Final Project

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.] Your paper should be written for an audience of your peers in the class, and should include example numerical results (by you) from application to a realistic problem (small-scale is fine), discussion of accuracy and performance characteristics (both theoretical and experimental), and a fair comparison to at least one competing algorithm for the same problem. Like any review paper, you should thoroughly reference the published literature (citing both original articles and authoritative reviews/books where appropriate [rarely web pages]), tracing the historical development of the ideas and giving the reader pointers on where to go for more information and related work and later refinements, with references cited throughout the text (enough to make it clear what references go with what results). Model your paper on academic review articles (e.g. read SIAM Review and similar journals for examples).

Frequently asked questions about the final project:

1. Does it have to be about numerical linear algebra? No. It can be any numerical topic (basically, anything where you are computing a conceptually real result, not integer computations), excluding algorithms for discretizing PDEs.
2. Can I use a matrix from a discretized PDE? Yes. You can take a matrix from the PDE as input and then talk about iterative methods to solve it, etcetera. I just don't want the paper to be about the PDE discretization technique itself.
3. How formal is the proposal? Very informal—one page describing what you plan to do, with a couple of references that you are using as starting points. Basically, the proposal is just so that I can verify that what you are planning is reasonable and to give you some early feedback.
4. How much code do I need to write? A typical project (there may be exceptions) will include a working proof-of-concept implementation, e.g. in Matlab, that you wrote to demonstrate that you understand how the algorithm works. Your code does not have to be competitive with "serious" implementations, and I encourage you to download and try out existing "serious" implementations (where available) for any large-scale testing and comparisons.
5. How should I do performance comparisons? Be very cautious about timing measurements: unless you are measuring highly optimized code or only care about orders of magnitude, timing measurements are more about implementation quality than algorithms. Better to measure something implementation independent (like flop counts, or matrix-vector multiplies for iterative algorithms, or function evaluations for integrators/optimizers), although such measures have their own weaknesses.