TY - JOUR
T1 - Neutrally Stable Fixed Points of the QR Algorithm
AU - D. M. Day & A. D. Huang
JO - International Journal of Numerical Analysis and Modeling
VL - 2
SP - 147
EP - 156
PY - 2004
DA - 2004/01
SN - 1
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/971.html
KW - Unsymmetric eigenvalue problem
KW - QR algorithm
KW - unreduced fixed point
AB - 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.