[SciPy-dev] Generic polynomials class (was Re: Volunteer for Scipy Project)
Wed Oct 7 00:09:01 CDT 2009
2009/10/7 Charles R Harris <firstname.lastname@example.org>:
> I wasn't able to get to the paper on division, which was one that really
> roused my curiosity. Maybe the library will have it.
It's less interesting than it looks; his abstract algorithm is really
just grade-school long division with different operations to extract
the head term and form the one-term-less quotients. It may also have
poorer numerical stability than previously known matrix methods.
> One of the papers didn't regard degree as important, which is probably
> because 1) the Bernstein polynomials aren't graded by degree and 2) neither
> are the Lagrange functions. And because it is difficult to define degree in
> those contexts. At some point I think we will have to decide whether we want
> "polynomial approximation", "orthogonal polynomials", or both. Both seem to
> have their virtues.
Well, degree is an issue even for monomial-basis polynomials once you
start adding and subtracting - the highest coefficient could be nearly
zero and you'd have to decide whether it was close enough to zero to
discard. Whether that's true depends on the x values you're interested
in... At least for Chebyshev-basis polynomials there's a clear
criterion for when it can be treated as zero
> Thanks for putting that page together, that helps get things organized. It
> also grows what started as a simple Chebyshev class into something more
> extensive ;) Oh well...
Heh. Well, I don't think you should delay releasing your Chebyshev
class at all. Putting code that works aside in favour of code that
doesn't exist yet seems like a mistake.
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