4 ms·
I know rational representations (with bignums) can work better for some purposes (albeit not high-perf numerical ones), but the rub is, like you say, you're try
by mjhay 4y ago
I know rational representations (with bignums) can work better for some purposes (albeit not high-perf numerical ones), but the rub is, like you say, you're trying to represent an uncountably infinite set with finite memory, so everything is going to be some kind of space/time/fidelity tradeoff.
The only concrete finite fields are the finite fields of size p^k, where p is prime. I'm kind of thinking that there must be some representations that at least can shove off the breaking of associativity, commutativity, etc more into the background. It's probably more of a computer science and usability problem, but hopefully backed up by enough theory to provide some guarantees.
You can make a "free" field as a programming interface (like in the OP) - and the floats would then be a leaky implementation of that interface. I think the question is how to make an implementation that better captures what we talk about when we talk about real numbers, while having good enough performance for big numerical tasks, and avoiding common numerical problems. Definitely not easy, and I think application dependent to a great extent - a place where one really has to be able to glide up and down levels of abstraction.
- bodhiandphysics 4y agoThe thing about floating point numbers is that while they're not associative... they are approximately associative (mod inf and nan). In other words (ab)c = a(bc) + e where e is is a number much smaller than max(a,b,c). This means that we can measure quite easily how bad an operation is in floating points, using what's called the condition number, and there are many many good ways of either estimating or bounding the condition number. So floating point numbers are easy to analyze, though they require a bit more machinery than most programmers get in intro to computer science. Really floats are the right way to do calculations involving reals on the computer. Of course, if you're not actually using reals, you shouldn't use them!
- mjhay 4y agoYeah the condition number helps in simple cases (or when solving single linear systems), but things rapidly spiral once you start getting into solving stiff ODEs, etc - to the point that only numerical specialists can judge it. I feel like there ought to be ways of dealing with these issues in a more fine-grained manner with better guardrails - something like a numeric-aware typesystem or automatic algorithmic scheme generator. Like I said though, maybe in 100 years.
- bodhiandphysics 4y agostiffness has nothing to do with fp numbers... stiff equations are just as hard to solve using arbitrary precision bigints since the issue is the limits of approximations of the integral.
- mjhay 4y agoThat's getting back to my point though - the integral is over measurable sets of reals, and we are approximating that with finite-difference floats. This is a complex problem with many guises.
- deleted 4y ago[deleted]