Lecture Notes on Iterative Methods

Lecture Notes:

1. Introduction (13 MB)
2. Synopsis of Numerical Linear Algebra (more extensive notes are below)
3. Fixed-Point Iterations, Krylov Spaces, and Krylov Methods
4. Comparing CG and GMRES for Various Model Problems
5. Convergence of CG – part I
6. Convergence of CG – part II: local convergence
7. Convergence of MINRES and GMRES
8. Generalizations of (restarted) GMRES
9. Methods based on the two-sided Lanczos algorithm
10. Preconditioners based on incomplete factorizations
11. Saddle-Point preconditioners
12. Domain decomposition preconditioners - for convergence theory see notes from class)
13. Multigrid 1 – basic iterative methods and error smoothing
14. Multigrid 2 – smooth and oscillatory modes, basic multigrid
15. Multigrid 3 – local mode analysis
16. Multigrid 4 – convergence proof and analysis

Background Material:

Lecture notes (by EdS) based on David Watkins, Fundamentals of Matrix Computations (2nd ed.), Wiley:

• Intro
• Overview
• Chapters 1 and 2 - Solving Linear Systems of Equations
• Chapter 3 - Overdetermined Systems
• Chapter 4 - Singular Value Decomposition
• Chapter 5 - Eigenvalue Problems Part I
• Chapter 5 - Eigenvalue Problems Part II (more fun stuff to be added)

Lecture notes (by Mike Heath) based on Michael T. Heath, Scientific Computing: An Introductory Survey (2nd ed.), McGraw-Hill:

• All Lecture Notes
• Systems of Linear Equations
• Linear Least Squares
• Eigenvalue Problems

Lecture notes (EdS) on Numerical Partial Differential Equations (soon)