sign in sign up
<<
Home , Numerical linear algebra

Math228 NumericalLinearAlgebra Spring 2014 Course Information

BLOCK: D+: 10:30a-11:45a, Tuesday and Thursday INSTRUCTOR: Kye Taylor EMAIL: [email protected] OFFICE: Bromfield-Pearson 217 OFFICE HOURS: Monday 10:00a - 12:00p, Wednesday 2:00p - 4:00p, Friday 1:20p-2:20p, and by appointment. COURSE PAGE: http://courses.math.tufts.edu/math228

PREREQUISITES: • Experience in a scientific programming language (e.g. , , C++, MATLAB, Python, etc.), though the expectation will be to program in MATLAB, C++, or Python. • MATH 70 and COMP 11, or equivalent.

REQUIRED TEXT: • “Fundamentals of Computations, by David S. Watkins, 3rd edition, Wiley. You may get by with the 2nd edition if available and cheaper.

BRIEF DESCRIPTION: We will study the and the relevant matrix theory for computing the solution to several problems of great interest in a wide va- riety of science and applications. The list of linear algebra problems we will consider includes solving linear systems through direct and iterative techniques, (orthog- onal) matrix factorization, and eigenvalue/eigenvector computation. We will build on the basic linear algebra concepts (i.e. range, null space, vector subspaces, orthogonal projections) and tools from a standard linear algebra course. While paying attention to storage, operations counts and finite precision arithmetic, we will learn how tools from linear algebra can be used to solve real-world problems. In-class examples and homework problems will feature some of these applications (e.g. optimization, , and image processing). Because this is a graduate course, we will cover state-of- the-art algorithms/theory and open research problems in the field.

LEARNING OBJECTIVES: Learning objectives relevant to a Master’s Degree or a Ph.D. in are available at http://ase.tufts.edu/faculty/committees/objectives/math.htm

HOMEWORKS: Typical homework assignments will include problems that can be worked through by hand, as well as programming tasks that must be coded on a computer using MATLAB, C++, or Python. If you prefer to program in another language, please speak with me as soon as possible. For each homework assignment, please provide a write-up that details the mathemat- ical manipulations required for those problems completed by hand. If applicable, the write-up should also address any homework problems that are related to the execution or design of relevant code. Homework problems that can be worked through by hand will be collected in class. Please mark your homework with the course and section numbers as well as an identifier to help you know that it is yours – something that is likely unique to your section and something that is pronounceable in case the homework is returned by calling out the identifiers. Please write it as clearly as possible and make sure to tell your instructor well before the end of the semester what your identifier is so credit associated with it can be counted towards your course grade. Feel free to use your name as your identifier, but expect that unless you are told other- wise, the homework will be handed off between instructor and grader in a way that does not ensure their confidentiality (usually by way of drawers in the lobby of the Bromfield- Pearson building). Your educational record is privileged information under the federal Family Educational Rights and Privacy Act (FERPA), and using your name as identifier means that you opt out of being guaranteed the confidentiality of the information on and in your homework. Any programs that you are asked to code will also be collected via email. Specific in- structions for email-submissions of homework will be given in the homework assignment itself. Note that 10% of each homework score will be for “style points,” based on the pre- sentation of your results. Imagine turning work into an employer; it should appear professional.

DISABILITY SERVICES: If you are requesting an accommodation due to a documented disability, you must register with the Disability Services Office at the beginning of the semester. To do so, call the Student Services Desk at 617-627-2000 to arrange an appoint- ment with Linda Sullivan, Program Director of Disability Services.

GRADES:

2In-classQuizzes = 30% 8 Homeworks = 40% 1Take-homeFinal = 30% CALENDAR:

What Assigned Due HW 1 Thurs. Jan. 23 Fri. Jan. 31 HW 2 Thurs. Jan. 30 Fri. Feb. 7 In-class Quiz Thurs. Feb 13 HW 3 Tues. Feb. 18 Fri. Feb. 28 HW 4 Thurs. Feb. 27 Fri. Mar. 7 HW 5 Thurs. Mar. 6 Tues. Mar. 25 In-class Quiz Thurs. Apr. 3 HW 6 Thurs. Apr. 3 Fri. Apr. 11 HW 7 Thurs. Apr. 10 Fri. Apr. 18 HW 8 Thurs. Apr. 17 Fri. Apr. 25 Take-home Final Thurs. Apr. 24 Tues. May. 6

SEMESTER OUTLINE:

• Ch 1: Basic Operations and Algorithms 1.1 – 1.3 , Linear systems of , Triangular solves 1.4 Positive Definite systems and 1.7 – 1.8 , LU Decomposition (with pivoting) 1.6, 1.9 Factorization of sparse matrices (and 8.6) • Ch 2: Sensitivity of Linear Systems 2.1 – 2.2 Vector and Matrix norms, Condition numbers 2.3 – 2.7 Perturbation analysis, roundoff error analysis • Ch 3: Least Squares Problem 3.1 The Discrete Least Squares problem 3.2 Orthogonal matrices, rotators, and reflectors 3.3 Solution of Least Squares with QR factorization 3.4 Gram-Schmidt (briefly), connection with projectors and numerical properties • Ch 4: The decomposition (SVD) • Ch 5: Eigenvalues and Eigenvectors Part I 5.2 Facts 5.3 Power Method Plus 5.4 – 5.5 Similarity transformations and reduction to tridiagonal form 5.6 – 5.7 Francis’s (aka QR with shift) 5.8 Computing the SVD • Ch 6: Eigenvalues and Eigenvectors Part II 6.1 Invariant subspaces 6.2 Subspace iteration/ simultaneous iteration 6.3 Krylov subspaces (and connection to 6.1 – 6.2) 6.6 Jacobi-Davidson and related algorithms • Ch 7: Eigenvalues and Eigenvectors Part III 7.1 Sensitivity of eigenvalues and vectors 7.2 The symmetric eigenvalues problems • Ch 8 and Supplemental: Iterative Methods for Linear Systems GMRES (connection to Krylov subpsaces, eigenvalues) Conjugate gradient Preconditioners (see 1.6, 1.9)