Loading [MathJax]/jax/output/HTML-CSS/config.js
Go to previous page

Neutrally Stable Fixed Points of the QR Algorithm

Neutrally Stable Fixed Points of the QR Algorithm

Year:    2004

Author:    D. M. Day, A. D. Hwang

International Journal of Numerical Analysis and Modeling, Vol. 1 (2004), Iss. 2 : pp. 147–156

Abstract

Practical QR algorithm for the real unsymmetric algebraic eigenvalue problem is considered. The global convergence of shifted QR algorithm in finite precision arithmetic is addressed based on a model of the dynamics of QR algorithm in a neighborhood of an unreduced Hessenberg fixed point. The QR algorithm fails at a "stable" unreduced fixed point. Prior analyses have either determined some unstable unreduced Hessenberg fixed points or have addressed stability to perturbations of the reduced Hessenberg fixed points. The model states that sufficient criteria for stability (e.g. failure) in finite precision arithmetic are that a fixed point be neutrally stable both with respect to perturbations that are constrained to the orthogonal similarity class and to general perturbations from the full matrix space. The theoretical analysis presented herein shows that at an arbitrary unreduced fixed point "most" of the eigenvalues of the Jacobian(s) are of unit modulus. A framework for the analysis of special cases is developed that also sheds some light on the robustness of the QR algorithm.

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/2004-IJNAM-971

International Journal of Numerical Analysis and Modeling, Vol. 1 (2004), Iss. 2 : pp. 147–156

Published online:    2004-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    10

Keywords:    Unsymmetric eigenvalue problem QR algorithm unreduced fixed point.

Author Details

D. M. Day Email

A. D. Hwang Email