On Jul 8, 11:36 am, Mark Dickinson <dicki...@gmail.com> wrote: > I think that's because we're talking at cross-purposes. > > To clarify, suppose you want to compute some value (pi; log(2); > AGM(1, sqrt(2)); whatever...) to 1000 significant decimal places. > Then typically the algorithm (sometimes known as Ziv's onion-peeling > method) looks like: > > (1) Compute an initial approximation to 1002 digits (say), with known > absolute error (given by a suitable error analysis); for the sake of > argument, let's say that you use enough intermediate precision to > guarantee an absolute error of < 0.6 ulps. > > (2) Check to see whether that approximation unambiguously gives you > the correctly-rounded 1000 digits that you need. > > (3) If not, increase the target precision (say by 3 digits) and try > again. > > It's the precision increase in (3) that I was calling small, and > similarly it's step (3) that isn't usually needed more than once or > twice. (In general, for most functions and input values; I dare say > there are exceptions.) > > Step (1) will often involve using significantly more than the target > precision for intermediate computations, depending very much on what > you happen to be trying to compute. IIUC, it's the extra precision in > step (1) that you don't want to call 'small', and I agree. > > IOW, I'm saying that the extra precision required *due to the table- > maker's dilemma* generally isn't a concern. Yes, though I think attributing only the precision added in step 3 to the table-maker's dilemma isn't entirely correct. While it'd be certainly less of a dilemma if we could precompute the necessary precision, it doesn't' help us if the precision is generally unbounded. As such, I think it's really two dilemmas for the price of one.
> > I actually agree with much of what you've said. It was just the > "impossible" claim that went over the top (IMO). The MPFR library > amply demonstrates that computing many transcendental functions to > arbitrary precision, with correctly rounded results, is indeed > possible. That's because you're latching onto that word instead of the whole sentence in context and making a much bigger deal out of than is appropriate. The fact that I may not be able to complete a given calculation for an arbitrary precision is not something that can be ignored. It's the same notional problem with arbitrary-precision integers: is it better to run out of memory or overflow the calculation? The answer, of course, is a trick question. Adam -- http://mail.python.org/mailman/listinfo/python-list
