Fast inversion of polynomial-vandermonde matrices for polynomial systems related to order one quasiseparable matrices

T. Bella, Y. Eidelman, I. GohbergZ, V. Olshevsky, E. Tyrtyshnikov

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

2 Scopus citations

Abstract

While Gaussian elimination is well known to require O(n 3) operations to invert an arbitrary matrix, Vandermonde matrices may be inverted using O(n 2) operations by a method of Traub [24]. While this original version of the Traub algorithm was noticed to be unstable, it was shown in [12] that with a minor modification, the Traub algorithm can typically yield a very high accuracy. This approach has been extended from classical Vandermonde matrices to polynomial-Vandermonde matrices involving real orthogonal polynomials [3], [10], and Szegő polynomials [19]. In this paper we present an algorithm for inversion of a class of polynomial-Vandermonde matrices with special structure related to order one quasiseparable matrices, generalizing monomials, real orthogonal polynomials, and Szegő polynomials. We derive a fast O(n 2) inversion algorithm applicable in this general setting, and explore its reduction in the previous special cases. Some very preliminary numerical experiments are presented, demonstrating that, as observed by our colleagues in previous work, good forward accuracy is possible in some circumstances, which is consistent with previous work of this type.

Original languageEnglish
Title of host publicationOperator Theory
Subtitle of host publicationAdvances and Applications
PublisherSpringer International Publishing
Pages79-106
Number of pages28
DOIs
StatePublished - 2013

Publication series

NameOperator Theory: Advances and Applications
Volume237
ISSN (Print)0255-0156
ISSN (Electronic)2296-4878

Keywords

  • Inversion of vandermonde matrices
  • Polynomial vandermonde matrices
  • Quasiseparable matrices

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