AMCS/MATH 602:   Algebraic Techiques I: Algebraic Techniques For Applied Math and Computational Science

Fall 2017
Instructor: Zhenfu Wang

The Course

AMCS 602 focuses on numerical aspects of Linear Algebra. These become extremely important when attempting to solve linear systems with thousands, hundreds of thousands, or even millions of unknowns. This situation arises in the approximate numerical solution of PDEs, and also in application of statisical tools to do analysis of large and/or high dimensional data sets. In addition to algorithms one also needs useful methods for characterizing accuracy, stability and efficiency. We will take material from a variety of sources, among them:
  1. Bau III, D., & Trefethen, L.N. (1997). Numerical Linear Algebra. Philadelphia, PA: Society for Industrial & Applied Mathematics. This is a book that focuses on numerical linear algebra. Required!
  2. Demmel, James,  (1997). Applied Numerical Linear Algebra. Philadelphia, PA: Society for Industrial & Applied Mathematics.
  3. Additional material for more advanced topics will be announced later in the semester.

A problem set will be assigned on Wednesday each week, which will be due in class on Wednesday next week. The problem sets appear at the bottom of this web-page.

Students can discuss problems with each other or seek tips from books, articles or online rescources. But it is wrong to simply copy from any source. See Penn Office of Student Conduct.

Syllabus (Preliminary)

  1. Matrix operations
  2. Floating point arithmetic
  3. QR and least squares
  4. Norms, conditioning and stability
  5. Direct methods for systems of linear equations, including
    • LU
    • QR
    • SVD factorizations
    • Gaussian Elimination
  6. Iterative methods
    1. Krylov
    2. Arnoldi
    3. GMRES
    4. Lanczos
  7. Matrix eigenvalue problems
    • Symmetric matrices
    • Non-symmetric matrices, spectrum and pseudospectrum
    • The Perron-Frobenius theorem

Additional topics taken from:

  1. Orthogonal polynomials and 3-term recurrence relations
  2. Numerical quadratures
  3. Numerical solution of PDEs
  4. Compression and randomized algorithms
  5. Discrete Fourier and wavelet transforms
  6. Linear programming


Homework Sets and Matlab Projects

  1. Homework 1 . Due Sept. 13, 2017
  2. Homework 2 . Due Sept. 20, 2017.
  3. Homework 3 . Due Oct. 2, 2017.
  4. Matlab project 1 . Due Oct. 9, 2017. You might find it helpful to refer to the worksheets worksheet 1 and worksheet 2 due to Dr. Epstein.
  5. Homework 4 . Due Oct. 18, 2017.
  6. Homework 5 . Due Oct. 30, 2017.
  7. Homework 6 . Due Nov. 15, 2017.
  8. Homework 7 . Due Nov. 29, 2017.
  9. Final Project/Exam . Due Dec. 6, 2017.

Some Additional References

Return to Zhenfu's home page.