David S. Watkins - Böcker

This new, modernized edition provides a clear and thorough introduction to matrix computations,a key component of scientific computing Retaining the accessible and hands-on style of its predecessor, Fundamentals of Matrix Computations, Third Edition thoroughly details matrix computations and the accompanying theory alongside the author's useful insights. The book presents the most important algorithms of numerical linear algebra and helps readers to understand how the algorithms are developed and why they work.Along with new and updated examples, the Third Edition features: A novel approach to Francis' QR algorithm that explains its properties without reference to the basic QR algorithmApplication of classical Gram-Schmidt with reorthogonalizationA revised approach to the derivation of the Golub-Reinsch SVD algorithmNew coverage on solving product eigenvalue problemsExpanded treatment of the Jacobi-Davidson methodA new discussion on stopping criteria for iterative methods for solving linear equationsThroughout the book, numerous new and updated exercises—ranging from routine computations and verifications to challenging programming and proofs—are provided, allowing readers to immediately engage in applying the presented concepts. The new edition also incorporates MATLAB to solve real-world problems in electrical circuits, mass-spring systems, and simple partial differential equations, and an index of MATLAB terms assists readers with understanding the basic concepts related to the software.Fundamentals of Matrix Computations, Third Edition is an excellent book for courses on matrix computations and applied numerical linear algebra at the upper-undergraduate and graduate level. The book is also a valuable resource for researchers and practitioners working in the fields of engineering and computer science who need to know how to solve problems involving matrix computations.

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.

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.