Iterative Methods for linear equations and eigen-problems (Math-6320)
Fall 2009

• Course description:
  This course will cover development and analysis of iterative methods used for solving large, sparse linear equations and eigenvalue problems. The emphasis will be on solving linear equations. It will cover basic iterative methods, Krylov subspace methods, preconditioning, and multigrid. [ Prerequisites:   Basic knowledge of matrix theory and linear algebra; basic knowledge of computer programming. ]

• Educational outcomes:
  Students will learn important iterative methods used for solving large sparse linear equations and eigen-problems. They will also learn the theoretical properties related to certain iterative methods. Students will be able to correctly code certain iterative algorithms (e.g., CG, Lanczos/Arnoldi, GMRES, Davidson) for solving linear equations and eigenvalue problems.

• Main Textbook:
  Iterative methods for sparse linear systems , written by Yousef Saad , Published by SIAM (2003, ISBN-13: 978-0-898715-34-7.)   Part of the lecture slides generously provided by Prof. Saad will be used. Lecture notes will also be prepared by the instructor as the class moves on, the lecture notes will be the main reference for this class besides the textbook.

• Supplementary textbooks:    
  • J. Demmel , Applied numerical linear algebra, SIAM, 1997.
  • Gene Golub and Charles Van Loan , Matrix computations, 3rd Ed., Johns Hopkins University Press, 1996.
  • C. T. Kelley , Iterative Methods for Linear and Nonlinear Equations SIAM, 1995. PDF version is available here .
  • G. W. Stewart , Matrix algorithms I: Basic decompositions, SIAM, 1998.
    G. W. Stewart, Matrix algorithms II: Eigensystems, SIAM, 2001.
  • Henk A. Van der Vorst ,   Iterative Krylov methods for large linear systems, Cambridge University Press, 2003.
    Lecture notes on large linear systems and eigenvalue problems

• Course content:
  
  • Basic iterative methods: Richardson, Jacobi, Gauss-Seidel, SOR
  • Krylov subspace methods: Arnoldi, Lanczos, CG, GMRES
  • Preconditioning: ILU, Approximate Inverse
  • Multigrid: Prologation, restriction, V-cycle, W-cycle, FMG
  • Davidson-type methods for eigenvalue problems

• Software:    the commercial   Matlab   and/or the open sourceware   Scilab , Octave , SciPy  
• Matlab primer (a bit outdated, but still a good starting point. Do a google search for other online tutorials)
• Scilab tutorial
• Further reading of interests:
  • Templates for the Solution of Linear Systems
  • Templates for the Solution of Algebraic Eigenvalue Problems

• Course Policy:
  • Class attendance is required (unless with permission from the instructor)
  • There will be two sets of projects (25% each) and a final (50%).