The International Doctoral School of Technology and Science

I use the computer algebra system Maple to illustrate various things in the course. If you wish to experiment with my files, you need to have Maple installed. You can get a copy of Maple here. Note that Aalborg University has a site license for Maple. Follow the instructions concerning installation give in the link above, in particular with respect to the license file.

A more precise description follows below.

**Pseudospectral methods**- Pseudospectral methods (also called collocation methods) are important fairly recent methods for solving differential equations numerically. We will describe them in detail and compare with other methods, such as finite difference and finite element methods. Depending on the background of the participants, a number of important results and techniques will be reviewed, including results from approximation theory, Fourier analysis, and differential equations. A number of examples of applications will be given.
**Pseudospectra of linear operators**- Although the name sounds like the previous topic, it is actually something quite different. Given a matrix, its eigenvalues and (generalized) eigenvectors characterize it completely. This is sometimes called the Jordan normal form of a matrix. As is well known, the set of eigenvalues is called the spectrum. If the matrix is nice (for example hermitean (also called selfadjoint), the spectrum gives valuable information on the matrix. But if the matrix is far from being hermitean, this information may be very misleading. To remedy this problem, the pseudospectrum has been introduced. I will introduce the pseudospectrum, show some of its properties, and describe the kind of information it contains. A number of computational examples will be given. They can be chosen to fit interests of the participants.

Lloyd N. Trefethen:This book covers both topics. There will also be some papers about the pseudospectrum that will be handed out during the course. An overview of the psesudospectrum can be found inSpectral Methods in Matlab, SIAM 2000. ISBN 0-89871-465-6. A link to the SIAM homepage is here. A link to the author's page for the book is here. Note that this page also contains a number of matlab m-files we will use during the course.

Lloyd N. Trefethen:You can access this paper electronically through the library.Pseudospectra of linear operators. SIAM Review 39 (1997), 383-406.

There will be a number of Matlab exercises posed during the course. You will be reguired to hand in solutions of some of them, as part of the evaluation of the course.

Beyond this some familiarity with Matlab is required. You should know the basic matrix operations, and how to write and edit an .m file, carrying out a sequence of computations.

Written by Arne Jensen.