Re: "Kantor dust"

[Steve Stevenson <fatmarauder@me.com>]

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.