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arXiv · 2205.06810

Global Convergence of Hessenberg Shifted QR II: Numerical Stability

Jess Banks·Jorge Garza-Vargas·Nikhil Srivastava
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Abstract

We develop a framework for proving rapid convergence of shifted QR algorithms which use Ritz values as shifts, in finite arithmetic. Our key contribution is a dichotomy result which addresses the known forward-instability issues surrounding the shifted QR iteration [Parlett and Le 1993]: we give a procedure which provably either computes a set of approximate Ritz values of a Hessenberg matrix with good forward stability properties, or leads to early decoupling of the matrix via a small number of QR steps. Using this framework, we show that the shifting strategy introduced in Part I of this series [Banks, Garza-Vargas, and Srivastava 2021] converges rapidly in finite arithmetic with a polylogarithmic bound on the number of bits of precision required, when invoked on matrices of controlled eigenvector condition number and minimum eigenvalue gap.

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Jess Banks, Jorge Garza-Vargas, Nikhil Srivastava. 2022-05-13. Global Convergence of Hessenberg Shifted QR II: Numerical Stability. https://arxiv.org/abs/2205.06810

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