tag:blogger.com,1999:blog-8288194986820249216.post8204897719486014732..comments2024-03-22T05:09:17.789-07:00Comments on Abstract Heresies: Groups, semigroups, monoids, and computersJoe Marshallhttp://www.blogger.com/profile/03233353484280456977noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-8288194986820249216.post-91265259745001037842020-11-18T13:10:53.783-08:002020-11-18T13:10:53.783-08:00Exact real arithmetic has the advantage that it ne...Exact real arithmetic has the advantage that it never produces an incorrect answer, but it might take an unbounded amount of time (for example, y > x might not be decidable). Floating point produces an answer in a bounded amount of time, but the answer might be wrong.<br />Joe Marshallhttps://www.blogger.com/profile/03233353484280456977noreply@blogger.comtag:blogger.com,1999:blog-8288194986820249216.post-2262479324461962412020-11-18T12:54:22.539-08:002020-11-18T12:54:22.539-08:00Thank you. I haven't thought about exact real...Thank you. I haven't thought about exact real arithmetic as "arbitrary precision" for years - placed it in a different category - anyway, not since H Boehm and someone else I forget who implemented it as a demonstration of the Russell language - which itself was what I was interested in. <br /><br />Probably because 1) I generally associated it with kinds of computational geometry problems I don't really know much about, and 2) because sometimes when you try `x > y` the calculator starts consuming CPU and never ever returns - yeech!<br /><br />I'll look into it now ...David Bakinhttps://www.blogger.com/profile/08596808011584860686noreply@blogger.comtag:blogger.com,1999:blog-8288194986820249216.post-40286782567253234922020-11-16T18:48:48.793-08:002020-11-16T18:48:48.793-08:00Look at Gosper Continued Fraction Arithmetic and P...Look at Gosper <a href="https://perl.plover.com/classes/cftalk/INFO/gosper.txt" rel="nofollow">Continued Fraction Arithmetic</a> and Potts <a href="https://www.doc.ic.ac.uk/~ae/papers/potts-phd.pdf" rel="nofollow">Exact Real Arithmetic using Mobius Transformations</a>Joe Marshallhttps://www.blogger.com/profile/03233353484280456977noreply@blogger.comtag:blogger.com,1999:blog-8288194986820249216.post-91736661823613727572020-11-16T16:45:21.068-08:002020-11-16T16:45:21.068-08:00I'm extremely curious to know how the group of...I'm extremely curious to know how the group of "invertible 2x2 matrices with integer components, a determinant of 1 or -1, and the operation of matrix multiply" "comes in handy for implementing arbitrary precision arithmetic". I'm unaware of that, plus couldn't find anything in a quick search of <i>TAOCP</i> (I suppose it could have been buried in an exercise) or <i>Hacker's Delight</i>. And I don't really recall anything like that in Bernstein's "Multidigit Multiplication for Mathematicians" either (though I could have easily missed it since I didn't understand most of that paper). <br /><br />Pointer to a reference, please?David Bakinhttps://www.blogger.com/profile/08596808011584860686noreply@blogger.com