Much recent research has concentrated on the efficient solution of large sparse or structured linear systems using iterative methods. A language loaded with acronyms for a thousand different algorithms has developed, and it is often difficult even for specialists to identify the basic principles involved. Here is a book that focuses on the analysis of iterative methods. The author includes the most useful algorithms from a practical point of view and discusses the mathematical principles behind their derivation and analysis. Several questions are emphasized throughout: Does the method converge? If so, how fast? Is it optimal, among a certain class? If not, can it be shown to be near-optimal? The answers are presented clearly, when they are known, and remaining important open questions are laid out for further study.
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Here is a book that focuses on the analysis of iterative methods for solving linear systems. The author includes the most useful algorithms from a practical point of view and discusses the mathematical principles behind their derivation and analysis.Review:
'This graduate-level textbook gives equal weights to iterative methods and preconditioning (including domain decomposition and multigrid), and it approaches Krylov space methods from a somewhat different angle. It also treats some subjects that appear for the first time in a textbook, like new results on roundoff effects in the Lanczos and conjugate gradient algorithms. This well-done introduction to the area can be strongly recommended. It is competently written by an author who has contributed much to the complete reshaping of this field in the last twenty years.' Martin H. Gutknecht, ETH Zurich
'For a course in matrix iterations, this is just the right book. It is wide-ranging, careful about details, and appealingly written - a major addition to the literature in this important area.' Nick Trefethen, Professor of Numerical Analysis, Oxford University
'This book differs substantially from other books on iterative methods, including those recently published, in that it concentrates on several principles behind the derivation and analysis of the most important methods and preconditioning techniques. Individual algorithms serve as examples illustrating the discussed ideas. Strong emphasis is given to motivation and its relation to problems in other areas of mathematics. The book speaks in clear language about principal problems in the area of iterative methods. It represents a comprehensive introduction to the field and stimulates the interest of the reader. It is valuable for students and also for experts working in the area of iterative methods.' Zdenek Strakos, Professor, Czech Academy of Sciences, Institute of Computer Science
'Anne Greenbaum is an admired authority in the field of iterative methods. Engineers and scientists often ask me about the puzzling behavior of iterative methods, which I almost always answer with a reference to Anne's work, now made easy to point to in her new book.' Paul Saylor, Department of Computer Science, University of Illinois, Urbana-Champaign
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