The Lanczos and Conjugate Gradient Algorithms. From Theory to Finite Precision Computations
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Author(s): G. Meurant
Publisher: SIAM
ISBN: 0898716160
Format: softback
365pp
Price: $107.00
Review Date: 03 October 2006
Review: The Lanczos algorithm is one of the most frequently used numerical methods for computing a few eigenvalues (and eventually eigenvectors) of a large sparse symmetric matrix, A. The Lanczos method is closely linked to the conjugate gradient (CG) method for solving linear systems Ax = b. Currently, CG is the method used for solving large sparse systems with a positive definite symmetric matrix A. This book presents some established, as well as a few new, results about the Lanczos and CG algorithms both in exact and in precision arithmetics. There are chapters on: the Lanczos and CG algorithms in exact arithmetic; a historical perspective on the Lanczos algorithm in finite precision; the Lanczos and CG algorithms in exact finite precision; the maximum attainable accuracy; estimates of norms of the error in finite precision; and the preconditioned CG algorithm.