Core-chasing algorithms for the eigenvalue problem

by Aurentz, Jared L.

Type Book

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Title Statements

Full title :Core-chasing algorithms for the eigenvalue problem / Jared L. Aurentz, Thomas Mach, Leonardo Robol, Raf Vandebril, David S. Watkins.

Summary

"For the past 50 years the workhorse algorithm for solving eigenvalue problems has been Francis's implictly shifted QR algorithm. We present here a new formulation of Francis's algorithm that operates on the QR factors of a matrix A rather than A itself. This is not in any existing book. A popular way to compute the roots of a polynomial is to form the companion matrix and compute its eigenvalues (which are exactly the roots of the polynomial). This is what Matlab's roots command does. Our new formulation leads to a better algorithm for solving the companion eigenvalue problem (which is unitary-plus-rank-one). Our algorithm is faster than all of the competing algorithms. We can prove it is backward stable, and it is stable in a stronger sense than the algorithm that Matlab uses. Thus our algorithm is faster and more accurate than Matlab's. In the final chapter we present a generalization of Francis's algorithm that we published (in a SIAM journal) a few years ago. This applies to all classes of eigenvalue problems, structured or not. This is not in any book. Our monograph presents a unified treatment of several classes of eigenvalue problems. We listed them above, but here they are again: unitary, unitary-plus-rank-one (companion matrices and pencils), symmetric, symmetic-plus-rank-one, matrix polynomials. We also provide new insights into matrix eigenvalue problems with no special structure. These are the results of our research of the past few years. This is hot off the press. Most of this material has appeared in our publications, but none of it is in any book. In recent conference presentations we have mentioned that we are working on a book. Informal feedback suggests there is significant interest in this book among researchers in numerical linear algebra"--

Edition Statements

Publication Date:2018

Publication Statements

Publisher(s):Society for Industrial and Applied Mathematics
Place of Publication:Philadelphia
Language:eng
Publication Date:2018

The Physical Object

Description:ix, 149 pages ; : 26 cm.

Edition Identifiers

ISBN:9781611975338, 1611975336
oclc:on1050141002
LC Classification:QA193
Dewey Decimal:512.9/436

Core transformations -- Francis's algorithm -- Francis's algorithm as a core-chasing algorithm -- Some special structures -- Generalized and matrix polynomial eigenvalue problems -- Beyond upper Hessenberg form.