Stable look-ahead versions of the Euclidean and Chebyshev algorithms

We first review the basic relations between the regular formal orthogonal polynomials (FOPs) for a sequence of moments (Markov parameters), the nonsingular leading principal submatrices of the moment matrix M (which is an infinite Hankel matrix), the distinct entries on the main diagonal of the Pade table for the symbol of M (which is the generating function or z-transform of the moments), the corresponding continued fraction (which is a J-fraction or a P-fraction), and the Euclidean algorithm for power series in ζ-1, which in the generic case is seen to reduce to the Chebyshev algorithm. The underlying recurrences are a special case of the general recurrences that are the basis of the Cabay-Meleshko algorithm which, in contrast to the aforementioned tools, is (weakly) stable. While, in the Toeplitz solver terminology, the Cabay-Meleshko algorithm is of Levinson type, we also outline the corresponding O(N 2) Schur-type algorithm and a related O(N log2 N) algorithm. Finally, we sketch three look-ahead strategies of which two are applicable to the O(N log2 N) algorithm also.

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