Courses:

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This course will consist of two parts. During the first two thirds of the course, we will concentrate on Numerical Linear Algebra. We will study the solutions of linear systems of equations, least square problems, eigenvalue problems, and singular value problems. Techniques for dense, sparse and structured problems will be covered. Students should still come to appreciate many state-of-the-art techniques and recognize when to consider applying them. We will also learn basic principles applicable to a variety of numerical problems and learn how to apply them. These principles include (1) matrix factorizations, (2) perturbation theory and condition numbers, (3) effect of roundoff on algorithms, including properties of floating point arithmetic, (4) analyzing the speed of an algorithm, (5) choosing the best algorithm for the mathematical structure of your problem, and (6) engineering numerical software. In addition to discussing established solution techniques, open problems will also be presented.

During the second part of the course, we will concentrate on numerical methods for solving ordinary differential equations. These methods are usually introduced in undergraduate numerical analysis courses such as Introduction to Numerical Analysis (18.330). Such courses, however, are not prerequisites for 18.335. This graduate-level exposition will be self contained and lecture notes will be provided.

Prerequisites

Differential Equations (18.03) and Linear Algebra (18.06) or equivalent. The course assumes familiarity with linear algebra and will involve a reasonable amount of programming in MATLAB®.

Textbooks

Required

Trefethen, Lloyd N., and David Bau. Numerical Linear Algebra. Philadelphia, PA: Society for Industrial and Applied Mathematics, May 1, 1997. ISBN: 0898713617.

Optional

Demmel, James W. Applied Numerical Linear Algebra. Philadelphia, PA: Society for Industrial and Applied Mathematics, September 1, 1997. ISBN: 0898713897.

Lecture notes for Numerical Ordinary Differential Equations will be provided.

Grading