# PhD Course 2009

## An Introduction to Pseudospectra and their Applications

### Files

Here you find a number of files to be used as starting points for your own experiments. They may have errors and omissions. Explanations are very limited.
#### m-files

The following files have been tested using Matlab R2009a and the most recent version of eigtool.
- randomperturb.m This file generates a Toeplitz matrix, plots its spectrum and then plot the spectra of a number of small random perturbations of this matrix.
- matrixexponentials1.m This file plots norm(exp(tA)) as a semilogarithmic plot, and then using the spectral abscissa and numerical abscissa, obtained from eigtool, plots the upper and lower bounds obtained from these two numbers. Thus you must obtaine these numbers from eigtool before executing this m-file.
- cheb.m This file generates a Chebyshev differentiation matrix. You
should change the size parameter N as appropriate.
- chebnor.m This file generates a
*normalized* Chebyshev
differentiation matrix. It uses the file cheb.m. You
should change the size parameter N as appropriate.
- equi-cheb.m This file generates plots comparing interpolation
based on equidistant points with interpolation based on Chebyshev
points. Do not choose the number of points larger than 24.
- Further m-files related to spectral differentiation and polynomial interpolation can be downloaded from this site.
- landau_demo.m contains a function defining a matrix coming from laser physics. It is a modified version of the function by the same name in eigtool.

#### mw-files

The following files have been tested using Maple13. They should work
also with Maple12. Note that you may have to configure your computer
to recognize files with the extension .mw as Maple
worksheets. Otherwise they may be recognized as xml files.
- chebpointplot.mw This file produces a plot illustrating the
significance of the Chebyshev points.
- chebyshev.mw This worksheet generates Chebyshev differentiation matrices.
- finitedifference.mw A worksheet showing how to generate finite
difference formulas based on polynomial interpolation.
- rungecheb.mw This worksheet compares polynomial interpolation
based on equidistant points with polynomial interpolation based on
Chebyshev points.

Webpages maintained by Arne Jensen, matarne at math.aau.dk

Updated 27/09/2009, 11:15