2011/9/17 Stéfan van der Walt <[email protected]> > On Sat, Sep 17, 2011 at 9:40 AM, Charles R Harris > <[email protected]> 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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