Eugene McDonnell <[EMAIL PROTECTED]> wrote: > On Jun 20, 2007, at 2:36 PM, Mark D. Niemiec wrote: > > > Granted, this value lies right along a line separating two regions, > > and > > floating-point inaccuracies would explain why a number might stray > > slightly > > to one side or other of this line, but tolerant comparisons should > > remove > > such artifacts. The anomaly looks particularly heinous in the > > second case, > > since it interrupts the linear progression of values. > > The numbers in a computer do not contain all the reals, much less all > the complex numbers. > > My papers assume a mathematical treatment of numbers. > > Given that: > > The complex plane is completely tessellated by slanted bricks, with > the upper left and lower left boundaries within the brick, and the > other sides of the brick not in the brick. > > If a Gaussian or complex number arises it must be on one of the > included sides or not. The plane is completely defined by these L- > shaped sides. > > Given this brick, then ceiling and residue follow -- defined > algebraically. > > Given this definition, the division theorem follows, and one of its > strengths is that it is the identical division theorem obtaining > among the reals, and that residues will always be strictly less than > the divisors. > > The reason I undertook this study was to answer Larry Breed's > question: how is complex residue defined?
I spent some time playing with the J implementation of this a few years ago until I had a reasonable understanding of what was going on. My problem is not with the definition, but with the observed artifacts of the implementation. In particular, the dictionary says that one of the desiderata is: Convexity: If (<.z1)=(<.z2) and z3 lies on the line between z1 to z2, then (<.z3)=(<.z1) This does not always seem to be the case. It appears that the edges of the boxes are slightly wobbly, so that a straight line running along the inside of one of the boxes can be within it most of the time, but pass outside for brief moments before coming back in. This is the kind of thing that comparison tolerance was designed to avoid (and it does, indeed, do so for the real portion of the floor funciton). -- Mark D. Niemiec <[EMAIL PROTECTED]> ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
