Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 1 Direct Solution Methods
- 2 Theory of Matrix Eigenvalues
- 3 Positive Definite Matrices, Schur Complements, and Generalized Eigenvalue Problems
- 4 Reducible and Irreducible Matrices and the Perron-Frobenius Theory for Nonnegative Matrices
- 5 Basic Iterative Methods and Their Rates of Convergence
- 6 M-Matrices, Convergent Splittings, and the SOR Method
- 7 Incomplete Factorization Preconditioning Methods
- 8 Approximate Matrix Inverses and Corresponding Preconditioning Methods
- 9 Block Diagonal and Schur Complement Preconditionings
- 10 Estimates of Eigenvalues and Condition Numbers for Preconditioned Matrices
- 11 Conjugate Gradient and Lanczos-Type Methods
- 12 Generalized Conjugate Gradient Methods
- 13 The Rate of Convergence of the Conjugate Gradient Method
- Appendices
- Index
5 - Basic Iterative Methods and Their Rates of Convergence
Published online by Cambridge University Press: 05 August 2012
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 1 Direct Solution Methods
- 2 Theory of Matrix Eigenvalues
- 3 Positive Definite Matrices, Schur Complements, and Generalized Eigenvalue Problems
- 4 Reducible and Irreducible Matrices and the Perron-Frobenius Theory for Nonnegative Matrices
- 5 Basic Iterative Methods and Their Rates of Convergence
- 6 M-Matrices, Convergent Splittings, and the SOR Method
- 7 Incomplete Factorization Preconditioning Methods
- 8 Approximate Matrix Inverses and Corresponding Preconditioning Methods
- 9 Block Diagonal and Schur Complement Preconditionings
- 10 Estimates of Eigenvalues and Condition Numbers for Preconditioned Matrices
- 11 Conjugate Gradient and Lanczos-Type Methods
- 12 Generalized Conjugate Gradient Methods
- 13 The Rate of Convergence of the Conjugate Gradient Method
- Appendices
- Index
Summary
Let us now consider various iterative methods for the numerical computation of the solution of a linear system of equations. Iterative methods for solving linear systems (originally by Gauss in 1823, Liouville in 1837, and Jacobi in 1845) embody an approach quite different from that behind direct methods such as Gaussian elimination (see Chapter 1). In 1823 Gauss wrote, “Fast jeden Abend mache ich eine neue Auflage des Tableau, wo immer leicht nachzuhelfen ist. Bei der Einförmigkeit des Messungsgeschäfts gibt dies immer eine angenehme Unterhaltung; man sieht daran auch immer gleich, ob etwas Zweifelhaftes eingeschlichen ist, was noch wünschenswert bleibt usw. Ich empfehle Ihnen diesen Modus zur Nachahmung. Schwerlich werden Sie je wieder direct eliminieren, wenigstens nicht, wenn Sie mehr als zwei Unbekannte haben. Das indirecte Verfahren läßt sich halb im Schlafe ausführen oder man kann während desselben an andere Dingen denken.” (Freely translated, ”I recommend this modus operandi. You will hardly eliminate directly anymore, at least not when you have more than two unknowns. The indirect method can be pursued while half asleep or while thinking about other things.”)
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- Iterative Solution Methods , pp. 158 - 199Publisher: Cambridge University PressPrint publication year: 1994
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