The Lanczos and Conjugate Gradient Algorithms: From Theory to Finite Precision Computations (Software, Environments and Tools)

✍ Scribed by Gérard Meurant

• Publisher: SIAM
• Year: 2006
• Tongue: English
• Leaves: 382
• Series: Software, Environments and Tools
• Category: Library

✦ Synopsis

"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. 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. 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.