[SciPy-dev] Generic polynomials class (was Re: Volunteer for Scipy Project)
Tue Oct 6 22:34:39 CDT 2009
2009/10/6 Charles R Harris <email@example.com>:
> On Tue, Oct 6, 2009 at 7:37 PM, Anne Archibald <firstname.lastname@example.org>
>> 2009/10/6 David Goldsmith <email@example.com>:
>> > IMO this thread has matured to the point where someone(s) should
>> > formally
>> > propose a feature spec/object structure for community vetting. :-)
>> I don't propose an object structure, because I'm not sure how that
>> should look, and I'm also not sure how reasonable my various
>> requirements and restrictions are. On the other hand I did list at the
>> end some references I found in a quick literature search; it appears
>> the wisdom is that for general polynomial manipulation it's best to
>> use the Bernstein basis (the same used in Bezier splines) or the
>> Lagrange basis (representing polynomials by (x,y) pairs). In either
>> case papers are available describing algorithms for all the basic
>> polynomial operations.
> Bernstein and Lagrange would both be defined on interval domains, [0,1] and
> [-1,1] respectively, and that would define both center and scaling if
> arbitrary intervals are mapped to those domains, just like for the Chebyshev
> polynomials. Hmm...
Yes, I haven't found any discussion of polynomial bases for unbounded
domains. It's not even clear to me what the right criterion for error
is, since any polynomial basis is going to become large far enough
away from the origin. If what one cared about were some sort of
weighted mean squared error, then some of the classical orthogonal
polynomials would be the right choice, but that has little relevance
for questions like root-finding or max-difference error. I suppose
some sort of weighted max-difference error might be usable - perhaps
weighted by exp(-x**2) or some such thing - but without some serious
numerical analysis it's not clear what to do there.
I'd guess that for most practical applications a specific interval can
be chosen. That does mean one needs to have parameterized bases, but
is otherwise inoffensive.
There's a comment in one of those references (I forget which just now)
to the effect that the Chebyshev basis is a poor choice if one wants
to find the roots of the resulting polynomial; on the other hand it's
an excellent basis for finding approximating polynomials. This is the
sort of thing that makes me want multiple representations.
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