Krylov subspace methods : principles and analysis
 Author/Creator
 Liesen, Jörg.
 Language
 English.
 Edition
 1st ed.
 Imprint
 Oxford, U.K. : Oxford University Press, 2013.
 Physical description
 xv, 391 p. : ill. ; 24 cm.
 Series
 Numerical mathematics and scientific computation.
Access
Available online

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QA188 .L547 2013

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QA188 .L547 2013
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Contributors
 Contributor
 Strakoš, Zdeněk.
Contents/Summary
 Bibliography
 Includes bibliographical references (p. [349]384) and index.
 Contents

 1. Introduction  2. Krylov subspace methods  3. Matching moments and model reduction view  4. Short recurrences for generating orthogonal Krylov subspace bases  5. Cost of computations using Krylov subspace methods.
 (source: Nielsen Book Data)
 Publisher's Summary
 The mathematical theory of Krylov subspace methods with a focus on solving systems of linear algebraic equations is given a detailed treatment in this principlesbased book. Starting from the idea of projections, Krylov subspace methods are characterised by their orthogonality and minimisation properties. Projections onto highly nonlinear Krylov subspaces can be linked with the underlying problem of moments, and therefore Krylov subspace methods can be viewed as matching moments model reduction. This allows enlightening reformulations of questions from matrix computations into the language of orthogonal polynomials, GaussChristoffel quadrature, continued fractions, and, more generally, of Vorobyev's method of moments. Using the concept of cyclic invariant subspaces, conditions are studied that allow the generation of orthogonal Krylov subspace bases via short recurrences. The results motivate the important practical distinction between Hermitian and nonHermitian problems. Finally, the book thoroughly addresses the computational cost while using Krylov subspace methods. The investigation includes effects of finite precision arithmetic and focuses on the method of conjugate gradients (CG) and generalised minimal residuals (GMRES) as major examples. There is an emphasis on the way algebraic computations must always be considered in the context of solving realworld problems, where the mathematical modelling, discretisation and computation cannot be separated from each other. The book also underlines the importance of the historical context and demonstrates that knowledge of early developments can play an important role in understanding and resolving very recent computational problems. Many extensive historical notes are included as an inherent part of the text as well as the formulation of some omitted issues and challenges which need to be addressed in future work. This book is applicable to a wide variety of graduate courses on Krylov subspace methods and related subjects, as well as benefiting those interested in the history of mathematics.
(source: Nielsen Book Data)
Subjects
 Subject
 Sparse matrices.
Bibliographic information
 Publication date
 2013
 Responsibility
 Jörg Liesen, Zdeněk Strakoš.
 Series
 Numerical mathematics and scientific computation
 ISBN
 9780199655410
 0199655413