# PhD Course 2009

## An Introduction to Pseudospectra and their Applications

### Lecture 6

The program for the sixth day is as follows:

12:30-13:00
Questions and answers. Review of results on spectral differentiation. An example.
13:00-14:00
Further results on pseudospectra. Perturbation theory and pseudospectra. An example of applications of pseudospectra.
14:00-14:20
Coffee break.
14:20-14:40
Presentation by Jesper Gadegaard
14:40-15:40
Exercises.
15:40-16:00
Discussion of the exercises.

#### Exercises

You should select some of the exercises below, depending on your interests.

Theoretical exercises

• Exercise 10.4
• Exercise 10.5
• Exercise 10.6
• Exercise 10.7

Eigtool exercises

• Chebyshev differentiation matrices. If you have not yet tried to investigate these matrices in eigtool, you should do so.
• One can impose a boundary condition u(1)=0 by deleting the first row and the first column in the Chebyshev differentiation matrix. Carry out numerical experiments with such matrices of different sizes.
• Similarly one can impose the boundary condition u(-1)=0 by deleting the last row and the last column in the Chebyshev differentiation matrix. Carry out some experiments and compare with the previous experiments. What is your explanation for the phenomenon observed?
• By deleting both the first and the last row, and the first and the last column, in the Chebyshev differentiation matrix one imposes the two boundary conditions u(1)=0 and u(-1)=0. Again. carry out the experiments and explain.
• Based on the material on perturbation theory, try to reproduce Figures 8 and 9 (on pages 32 and 33) in the Lecture Notes, and then try the same for some other matrices. You can compute the eigenvalue condition numbers in Matlab by imitating the computations explained in the Lecture Notes.

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Updated 27/09/2009, 11:25