@Article{IJNAM-1-2,
author = {M., Day, D. and Hwang, D., A.},
title = {Neutrally Stable Fixed Points of the QR Algorithm},
journal = {International Journal of Numerical Analysis and Modeling},
year = {2004},
volume = {1},
number = {2},
pages = {147--156},
abstract = {<p style="text-align: justify;">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 &quot;stable&quot; 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 &quot;most&quot; 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.</p>},
issn = {2617-8710},
doi = {https://doi.org/2004-IJNAM-971},
url = {https://global-sci.com/article/83898/neutrally-stable-fixed-points-of-the-qr-algorithm}
}