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The Lanczos and Conjugate Gradient Algorithms: From Theory to Finite Precision ComputationsThe Lanczos and Conjugate Gradient Algorithms: From Theory to Finite Precision Computations

Gérard Meurant


Software, Environments, and Tools 19

"No present book comes near this one in the range and depth of treatment of these two extremely important methods—the Lanczos algorithm and the method of conjugate gradients." —Chris Paige, School of Computer Science, McGill University


The Lanczos and conjugate gradient (CG) algorithms are fascinating numerical algorithms. This book presents the most comprehensive discussion to date of the use of these methods for computing eigenvalues and solving linear systems in both exact and floating point arithmetic. The author synthesizes the research done over the past 30 years, describing and explaining the "average" behavior of these methods and providing new insight into their properties in finite precision. Many examples are given that show significant results obtained by researchers in the field.

The author emphasizes how both algorithms can be used efficiently in finite precision arithmetic, regardless of the growth of rounding errors that occurs. He details the mathematical properties of both algorithms and demonstrates how the CG algorithm is derived from the Lanczos algorithm. Loss of orthogonality involved with using the Lanczos algorithm, ways to improve the maximum attainable accuracy of CG computations, and what modifications need to be made when the CG method is used with a preconditioner are addressed.

Audience This book is intended for applied mathematicians, computational scientists, engineers, and physicists who have an interest in linear algebra, numerical analysis, and partial differential equations. It will be of interest to engineers and scientists using the Lanczos algorithm to compute eigenvalues and the CG algorithm to solve linear systems, and to researchers in Krylov subspace methods for symmetric matrices, especially those concerned with floating point error analysis. Moreover, it can be used in advanced courses on iterative methods or as a comprehensive presentation of a well-known numerical method in finite precision arithmetic.

About the Author Gérard Meurant is Director of Research in the military applications division at Commissariat à l'Énergie Atomique (CEA) in Bruyères le Châtel, France. He is the author of Computer Solution of Large Linear Systems (North–Holland, 1999) and serves on the editorial boards of the International Journal of High Speed Computing and Numerical Algorithms. In 1988 Meurant was awarded the Prix CEA and in 1995 the Palmes Académiques, an honor presented each year by the French Ministry of Education.

Contents
Preface
Chapter 1: The Lanczos algorithm in exact arithmetic
Chapter 2: The CG algorithm in exact arithmetic
Chapter 3: A historical perspective on the Lanczos algorithm in finite precision
Chapter 4: The Lanczos algorithm in finite precision
Chapter 5: The CG algorithm in finite precision
Chapter 6: The maximum attainable accuracy
Chapter 7: Estimates of norms of the error in finite precision
Chapter 8: The preconditioned CG algorithm
Chapter 9: Miscellaneous
Appendix
Bibliography
Index

2006 / xvi + 365 pages / Softcover
ISBN-13: 978-0-898716-16-0 / ISBN-10: 0-89871-616-0
List Price $116.00 / SIAM Member Price $81.20 / Order Code SE19
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