I forgot to paste in the one _useful_ part of my intended message. Sorry about that. Here's a link to some work attempting to characterize a numerically unstable algorithm using a superaccumulator as a stabilizer:
https://www.comp.nus.edu.sg/~wongwf/papers/HiPC-2018.pdf -Wm On Fri, Mar 29, 2019 at 1:30 PM William Tanksley, Jr <[email protected]> wrote: > > Ian Clark <[email protected]> wrote: > > One idea to come out of all this, which may or may not be original, but is > > impossible to contemplate addressing with floating-point numbers, is that > > inside a tool which is working with real-world observations, the numbers > > chiefly of interest will be small ones, in terms of the amount of storage > > they occupy, so that when designing iterative methods there may be some > > traction in steering away from large numbers into neighbouring small ones. > > Finding how to steer of this is one of the points of the > super-accumulator -- it keeps a finite bound on the error bars, which > allows for well-bounded speed, while allowing intermediates to include > both very small and very large values, with rounding to the final > format only when it's time to output. > > > In opposition to that idea, it's worth remembering that between any two > > rational numbers there is an uncountable infinity of irrational ones, > > Not particularly relevant; only a countable number of irrationals can > appear as the limit of computations. And some pairs of rationals have > no irrationals between them (for example, 0 and the nearest rational > to it, which is of the form 1/x, cannot have any irrationals between > them, even though (1/x and 1/(x-1)) has an countable infinity of > rationals between _them_). > > Chaitin proved that all of the accessible irrationals are a countable > set. (That is, the ones that can be named, defined, specified, or in > any way thought about, even when including the ones which can be shown > to be uncomputable such as his own "Omega" constant.) "The Tao that > can be named is not the true Tao." > > > and > > any attempt to avoid them might reintroduce the very sort of noise I'm > > aiming to eliminate. > > Don't worry -- you can't compute irrationals by a finite process. So > you don't have to avoid them, they'll automatically avoid you. :) > > > But the overwhelming sensation I have at the moment is one of admiration > > plus gratitude for the originators (and improvers) of J, for all that > > hidden work where nobody imagined it would matter. > > I am continually amazed. > > It reminds me of my first real math professor (after junior college). > He taught abstract algebra, and in particular he used a language > called "Forth" to explore finite group theory. It was very impressive > to be able to get my hands on those very abstract concepts. Even more > so because I knew Forth before I'd ever thought of learning abstract > algebra, and would have never guessed it would become a tool in a > mathematician's bag of tricks. J, on the other hand... I can see why > people don't learn it, but it's an amazing force multiplier once you > do. > > > Ian Clark > > -Wm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
