Thomas Mach

Matrix eigenvalue problems arise in a wide variety of fields in science and engineering, so it is important to have reliable and efficient methods for solving them. Of the methods devised, bulge-chasing algorithms, such as the famous QR and QZ algorithms, are the most important. This book focuses on pole-swapping algorithms, a new class of methods that are generalizations of bulge-chasing algorithms and a bit faster and more accurate owing to their inherent flexibility. The pole-swapping theory developed by the authors sheds light on the functioning of the whole class of algorithms, including QR and QZ.The only book on the topic, Pole-Swapping Algorithms for the Eigenvalue Problem describes the state of the art on eigenvalue methods and provides an improved understanding and explanation of why these important algorithms work.AudienceThis book is for researchers and students in the field of matrix computations, software developers, and anyone in academia or industry who needs to understand how to solve eigenvalue problems, which are ubiquitous in science and engineering.

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