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Combinatorics, Linear Algebra and Number Theory (CLaN) Seminar

Department of Mathematics and Statistics

WEBS 11

September 9, Monday, 4:10 - 5:00 PM

David Watkins

Department of Mathematics and Statistics

Washington State University

Combinatorics, Linear Algebra and Number Theory (CLaN) Seminar

Department of Mathematics and Statistics

WEBS 11

September 9, Monday, 4:10 - 5:00 PM

David Watkins

Department of Mathematics and Statistics

Washington State University

Title: Solving
the Matrix Eigenvalue Problem: chasing bulges,
cores, or pole

Abstract: Sixty years ago John Francis invented the winning general-purpose procedure for computing eigenvalues of a matrix, the implicitly-shifted QR algorithm. Variants of this algorithm can also be applied to related problems, including the generalized eigenvalue problem $Ax = \lambda Bx$. One might think that after so many years, everything that can be said on this topic has already been said, but this turns out not to be the case. A recent burst of activity has resulted in interesting new variants of and insights into Francis's algorithm. It is normally implemented as a bulge-chasing algorithm, but recently we have shown that there are some advantages to implementing it by core chasing instead. Another new and interesting variant, which is most naturally applied to the generalized eigenvalue problem, is the pole-swapping algorithm of Camps, Meerbergen, and Vandebril. We will briefly describe, compare, and contrast the bulge-chasing, core-chasing, and pole-swapping approaches to solving the eigenvalue problem.

Much of the material of this talk is from joint work with others, including Jared Aurentz, Daan Camps, Thomas Mach, Leonardo Robol, and Raf Vandebril.

Abstract: Sixty years ago John Francis invented the winning general-purpose procedure for computing eigenvalues of a matrix, the implicitly-shifted QR algorithm. Variants of this algorithm can also be applied to related problems, including the generalized eigenvalue problem $Ax = \lambda Bx$. One might think that after so many years, everything that can be said on this topic has already been said, but this turns out not to be the case. A recent burst of activity has resulted in interesting new variants of and insights into Francis's algorithm. It is normally implemented as a bulge-chasing algorithm, but recently we have shown that there are some advantages to implementing it by core chasing instead. Another new and interesting variant, which is most naturally applied to the generalized eigenvalue problem, is the pole-swapping algorithm of Camps, Meerbergen, and Vandebril. We will briefly describe, compare, and contrast the bulge-chasing, core-chasing, and pole-swapping approaches to solving the eigenvalue problem.

Much of the material of this talk is from joint work with others, including Jared Aurentz, Daan Camps, Thomas Mach, Leonardo Robol, and Raf Vandebril.