Polynomial and Rational Evaluation and Interpolation (with Structured Matrices)

Polynomial and rational interpolation and multipoint evaluation are classical subjects, which remain central for the theory and practice of algebraic and numerical computing and have extensive applications to sciences, engineering and signal and image processing. In spite of long and intensive study of these subjects, several major problems remained open. We achieve substantial progress, based on our new matrix representations of the problems with the use of node polynomials. In particular: 1. We show strong correlation between rational and polynomial problems as well as between evaluation and interpolation, which enables our unified algorithmic treatment of all these subjects. 2. In applications of real polynomial evaluation and interpolation to sciences, engineering, and signal and image processing, most important is the case where input/output is represented in Chebyshev bases. In this case we rely on fast cosine/sine transforms (FCT/FST) to decrease the arithmetic cost of the known solutions from order of n to O(log2 n) per node point. 3. In the general complex case, we devise new effective approximation algorithms for polynomial and rational evaluation and interpolation, for all input polynomials of degree n - 1 and all sets of n nodes. The algorithms support the arithmetic complexity bounds of O(log n) per node point for approximate solution of these classical problems within the output error bound Ɛ = 2-b, log b = O(log n), taken relative to the specified input parameters. This substantially improved the known estimate of O(log2 n) per point. Our algorithms supporting the cited complexity bounds allow their NC and work optimal parallelization. Our results also include new exact solution algorithms with arithmetic cost O(log2 n) per node point for a) Trummer's problem of rational evaluation, which (unlike Gerasoulis algorithm) is interpolation-free, and b) rational interpolation with unknown poles, which exploits the matrix structure instead of the customary reduction to Euclidean algorithm. Technically, we exploit correlation among various problems of polynomial and rational interpolation, their matrix representations and transformations (mappings) into each other of both problems and the associated structured matrices.

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