Focusing on the analysis of iterative methods, the author includes the most useful algorithms from a practical point of view and discusses the mathematical principles behind their derivation and analysis. Several questions are emphasized throughout: Does the method converge? If so, how fast? Is it optimal, among a certain class? If not, can it be shown to be near-optimal?
Focusing on the analysis of iterative methods, the author includes the most useful algorithms from a practical point of view and discusses the mathematical principles behind their derivation and analysis. Several questions are emphasized throughout: Does the method converge? If so, how fast? Is it optimal, among a certain class? If not, can it be shown to be near-optimal?
1. List of Algorithms 2. Preface 3. Chapter 1: Introduction. Brief Overview of the State of the Art 4. Notation 5. Review of Relevant Linear Algebra 6. Part I: Krylov Subspace Approximations. Chapter 2: Some Iteration Methods. Simple Iteration 7. Orthomin(1) and Steepest Descent 8. Orthomin(2) and CG 9. Orthodir, MINRES, and GMRES 10. Derivation of MINRES and CG from the Lanczos Algorithm 11. Chapter 3: Error Bounds for CG, MINRES, and GMRES. Hermitian Problems—CG and MINRES 12. Non-Hermitian Problems—GMRES 13. Chapter 4: Effects of Finite Precision Arithmetic. Some Numerical Examples 14. The Lanczos Algorithm 15. A Hypothetical MINRES/CG Implementation 16. A Matrix Completion Problem 17. Orthogonal Polynomials 18. Chapter 5: BiCG and Related Methods. The Two-Sided Lanczos Algorithm 19. The Biconjugate Gradient Algorithm 20. The Quasi-Minimal Residual Algorithm 21. Relation Between BiCG and QMR 22. The Conjugate Gradient Squared Algorithm 23. The BiCGSTAB Algorithm 24. Which Method Should I Use? 25. Chapter 6: Is There A Short Recurrence for a Near-Optimal Approximation? The Faber and Manteuffel Result 26. Implications 27. Chapter 7: Miscellaneous Issues. Symmetrizing the Problem 28. Error Estimation and Stopping Criteria 29. Attainable Accuracy 30. Multiple Right-Hand Sides and Block Methods 31. Computer Implementation 32. Part II: Preconditioners. Chapter 8: Overview and Preconditioned Algorithms. Chapter 9: Two Example Problems. The Diffusion Equation 33. The Transport Equation 34. Chapter 10: Comparison of Preconditioners. Jacobi, Gauss--Seidel, SOR 35. The Perron--Frobenius Theorem 36. Comparison of Regular Splittings 37. Regular Splittings Used with the CG Algorithm 38. Optimal Diagonal and Block Diagonal Preconditioners 39. Chapter 11: Incomplete Decompositions. Incomplete Cholesky Decomposition 40. Modified Incomplete Cholesky Decomposition 41. Chapter 12: Multigrid and Domain Decomposition Methods. Multigrid Methods 42. Basic Ideas of Domain Decomposition Methods.
1. List of Algorithms 2. Preface 3. Chapter 1: Introduction. Brief Overview of the State of the Art 4. Notation 5. Review of Relevant Linear Algebra 6. Part I: Krylov Subspace Approximations. Chapter 2: Some Iteration Methods. Simple Iteration 7. Orthomin(1) and Steepest Descent 8. Orthomin(2) and CG 9. Orthodir, MINRES, and GMRES 10. Derivation of MINRES and CG from the Lanczos Algorithm 11. Chapter 3: Error Bounds for CG, MINRES, and GMRES. Hermitian Problems—CG and MINRES 12. Non-Hermitian Problems—GMRES 13. Chapter 4: Effects of Finite Precision Arithmetic. Some Numerical Examples 14. The Lanczos Algorithm 15. A Hypothetical MINRES/CG Implementation 16. A Matrix Completion Problem 17. Orthogonal Polynomials 18. Chapter 5: BiCG and Related Methods. The Two-Sided Lanczos Algorithm 19. The Biconjugate Gradient Algorithm 20. The Quasi-Minimal Residual Algorithm 21. Relation Between BiCG and QMR 22. The Conjugate Gradient Squared Algorithm 23. The BiCGSTAB Algorithm 24. Which Method Should I Use? 25. Chapter 6: Is There A Short Recurrence for a Near-Optimal Approximation? The Faber and Manteuffel Result 26. Implications 27. Chapter 7: Miscellaneous Issues. Symmetrizing the Problem 28. Error Estimation and Stopping Criteria 29. Attainable Accuracy 30. Multiple Right-Hand Sides and Block Methods 31. Computer Implementation 32. Part II: Preconditioners. Chapter 8: Overview and Preconditioned Algorithms. Chapter 9: Two Example Problems. The Diffusion Equation 33. The Transport Equation 34. Chapter 10: Comparison of Preconditioners. Jacobi, Gauss--Seidel, SOR 35. The Perron--Frobenius Theorem 36. Comparison of Regular Splittings 37. Regular Splittings Used with the CG Algorithm 38. Optimal Diagonal and Block Diagonal Preconditioners 39. Chapter 11: Incomplete Decompositions. Incomplete Cholesky Decomposition 40. Modified Incomplete Cholesky Decomposition 41. Chapter 12: Multigrid and Domain Decomposition Methods. Multigrid Methods 42. Basic Ideas of Domain Decomposition Methods.
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