[SciPy-user] Polynomial interpolation
Stéfan van der Walt
Sun Apr 20 12:18:40 CDT 2008
I'm surprised there haven't been more responses to this thread. I
would certainly like to see this code included. There's been some
other work on the fitpack2 module (we're still waiting for an update
on that front from John Travers).
We should also think of including Stineman-
and Lanczos/Sinc interpolation.
On 19/04/2008, Anne Archibald <firstname.lastname@example.org> wrote:
> It appears that scipy does not have a facility for using the Lagrange
> polynomial to interpolate data. (Or did I miss it?) I am well aware of
> the numerical difficulties this poses, but it (and its generalization,
> the Hermite polynomial) does prove useful on occasion. I have written
> prototype implementations of two algorithms for evaluating this
> polynomial, and I'd like comments before submitting them for inclusion
> in scipy.
> One implementation is based on Krogh 1970, "Efficient Algorithms for
> Polynomial Interpolation and Numerical Differentiation"; it allows the
> construction of Hermite polynomials (where some derivatives at each
> point may also be specified) and the evaluation of arbitrary
> derivatives. It is based on a Neville-like algorithm, so that it does
> O(n^2) work when constructed but only O(n) per point evaluated, or
> O(n^2) per point for which all derivatives must be evaluated. (n here
> is the degree of the polynomial.)
> The other implementation is based on Berrut and Trefethen 2004,
> "Barycentric Lagrange Interpolation". This implementation uses a
> rewriting of the Lagrange polynomial as a rational function, and is
> efficient and numerically stable. It also allows efficient updating of
> the y values at which interpolation occurs, as well as addition of new
> x values. It does not support evaluation of derivatives or
> construction of Hermite polynomials.
> Finally, I have written a "PiecewisePolynomial" class for constructing
> splines in which each piece may have arbitrary degree, and for which
> the function values and some derivatives are specified at each knot.
> The intent is that this be used to represent solutions obtained from
> ODE solvers, using one polynomial for each solver step, with the same
> order as the solver's internal polynomial solution, and with (some)
> derivatives matching at the ends. Such a Trajectory object would
> contain additional information about the solution (for example
> stiffness or error estimates) beyond what is in PiecewisePolynomial.
> (I tried implementing Trajectory using splines, which are evaluated in
> compiled code, but their maximum degree is 5 while the solvers will go
> up to degree 12.)
> They all need work, in particular, efficiency would be improved by
> making the y values vectors, error checking needs to be more robust,
> and documentation is not in reST form. Ultimately, too, the evaluation
> functions at least should be written in a compiled language (cython?).
> But I thought I'd solicit comments on the code first - is the
> object-oriented design cumbersome? Is including the algorithm in the
> name confusing to users? Is the calling convention for Hermite
> polynomials too confusing or error-prone?
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