Thanks for the ref to the pdf, William. I recall there's a result-caching facility in J which would loosely answer to the description "superaccumulator", but I've forgotten offhand where to look for its write-up. In amongst the Foreigns, I'd guess.
> > 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. I guess I'd agree. I was half remembering an old pure math course on measure theory and Lebesgue integration. But a functional analyst's "rational" isn't quite the same as a J-er's "rational", which includes _r1 and 1r_ . I'm more worried about the theoretically irrational constants I need a lot, which I'm going to have to approximate, like √2 and π. So I ought to stop expecting unlimited precision and start thinking in terms of setting fixed limits to the length of the denominator that can arise. I can just see myself salting my code with: assert. 1000>#":saveit__=: myworkrational That was supposed to be a joke but I'm not sure it was. On Fri, 29 Mar 2019 at 20:46, William Tanksley, Jr <[email protected]> wrote: > 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 ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
