# Faculty of Engineering and Science

The International Doctoral School of Technology and Science

#
PhD Course

## Topics in Modern Applied Mathematics

**Web page updated May 30, 2006**
.
## Course files

Below are links to files used in demonstrations during the lectures
- Maple file illustrating the derivation of finite difference formulas through interpolation.
- Maple file showing the structure of the periodic spectral differentiation matrix.
- Maple file computing the derivatives needed in the infinte spectral differentiation formula.
- Maple file computing the derivatives needed for the periodic
spectral differentiation matrix.
- Maple file illustrating Exercise 3.1
from [T].
- Maple file illustrating the
location of the Chebyshev points.
- Maple file
ilustrating the difference between interpolation based on
the equidistant points and Chebyshev points. The function being
approximated is the famous one called the Runge function.
- Matlab toolbox to compute pseudospectra can be downloaded from here.

### m-files concerning pseudospectra

Here are some m-files with examples and exercises concerning
pseudospectra. The files contain as comments some explanations.
## Lecture Notes

- A set of Lecture Notes on Complex Analysis can be found here. They
are intended for mathematics students (third semester), but the
statements of the results should be accessible without studying the
proofs in detail.
- Notes on Chapter 5 in Trefethen's book can be found here. The
error estimate in polynomial approximation is derived, as is the
error estimate for spectral differentiation based on the Chebyshev points.

Updated May 30, 2006, by Arne Jensen.