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. Pole-Swapping Algorithms for the Eigenvalue Problem is the only book on the topic. It describes the state of the art on eigenvalue methods and provides an improved understanding and explanation of why these important algorithms work. This 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.