Implicit QR for Companion-like Pencils
Paola Boito, Yuli Eidelman, Luca Gemignani

TL;DR
This paper introduces a fast implicit QR algorithm tailored for eigenvalue problems involving low rank corrections of unitary matrices, specifically applied to polynomial zero-finding, achieving efficiency and stability.
Contribution
It presents a novel modified QZ algorithm that computes eigenvalues of structured matrix pencils with reduced computational complexity and memory usage.
Findings
Algorithm computes eigenvalues in O(N^2) operations
Numerical experiments confirm effectiveness and stability
Method is suitable for polynomial zero-finding problems
Abstract
A fast implicit QR algorithm for eigenvalue computation of low rank corrections of unitary matrices is adjusted to work with matrix pencils arising from polynomial zerofinding problems . The modified QZ algorithm computes the generalized eigenvalues of certain NxN rank structured matrix pencils using O(N^2) ops and O(N) memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.
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Taxonomy
TopicsMatrix Theory and Algorithms · Electromagnetic Scattering and Analysis · Electromagnetic Simulation and Numerical Methods
