Syllabus Detail

Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 226B: Numerical Methods: Large-Scale Matrix Computations

Approved: 2008-06-01, Roland Freund
Units/Lecture:
Winter, alt years; 1st LEC 3.0 hrs/wk; 2nd T-D 1.0 hrs/wk
Suggested Textbook: (actual textbook varies by instructor; check your instructor)

Prerequisites:
1MAT 167; Or equivalent, or consent of instructor; familiarity with some programming language
Course Description:
Numerical methods for large-scale matrix computations, including direct and iterative methods for the solution of linear systems, the computation of eigenvalues and singular values, the solution of least-squares problems, matrix compression, methods for the solution of linear programs.
Suggested Schedule:
  • Direct Methods for the Solution of Linear Systems
    • Sparse Cholesky factorization
    • Sparse LU factorization
    • Fast elliptic solvers
  • Iterative Methods for the Solution of Linear Systems
    • Krylov subspace methods for symmetric systems
    • Krylov subspace methods for nonsymmetric systems
    • Preconditioning
    • Multigrid methods
  • Eigenvalue and Singular Value Problems
    • The power method
    • Krylov subspace methods
    • Applications in information retrieval
    • Applications in image processing
  • Solution of Least-Squares Problems
    • Sparse QR factorization
    • Matrix Compression of Low-Rank Matrices
    • Randomized algorithms
    • FMM/HSS algorithm
  • Linear Programming
    • Simplex method
    • Interior-point methods
Additional Notes:
No required textbook. Optional references:
  • G.H. Golub and C.F.Van Loan, Matrix Computations, 3rd Ed., Johns Hopkins University Press, 1996
  • Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, 2003
  • T.A. Davis, Direct Methods for Sparse Linear Systems, SIAM, 2006
A few topics of this proposed course are also covered in ECS 231 (Large-Scale Scientific Computation), but the focus in ECS 231 is on applications and software related aspects of these topics. The focus of this course, however, is on a rigorous mathematical treatment of these topics. Therefore, the potential overlap of ECS 231 and this proposed course is minimal.
Assessment:
Homework assignments, covering both theory and computational problems: 50%. Final project and report: 50%