htm >> > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
