On Feb 8, 2009, at 7:31 PM, Toby Bartels wrote:
Steve Stevenson wrote in part:
How about IEEE 754 reals? They're really "scientific notation". <snip>
Floating-point reals have terrible theoretical properties; they're not even a ring (not even classically). This is why even after all of Kahan's good work on algorithms, rounding errors are unavoidable (the "Table-Maker's Dilemma"). <snip>
Being left-handed and old, I'll propose in my dotage that we may be asking the wrong question. In a rewording, what constructive real numbers are there for the purpose of 1. Being a model of an axiomatic characterization of the reals. 2 Being usable in supercomputing to compute values needed for modeling and simulation. Number 1 requires that we have nice theoretical properties. Number 2 requires something that is bounded only the dollars and life span. Those interested in either purpose have (presumedly) a solution for themselves. The first fix for number 2 might be to go to interval arithmetic --- now what do I need to guarantee that the "real number (in 1)" is trapped between two #2 numbers? Given the bandwidth and memory capacity, we should be able to do those things worth doing: H5N1 infection prediction, climate modeling, malaria control, food production ... I'm willing to live with a demonstrably correct approximation given that we are in an uncertain world. I'll never get the exact answer, I'll only get an approximation. Interval guarantees seems interesting to me. --- Steve Stevenson It's not that people don't know, it's so much of what they know ain't so - Josh Billings.