Eigenvalue computations are ubiquitous in science and engineering. John Francis’s implicitly shifted QR algorithm has been the method of choice for small to medium sized eigenvalue problems since its invention in 1959. This book presents a new view of this classical algorithm. While Francis’s original procedure chases bulges, the new version chases core transformations, which allows the development of fast algorithms for eigenvalue problems with a variety of special structures. This also leads to a fast and backward stable algorithm for computing the roots of a polynomial by solving the companion matrix eigenvalue problem. The authors received a SIAM Outstanding Paper prize for this work.

Audience
This book will be of interest to researchers in numerical linear algebra and their students.

About the Author

Jared L. Aurentz is a Severo Ochoa postdoctoral fellow at the Institute of Mathematical Sciences in Madrid and was previously a postdoctoral researcher in the numerical analysis group at the Mathematical Institute, University of Oxford. His research interests include developing fast algorithms for sparse and rank-structured matrices and algorithms at the intersection of linear algebra and approximation theory.

Thomas Mach is an assistant professor at Nazarbayev University, Kazakhstan. He was one of the LAA Early Career Speakers at the 21st Conference of the International Linear Algebra Society in 2017. The author of more than 20 publications, his research interest is structure-preserving algorithms, especially for eigenvalue problems.

Leonardo Robol is a researcher at the Institute of Information Science and Technologies (ISTI) of the National Research Council in Italy (CNR). He was the recipient of the honorable mention for the Householder Prize award in 2017. His research interests include the use of rank structures in matrix computations, particularly eigenvalue problems and matrix equations.

Raf Vandebril is a professor at KU Leuven. His thesis led to two books on semiseparable matrices (eigenvalue problems and system solving) co-authored with Marc Van Barel and Nicola Mastronardi. His primary research interests are eigenvalue computations and structured matrices.

David S. Watkins is professor emeritus of mathematics at Washington State University. He is the author of three books on matrix computations and numerous scientific publications, including several articles in SIAM Review.

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