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Special Sessions in Applications of Computer Algebra

ACA 2015: Applications of Computer Algebra pp 341–383Cite as

A Fast Schur–Euclid-Type Algorithm for Quasiseparable Polynomials

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 198))

Abstract

In this paper, a fast \(\mathscr {O}(n^2)\) algorithm is presented for computing recursive triangular factorization of a Bezoutian matrix associated with quasiseparable polynomials via a displacement equation. The new algorithm applies to a fairly general class of quasiseparable polynomials that includes real orthogonal, Szegö polynomials, and several other important classes of polynomials, e.g., those defined by banded Hessenberg matrices. While the algorithm can be seen as a Schur-type for the Bezoutian matrix it can also be seen as a Euclid-type for quasiseparable polynomials via factorization of a displacement equation. The process, i.e., fast Euclid-type algorithm for quasiseparable polynomials or Schur-type algorithm for Bezoutian associated with quasiseparable polynomials, is carried out with the help of a displacement equation satisfied by Bezoutian and Confederate matrices leading to \(\mathscr {O}(n^2)\) complexity.

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Author information

Authors and Affiliations

  1. Embry-Riddle Aeronautical University, Daytona Beach, USA

    Sirani M. Perera

  2. University of Connecticut, Storrs, USA

    Vadim Olshevsky

Authors
  1. Sirani M. Perera
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  2. Vadim Olshevsky
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Correspondence to Sirani M. Perera .

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Editors and Affiliations

  1. Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Ilias S. Kotsireas

  2. Institute of Mathematics, University of Valladolid, Valladolid, Spain

    Edgar Martínez-Moro

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Perera, S.M., Olshevsky, V. (2017). A Fast Schur–Euclid-Type Algorithm for Quasiseparable Polynomials. In: Kotsireas, I., Martínez-Moro, E. (eds) Applications of Computer Algebra. ACA 2015. Springer Proceedings in Mathematics & Statistics, vol 198. Springer, Cham. https://doi.org/10.1007/978-3-319-56932-1_24

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