Iterative Solution Methods (Paper)

This book deals primarily with the numerical solution of linear systems of equations by iterative methods. The first part of the book is intended to serve as a textbook for a numerical linear algebra course. The material assumes the reader has a basic knowledge of linear algebra, such as set theory and matrix algebra, however it is demanding for students who are not afraid of theory. To assist the reader, the more difficult passages have been marked, the definitions for each chapter are collected at the beginning of the chapter, and numerous exercises are included throughout the text. The second part of the book serves as a monograph introducing recent results in the iterative solution of linear systems, mainly using preconditioned conjugate gradient methods. This book should be a valuable resource for students and researchers alike wishing to learn more about iterative methods.

Preface, Acknowledgements, 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 preconditional matrices, 11. Conjugate gradient and Lanczos-type methods, 12. Generalized conjugate gradient methods, 13. The rate of convergence of the conjugate gradient method, Appendices.