Krylov solvers for linear algebraic systems

Krylov Solvers

Charles George Broyden, Maria Teresa Vespucci - Collection Studies in computational mathematics 11

330 pages, parution le 27/09/2004

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Résumé

The first four chapters of this book give a comprehensive and unified theory of the Krylov methods. Many of these are shown to be particular examples of the block conjugate-gradient algorithm and it is this observation that permits the unification of the theory. The two major sub-classes of those methods, the Lanczos and the Hestenes-Stiefel, are developed in parallel as natural generalisations of the Orthodir (GCR) and Orthomin algorithms. These are themselves based on Arnoldi's algorithm and a generalised Gram-Schmidt algorithm and their properties, in particular their stability properties, are determined by the two matrices that define the block conjugate-gradient algorithm. These are the matrix of coefficients and the preconditioning matrix. In Chapter 5 the "transpose-free" algorithms based on the conjugate-gradient squared algorithm are presented while Chapter 6 examines the various ways in which the QMR technique has been exploited. Look-ahead methods and general block methods are dealt with in Chapters 7 and 8 while Chapter 9 is devoted to error analysis of two basic algorithms. In Chapter 10 the results of numerical testing of the more important algorithms in their basic forms (i.e. without look-ahead or preconditioning) are presented and these are related to the structure of the algorithms and the general theory. Graphs illustrating the performances of various algorithm/problem combinations are given via a CD-ROM. Chapter 11, by far the longest, gives a survey of preconditioning techniques. These range from the old idea of polynomial preconditioning via SOR and ILU preconditioning to methods like SpAI, AInv and the multigrid methods that were developed specifically for use with parallel computers. Chapter 12 is devoted to dual algorithms like Orthores and the reverse algorithms of Hegedus. Finally certain ancillary matters like reduction to Hessenberg form, Chebychev polynomials and the companion matrix are described in a series of appendices.

Sommaire

Introduction
The long recurrences
The short recurrences
The Krylov aspects
Transpose-free methods
More on QMR
Look-ahead methods
General block methods
And in practice
Preconditioning
Duality

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Caractéristiques techniques

	PAPIER
Éditeur(s)	Elsevier
Auteur(s)	Charles George Broyden, Maria Teresa Vespucci
Collection	Studies in computational mathematics 11
Parution	27/09/2004
Nb. de pages	330
Format	17 x 24,5
Couverture	Relié
Poids	783g
Intérieur	Noir et Blanc
EAN13	9780444514745
ISBN13	978-0-444-51474-5