Stable Look-Ahead Versions of the Euclidean and Chebyshev Algorithms

  • William B. Gragg
  • Martin H. Gutknecht
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 119)


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 Padé 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.


Continue Fraction Hankel Matrix Moment Matrix Euclidean Algorithm Pade Approximants 
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Copyright information

© Birkhäuser 1994

Authors and Affiliations

  • William B. Gragg
    • 1
  • Martin H. Gutknecht
    • 2
  1. 1.Department of MathematicsNaval Postgraduate SchoolMontereyUSA
  2. 2.Interdisciplinary Project Center for Supercomputing, ETH Zurich, ETH-ZentrumZurichSwitzerland

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