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:

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

The linear algebra chapters of this book provide some useful background material.

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

Discretized PDEs will be used as test problems and benchmarks for understanding various aspects of linear solvers and solvers for eigenvalue problems.