Transcription of Iterative Methods for Sparse Linear Systems
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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 .. 33. Matrix Representations .. 35. Orthogonal and Oblique Projectors .. 35. Properties of Orthogonal Projectors.
and the increased need for solving very large linear systems triggered a noticeable and rapid shift toward iterative techniques in many applications. This trend can be traced back to the 1960s and 1970s when two important develop-ments revolutionized solution methods for large linear systems. First was the realization
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