CME 324/CS 336. Advanced Methods in Matrix Computations —
Iterative Methods
Institute for Computational and Mathematical Engineering
and the Department of Computer Science
Stanford University
Spring 2006
This is a course on Matrix Computations with emphasis on iterative
methods for solving linear systems. The course will include such methods
as the Conjugate Gradient Method and GMRES. There will be some discussion
of eigenvalue methods too. Prerequisite: CME 302/CS 237A. Numerical Linear
Algebra.
Announcements
 05/17/06: Instructions for term paper have been posted.
 05/21/06: There will not be a lecture on Friday, May 26.
 05/16/06: Problem 3 added to Problem Set 2. Please see updated
file below.
 05/11/06: Problem Set 2 has been posted.
 05/10/06: There will be lectures on the following Fridays: May 12, 19
& 26. The last day of class is May 26.
 04/19/06: Problem Set 1 has been posted.
 04/10/06: Future classes will be held in Gates 100 instead of Gates
260.
 04/06/06: Please send LekHeng an email if you are not registered
for the class but want your email to be included in the class mailing
list.
 04/05/06: There will be no Friday lectures unless announced
otherwise.
Lectures
Location: Gates Building, Room 100
Times: 11:00 AM–12:15 PM on Mon/Wed
Course staff
Instructor: Gene
Golub (golub@stanford.edu).
Gates Building 2B, Room 280
(650) 7233124
Office hours by appointment.
Teaching assistant: LekHeng Lim (lekheng@stanford.edu).
Gates Building 2B, Room 286
(650) 7234101
Office hours by appointment.
Topics
 Introduction
 Sparse and structured matrices
 Model problem: Poisson equation
 Stationary Methods
 SOR: Successive overrelaxation
 SSOR: Symmetric SOR
 Property A
 Checkerboard/redblack ordering
 Block methods
 Chebyshev polynomials
 LebedevFinogenov parameter ordering
 Regular splitting
 Ostrowski theorem
 Splitting and preconditioning
 Chebyshev semiiterative method
 Chebyshev acceleration and CG
 ForsytheStraus theorem
 Fast Poisson solver
 Domain decomposition
 Incomplete factorization
 Banded and blocktridiagonal matrices
 M matrix
 Meijerinkvan der Vorst theorem
 KKT systems
 Real positive matrices
 Additive polar decomposition
 Cayley transform
 Complex symmetric matrices
 ArrowHurwiczUzawa algorithm
 Krylov Subspace Methods
 Steepest descent
 Kantorovich inequality
 CG: conjugate gradient method
 Preconditioned CG from Chebyshev acceleration
 Classical CG method
 Lanczos tridiagonalization
 Unsymmetric and singular systems
 Least squares problems
 GolubKahan bidiagonalization
 PaigeSaunders LSQR
 SaadSchultz GMRES: generalized minimal residual
 Unsymmetric Lanczos tridiagonalization
 BiCG method
 Parlett's lookahead strategies
 FreundNachtigal QMR: quasiminimal residual
 Other Topics
 Moments
 Quadrature: Gauss, GaussRadau, GaussLobatto
 Stieljes integrals
 Orthogonal polynomials
 Estimating values of bilinear forms of analytic
functions of symmetric matrices
Lecture notes, etc
 Lecture 10 (05/08/06):

Transcript (posted: 05/10/06)

G.H. Golub, "Matrices, moments, and quadrature with
applications," Symposium Optim. Data Anal., September
21–23, 2005, Canberra, Australia.

G.H. Golub, "Matrix computation and the theory of
moments," Bull. Belg. Math. Soc. Simon Stevin,
suppl. (1996), pp. 19.

G.H. Golub and G. Meurant, "Matrices, moments and
quadrature," pp. 105–156 in Numerical analysis 1993,
Pitman Research Notes in Mathematics, 303, Longman, Harlow, UK,
1994.
Term paper
Read a paper or report which discusses some method for solving a linear
system. It can be a wellknown method or some new methods, possibly
illustrating a new preconditioner. A good place to start is some of our
SCCM
technical reports. If you prefer older papers, check out the
list here.
Gene and LekHeng will be pleased to help select if you come by (do not
send email messages to Gene though).
A report of about eight pages is due on Friday, May 26 at 5PM. The
report should have some descriptive material and a summary of numerical
experiments.
Grades and assignments
A grade will be assessed based on about three homework problem sets and a
term paper.
Textbooks
 G. Golub and C. Van Loan, Matrix Computations, 3rd Ed., Johns
Hopkins Series in the Mathematical Sciences, 3, Johns Hopkins University Press, Baltimore, MD, 1996. ISBN: 0801854148
 J.C. Strikwerda, Finite Difference Schemes and Partial
Differential Equations, 2nd Ed., SIAM, Philadelphia, PA, 2004.
ISBN: 0898715679
 H.A. van der Vorst, Iterative Krylov Methods for Large Linear
Systems,
Cambridge Monographs on Applied and Computational Mathematics, 13
Cambridge University Press, Cambridge, UK, 2003. ISBN:
0521818281
 R.S. Varga, Matrix Iterative Analysis, 2nd Ed., Springer Series
in Computational Mathematics, 27,
SpringerVerlag, Berlin, 2000. ISBN: 3540663215
Softwares
About this page
You may reach this page through either http://cme324.stanford.edu or http://www.stanford.edu/class/cme324.
Contact: Comments on these pages are welcome. Please write to
lekheng@stanford.edu