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Iterative Methods for Sparse Linear Systems

Iterative Methods for Sparse Linear Systems Yousef Saad 6. 15 12. 4 9 5. 11 14 8. 2 10 3. 13 7. 1. Copyright c 2000 by Yousef Saad. S ECOND EDITION WITH CORRECTIONS . JANUARY 3 RD , 2000.. PREFACE xiii Acknowledgments .. xiv Suggestions for Teaching .. xv 1 BACKGROUND IN Linear ALGEBRA 1. Matrices .. 1. Square Matrices and Eigenvalues .. 3. Types of Matrices .. 4. Vector Inner Products and Norms .. 6. Matrix Norms .. 8. Subspaces, Range, and Kernel .. 9. Orthogonal Vectors and Subspaces .. 10. Canonical Forms of Matrices .. 15. Reduction to the Diagonal Form .. 15. The Jordan Canonical Form .. 16. The Schur Canonical Form .. 17. Application to Powers of Matrices .. 19. Normal and Hermitian Matrices .. 21. Normal Matrices .. 21. Hermitian Matrices .. 24. Nonnegative Matrices, M-Matrices .. 26. Positive-Definite Matrices .. 30. Projection Operators .. 33. Range and Null Space of a Projector.

Iterative methods for solving general, large sparse linear systems have been gaining popularity in many areas of scientific computing. Until recently, direct solution methods were often preferred to iterative methods in real applications because of their robustness and predictable behavior.

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