Afast bjorck-pereyra-type algorithm for solving essenberg-quasiseparable- vandermonde systems

T. Bella*, Y. Eidelman, I. Gohberg, I. Koltracht, V. Olshevskyl

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Afast O(n2) algorithm is derived for solving linear systems where the coefficient matrix is a polynomial-Vandermonde matrix Vr(x) = [rj1(xi)] with polynomials {rk(x)} defined by a Hessenberg matrix with quasiseparable structure. The result generalizes the well-known Bjorck- ereyra algorithm for classical Vandermonde systems involving monomials. It also generalizes the algorithms of Reichel-Opfer for Vr(x) involving Chebyshev polynomials of Higham for Vr(x) involving real orthogonal polynomials, and a recent algorithm of the authors for Vr (x) involving Szegö polynomials. The new algorithm applies to a fairly general new class of (H, k)-quasiseparable polynomials (Hessenberg, order k quasiseparable) that includes (along with the above mentioned classes of real orthogonal and Szegö polynomials) several other important classes of polynomials, e.g., defined by banded Hessenberg matrices. Numerical experiments are presented that coincide with previous experiences with Björck-Pereyra- type algorithms giving better forward error than Gaussian elimination, and this accuracy is consistent with the so-called Chan-Foulser conditioning of the system.

Original languageEnglish
Pages (from-to)790-815
Number of pages26
JournalSIAM Journal on Matrix Analysis and Applications
Issue number2
StatePublished - 2009


  • Bjorck-pereyra algorithm
  • Quasiseparable matrices
  • Vandermonde matrices


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