Rational QR-iteration without inversion

Abstract

In this manuscript a new method will be presented for performing a QR-iteration with (A − σI)(A − κI) −1 = QR without explicit inversion of the factor (A − κI) −1 . A QR-method driven by a rational function is attractive since convergence can occur at both sides of the matrix.Each step of this new iteration consists of two substeps. In the explicit version, first an RQ-factorization of the initial matrix A − κI = RQ will be computed, followed by a QR-factorization of the matrix (A − σI)Q H . The factorization of (A − σI)Q H can be computed in an intelligent manner, exploiting properties of the already known RQ-factorization of A − κI. The similarity transformation yielding the QR-step is defined by the unitary factor Q in the QR-factorization of the transformed matrix (A − σI)Q H . Examples will be given, illustrating how to efficiently compute the factorization for some specific classes of matrices. The novelties of this approach with respect to these matrix classes will be discussed. Article information• Vandebril, Raf; Van Barel, Marc; Mastronardi, Nicola. Rational QR-iteration without inversion, Numerische Mathematik, volume 110, issue 4, pages 561-575, 2008.• The content of this article is identical to the content of the published paper, but without the final typesetting by the publisher.• Journal's homepage: http://link.springer.com/journal/211• Abstract In this manuscript a new method will be presented for performing a QR-iteration with (A − σI)(A − κI) −1 = QR without explicit inversion of the factor (A − κI) −1 . A QR-method driven by a rational function is attractive since convergence can occur at both sides of the matrix. Each step of this new iteration consists of two substeps. In the explicit version, first an RQfactorization of the initial matrix A − κI = RQ will be computed, followed by a QR-factorization of the matrix (A − σI)Q H . The factorization of (A − σI)Q H can be computed in an intelligent manner, exploiting properties of the already known RQ-factorization of A − κI. The similarity transformation yielding the QR-step is defined by the unitary factor Q in the QR-factorization of the transformed matrix (A − σI)Q H .Examples will be given, illustrating how to efficiently compute the factorization for some specific classes of matrices. The novelties of this approach with respect to these matrix classes will be discussed.

Chasing Bulges or Rotations? A Metamorphosis of the QR-Algorithm

“…Rewriting the formulas from (6.5) gives usR =R and thus A − μI = IQ aR − μI = R r − μQ H a Q a = QRQ a . The new iterate corresponds to a step of the QH-algorithm [30,31] or the rational driven QR-iteration on the matrix A.…”

Algorithms for the Geronimus transformation for orthogonal polynomials on the unit circle

“…This strategy is not new (see e.g. [34,35]), but it will give a groundwork for the following sections.…”

A numerical example showing that deflations based on rotation leads to higher relative accuracy in QR algorithm

“…In a recent series of papers [1][2][3] Vandebril and Watkins (and related with Aurentz in [4,5]) generalized Francis's implicitly shifted QR algorithm [6,7]. In implicitly shifted QR algorithm the Hessenberg matrix plays the key role, on which the implicit QR steps are performed.…”