Introduction to numerical methods

Lectures: 3 sessions / week, 1 hour / session

18.06 Linear Algebra, 18.700 Linear Algebra and 18.03 Differential Equations or 18.034 Honors Differential Equations.

Advanced introduction to numerical linear algebra and related numerical methods. Topics include direct and iterative methods for linear systems, eigenvalue decompositions and QR/SVD factorizations, stability and accuracy of numerical algorithms, the IEEE floating-point standard, sparse and structured matrices, and linear algebra software. Other topics may include memory hierarchies and the impact of caches on algorithms, nonlinear optimization, numerical integration, FFTs, and sensitivity analysis. Problem sets will involve use of MATLAB® (little or no prior experience required; you will learn as you go).

Bau III, David, and Lloyd N. Trefethen. Numerical Linear Algebra. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1997. ISBN: 9780898713619.

Additional readings include:

Bai, et al. Templates for the Solution of Algebraic Eigenvalue Problems: a Practical Guide. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2000. ISBN: 9780898714715.

Barrett, et al. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1993. ISBN: 9780898713282.

Talk to anyone you want to and read anything you want to, with three exceptions: First, you may not refer to homework solutions from the previous terms. Second, make a solid effort to solve a problem on your own before discussing it with classmates or doing an Internet search. Third, no matter whom you talk to or what you read, write up the solution on your own, without having their answer in front of you.

The final project will be a 5–15 page paper (single-column, single-spaced, ideally using the style template from the SIAM Journal on Numerical Analysis), reviewing some interesting numerical algorithm not covered in the course. [Since this is not a numerical PDE course, the algorithm should not be an algorithm for turning PDEs into finite/discretized systems; however, your project may take a PDE discretization as a given "black box" and look at some other aspect of the problem, e.g. iterative solvers.]

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