Additional Material

Gaussian elimination

Here and here, you can find the slides I presented in the lecture on 23rd of October with an example of Gaussian elimination and Gaussian elimination with partial pivoting.

Newton's Method in Higher Dimensions

In these lecture notes, you find a discussion of Newton's method in higher dimensions (cf. the lecture on 15th October).

Fixed Point iteration

The following images show what happens if one applies fixed point iteration to the function x = exp(x)/3 (cf. the lecture on 9th October).

Starting Value No. of Iterations
0 3
1 10
1.5 20
1.55 10

Some Code

Here is the code that I have used for producing the images of the solution of the heat equation that I have shown in the lecture on September 25: this produces the solution for a (truncated) sine as initial condition, this produces only the "heat kernel", that is, the function with which the initial value gets convolved. If you want to obtain results for different initial conditions, you will have to change the necessary parts of the code yourselves. Also: be aware that some of the comments in the code have been actually written for different versions of the code, which may at some points maybe lead to minor confusions.

The scripts are written in Python, which is a somehow easy to use scripting language quite well suited for mathematical applications (especially with the libraries that are loaded at the start of these scripts).

Heat Equation

One question that came up during the lecture on September 11 was concerned with the derivation of the heat equation. One possible derivation for this equation can be found in Kreyszig, Chapter 12.6 (10th edition). Alternatively, you may also look at this note (a translation of lecture notes for a course I didn't give in Vienna a few years ago) concerning the derivation of the one-dimensional heat equation for a possibly inhomogeneous material.

Fourier series of a rectangular wave

Here are the slides I showed on August 21 showing the approximation of a rectangular wave by the first few terms of its Fourier series.

One can see in the images that the approximation gets better as the number of terms increases, in particular in regions that are far away from the jumps of the function. Note that the approximation in the neighbourhood of the jumps is quite bad, however, and that the approximation has significant overshoots and undershoots near the jump (the maximum/minimum of the approximation is significantly larger/smaller than the function itself near the jump). It turns out that these overshoots are a "feature" of Fourier series, which always appears when one approximates a discontinuous function. This strange behaviour of the Fourier series is usually known under the name Gibbs phenomenon.

2015-10-23, Markus Grasmair