[Numpy-discussion] Polynomial implementation in numpy.ma
Charles R Harris
charlesr.harris at gmail.com
Sat Sep 17 15:39:10 EDT 2011
2011/9/17 Stéfan van der Walt <stefan at sun.ac.za>
> On Sat, Sep 17, 2011 at 9:40 AM, Charles R Harris
> <charlesr.harris at gmail.com> wrote:
> > The Vandermonde matrix needs to be used for the fitting so nothing should
> be
> > changed there.
>
> I remember now where I heard this:
>
> http://people.maths.ox.ac.uk/trefethen/mythstalk.pdf
>
> Specifically, the statement: "Some well-known algorithms take O(n^2)
> operations (e.g. naïve Lagrange formula)
> or are exponentially unstable (e.g. Vandermonde matrices based on
> monomials x^k )."
>
> I wasn't sure whether to interpret that as the Vandermonde-based
> algorithm being flawed, or the polynomial fitting problem on an
> equidistant grid being ill-conditioned in general.
>
>
Equispaced grid points are bad for power series fits -- the Chebyshev points
are better -- but power series also tend to have problems even when
Chebyshev points are used since they never become particularly orthogonal
over the sample points. Alternatives like Bernstein polynomials, Legendre
polynomials, and Chebyshev polynomials are much better in that regard but
not generally as easy to hand as power series, which one reason I wanted to
make available in numpy. However, doing a fit using Chebyshev polynomials
and then converting it to a power series throws away the Chebyshev advantage
as the Chebyshev series is much less sensitive to numerical errors in the
coefficients.
Chuck
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