AMCS/MATH 602: Algebraic Techiques I: Algebraic Techniques For Applied Math and Computational Science
Fall 2018
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:
- 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!
- Demmel, James, (1997). Applied
Numerical Linear Algebra.
Philadelphia, PA: Society for Industrial & Applied Mathematics.
- Additional material for more advanced topics will be announced
later in the semester.
A problem set will be assigned on Thursday each week, which will be due
in class on Thursday next week. The problem sets from Fall 2017 appear at the
bottom of this web-page.
This semester we use canvas to upload grades, post homework, make announcements, and share course notes. "https://canvas.upenn.edu/" .
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.
- The class meets on Tuesdays and Thursdays from 1:30 pm to 3:00 pm in room 4C4 in the David Rittenhouse Labs.
- My office hour will be Tuesdays and Thursdays 4:15 pm-5:15 pm or by appointment via email if you can not come during this time.
- My office in the Math Department is 4N63 DRL, tel. (215) 898-7844.
- My email: zwang423@math.upenn.edu.
Send e-mail if you have a question or need to contact me.
- The grader is Lingxi Lu. Office Hour: TBA. Her email: lulingxi@sas.upenn.edu.
Syllabus (Preliminary)
- Matrix operations
- Floating point arithmetic
- QR and least squares
- Norms, conditioning and stability
- Direct methods for systems of linear equations, including
- LU
- QR
- SVD factorizations
- Gaussian Elimination
- Iterative methods
- Krylov
- Arnoldi
- GMRES
- Lanczos
- Matrix eigenvalue problems
- Symmetric matrices
- Non-symmetric matrices, spectrum and pseudospectrum
- The Perron-Frobenius theorem
Additional topics taken from:
- Orthogonal polynomials and 3-term recurrence relations
- Numerical quadratures
- Numerical solution of PDEs
- Compression and randomized algorithms
- Discrete Fourier and wavelet transforms
- Linear programming
Announcements
- This class will have its first meeting on Tuesday, August 28, 2018.
Good references for the necessary background material are Linear Algebra by Peter Lax, Linear Algebra and Its Applications by Gilbert Strang, and Introduction to Matrix Analysis by Richard Bellman.
Homework Sets and Matlab Projects
- Homework 1 . Due Sept. 13, 2017
- Homework 2 . Due Sept. 20, 2017.
- Homework 3 . Due Oct. 2, 2017.
- 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.
- Homework 4 . Due Oct. 18, 2017.
- Homework 5 . Due Oct. 30, 2017.
- Homework 6 . Due Nov. 15, 2017.
- Homework 7 . Due Nov. 29, 2017.
- Final Project/Exam . Due Dec. 6, 2017.
Some Additional References
- A link to A Randomized
Algorithm for the Decomposition of Matrices, by Martinsson, Roklin
and Tygert.
- A link to Randomized
Algorithms for the low-rank Approximation of Matrices, by Liberty,
Woolfe, Martinsson, Roklin and Tygert.
- A link to The
Toda Lattice, Old and New, by Carlos Tomei
- A link to ON
THE COMPRESSION OF LOW RANK MATRICES, by H. CHENG,
Z. GIMBUTAS, P. G. MARTINSSON, AND V.
ROKHLIN.
- A page
discussing the QBX method for evaluation of layer potentials described
in class.
- The PSEUDOSPECTRUM web-page of Mark Embree and Nick Trefethen.
Return to Zhenfu's home
page.