[SciPy-dev] Difference between polynomial.trimcoef and trimseq

David Goldsmith d.l.goldsmith@gmail....
Sun Jan 24 03:09:09 CST 2010


fromCharles R Harris <charlesr.harris@gmail.com>
On Sat, Jan 23, 2010 at 11:08 PM, David Goldsmith
<d.l.goldsmith@gmail.com>wrote:

> Do you think a typical user would ever use both?  (Or is this an efficiency
> that most can live w/out?  I'm just curious how much we should "explain
> ourselves" in their docstrings.)
>
>
> Hard to say ;) I wrote the docstrings for the helper funtions

And in this case the "helper function" is the trimseq, correct?

> mostly for my own use and think of those helper functions as private. They
are in the standard import just in case anyone wants
> to do their own stuff.


> PS: If I were to use chebyshev as my "template," what would you say is the
> next most useful/algorithmically-studied polynomial basis to implement?
>
>
> The power/Chebyshev series have the special property that it is easy to
multiply/divide them, so the template needs to lose a
> few features to be useful for functions where that is far more difficult.

Yeah, that's what I meant by "algorithmically-studied": AFAYK, numericists
haven't derived/discovered nearly as efficient "tricks" for operating on the
other orthos/classes as they have for the standard and Chebyshev bases?
BTW: on the subject of "numerical tricks," are there such for trigonometric
polynomials?

> Multiplication by x should be sufficient for most things, in

Which, in the standard basis at least, is just a prepending of a zero, of
course...

> particular evaluation and conversion to/from other series.
> Apart from that, I think Legendre polynomials would fit in well. There

That was my first guess as to what to do next.

> was a request for Hermite polynomials,

Hermite requester: if you're reading this, did you already "roll your own"?

> which shouldn't be difficult in principle, but perhaps more so in practice
because there
> are two versions that go under that name but have different scalings. It
is also more difficult to assign a fixed domain for them
> because the domain essentially expands with the degree. But I don't think
those difficulties are fundamental.

No, I agree (I think): the issue is more one of "given a particular basis,
what can one do to stay in the basis while only manipulating the
coefficients" - if your multiplying Legendre polynomials, e.g., the natural
default is that you want the result to also be Legendre.  If good algorithms
for this have been worked out for standard and Cheby, but no others...well,
at the very least, it helps me see why you stopped where you did. ;-)

> Chuck

On Sat, Jan 23, 2010 at 11:44 PM, Anne Archibald
<peridot.faceted@gmail.com>wrote:

> 2010/1/24 David Goldsmith <d.l.goldsmith@gmail.com>:
>
> > PS: If I were to use chebyshev as my "template," what would you say is
> the
> > next most useful/algorithmically-studied polynomial basis to implement?
>
> There was extensive (and occasionally heated) discussion of other
> polynomial representations around the time the Chebyshev routines were
> being introduced. My point of view in that discussion was that there
> should be a general framework for working with polynomials in many
> representations, but the representations I thought might be worth
> having were:
>
> (a) Power basis.
> (b) Chebyshev basis.
> (c) Bases of other families of orthogonal polynomials.
> (d) Lagrange basis (polynomials by value).
> (e) Spline basis.
>
> The need for polynomials expressed in terms of other families of
> orthogonal polynomials is to some degree alleviated by the improved
> orthogonal polynomial support that came in a little after the
> discussion. Polynomials by value are a useful tool;


This was my other leading candidate for "Next!"


> if you choose the
> right evaluation points they are competitive with Chebyshev
> polynomials for many purposes, and they can do other things as well.
> The spline basis would be nice, in that it would give people good
> tools for manipulating functions represented by splines, but the
> issues of numerical instability with degree raising and lowering
> suggest to me that they're not going to be that useful as a generic
> polynomial library.
>
> So I think my vote would be for polynomials by value. Not that I'm
> unbiased! I have a mostly-functional implementation:
> http://github.com/aarchiba/scikits.polynomial
> I can't vouch for its consistency with the current implementations, or
> its completeness; it's been a while since I worked on it.
>

Great, I'll give it a look-see! :-)

DG


>
> Anne
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