  # Contents

## Chapter 1. Introduction

### §1.1 What is Numerical Analysis (NA) and how does it affect our daily life

(1.1) Example 1: MP3 Audio encoding (see also the presentation slides, PDF 2.8 MB)
(1.3) Example 3: Numerical Weather prediction

### §1.2 What if (NA) fails

(1.4) Example 1: Patriot Missile failure (see also the presentation slides, PDF 2.8 MB)
(1.5) Example 2: Sleipner Platform failure

### §1.3 Tools for NA

(1.6) Some remarks on MATLAB

### §1.4 Floating point representation of real numbers

(1.7) (IEEE) Floating point representation (see also floatgui.m in a package on MATLAB Central )

#### Thu, 10/27/2005:

(1.8) Resolution and machine precision (eps)
(1.10) Overflow and Underflow

### §1.5 Using numerical software

(1.11) NETLIB, GAMS and TOMS (see also the presentation slides, PDF 400 KB)
(1.12) Beyond MATLAB: Mathematica, Maple, Octave and SciLab

### §1.6 Aims of course

(1.13) What you should be able to do after completing the course

## Chapter 2. Solving linear systems of equations

### §2.1 Problem definition and introductory examples

(2.1) Example from hydraulics (presentation slides, PDF 37 KB)

#### Thu, 11/03/2005:

(2.2) Questions to ask: solvability, algorithm, efficiency, accuracy
(2.3) Solvability
(2.4) A naive idea: Cramer's rule

### §2.2 The Gauss-Algorithm

(2.5) A better idea based on LU-Decomposition
(2.6) Getting insight from a simple example
(2.7) Extending the idea to arbitrary n: Gauss-Algorithm
(2.8) Operation count
(2.9) Solve the hydraulic example (2.1)

#### Tue, 11/08/2005:

(2.10) Pivoting with examples
(2.11) Permutation matrix
(2.12) The effect of roundoff - example

### §2.3 Error analysis (Accuracy of the Gauss-Algorithm)

(2.13) Abstract formulation - condition number
(2.14) Norms

#### Thu, 11/10/2005:

(2.15) Forward and backward error analysis
(2.16) Condition number
(2.17) Accuracy analysis for linear systems - condition of a matrix
(2.18) Properties of the condition
(2.19) Numerically singular matrices

#### Tue, 11/15/2005:

(2.20) Distance from Singularity (Kahan 1966)
(2.21) Evaluation of the solution of Ax=b - a priori vs. a posteriori concepts
(2.22) A posteriori evaluation of Gaussian Elimination (Wilkinson 1954)
(2.23) A priori backward analysis of Gaussian Elimination (Sautter 1971, Wilkinson 1961)

#### Thu, 11/17/2005:

(2.24) Remarks on backward analysis
(2.25) Iterative refinement (Result by Skeel1980)
(2.26) Symmetric positive matrix - Cholesky factorization

### §2.4 How-To-Session: How to solve Ax=b

(2.27) Structure of the matrix
(2.28) Condition number - what does it tell us?
(2.29) Choice of algorithm
(2.30) Stability - what does it mean?

## Chapter 3. Approximation of functions and data

(3.1) Introduction and problem definition (interpolation vs. approximation)

### §3.1 Polynomial interpolation

(3.2) Examples (presentation slides, PDF 125 KB)
(3.3) Uniqueness of polynomial interpolation
(3.4) Lagrange representation
(3.5) Interpolation error (see the proof of the theorem, PDF 72 KB)

#### Thu, 11/24/2005:

(3.6) Condition - Lebesgue Constant
(3.7) Exploring the Lebesgue constant
(3.8) Chebyshev interpolation

### §3.2 Trigonometric interpolation

(3.9) Interpolation of periodic functions

#### Tue, 11/29/2005:

(3.10) Coefficients of the DFT
(3.11) Fast Fourier Transform (FFT) - The Cooley-Tuckey idea (divide et impera)
Have a look at the m-file Cooley-Tukey Radix-2-FFT (1965, Gauß 1866)
(3.12) Signal analysis
(3.13) Sampling theorem (Shannon/Nyquist)
Have a look at the Aliasing Java applet ### §3.3 Piecewise polynomial interpolation

(3.14) Piecewise linear interpolation
(3.15) Cubic spline interpolation (see slide, PDF 84 KB)

### §3.4 Least squares approximation

(3.16) Problem formulation

#### Thu, 12/08/2005:

(3.17) Linear models
(3.18) Theoretical background - abstract approximation problem
(3.19) Normal equations
(3.20) Orthogonalization methods (QR-Idea according to Golub 1965)

#### Tue, 12/13/2005:

(3.21) Householder reflection
(3.22) Givens rotation
(3.23) Final remarks

## Chapter 4. Numerical Differentiation and Integration

(4.1) Introduction
See the Maple work sheet, 2KB
or the Mathematica note book, 3KB
and view the slides of the sattellite example, PDF 113 KB

### §4.1 Numerical Differentiation

(4.2) Finite differences
(4.3) Functions on equidistant nodes

#### Thu, 12/15/2005:

(4.5) Interpolatory formulae

### §3.5 How-To-Session: How to interpolate function values

(3.24) The polynomial interpolation problem
(3.25) Condition - the Lebesgue constant
(3.26) Chebyshev nodes
(3.27) Piecewise interpolation

### §3.6 How-To-Session: How to set up and solve least squares problems (LSP)

(3.28) Problem formulation
(3.29) Example: set up a LSP
(3.30) Solve the LSP

#### Tue, 12/20/2005:

(4.6) Introduction
(4.7) Structure of the problem
(4.10) Properties of consistent quadrature rules
(4.11) Newton-Cotes quadrature (see the slide of different weight sets, PDF 33 KB)
(4.12) Clenshaw-Curtis or Féjer formulae

#### Thu, 12/22/2005:

(4.13) Local error for the trapezoidal rule - general structure of local error
(4.14) Composite rules
(4.15) Global error for composite rules
(4.16) Procision-cost diagram (see the slides for examples, PDF, 65 KB)

#### Tue, 01/10/2006:

(4.19) Error estimation

### §4.4 How-To-Session: How to integrate a function numerically

(4.21) Structure of the problem
(4.23) Approximation error
(4.24) Composite rules

## Chapter 5. Eigenvalue Problems

### §5.1 Introduction

(5.1) The Problem
(5.2) Applications
(5.3) Solution properties/ uniqueness
(5.4) The naive approach
(5.5) Roots of polynomials as eigenvalue problems

### §5.2 Recapitulation from linear algebra

(5.6) Multiplicities
(5.7) Similarity transformation and diagonalization
(5.8) Hermitian and normal matrix

### §5.3 Normal forms

(5.9) Normal forms and factorizations
(5.10) Instability of the Jordan form
(5.11) Good behaviour of the normal form
(5.12) Schur decomposition (Schur 1909) - deflation

### §5.4 Sensitivity analysis

(5.13) Eigenvalue decomposition
(5.14) Condition of the eigenvalue problem
(5.15) Sensitivity of single eigenvalues
(5.16) Condition for hermitian matrices

### §5.5 Vector iteration

(5.17) Iteration (Abel's theorem)
(5.18) Rayleigh-quotient
(5.19) Vector iteration (von Mises 1929) (see the MATLAB example algorithm, 2kB, together with a DEMO script, 1kB)
(5.20) Inverse vector iteration (Wielandt 1944) (see the MATLAB example algorithm, 2kB, together with a DEMO script, 1kB)
(5.21) Is a badly conditioned (µI-A) a problem?
(5.22) Conclusion

### §5.6 The QR-Algorithm

(5.23) Structure of the algorithmic idea
(5.24) Iterative deflation
(5.25) Idea: inverse vector iteration
(5.26) The QR-trick
(5.27) The QR-algorithm with shift (Francis, Kublanovskaya 1961)
(5.28) Convergence behaviour
(5.29) Problems with global convergence
(5.30) Wilkinson-shift
(5.31) Cost for the QR-algorithm
(5.32) Idea of the 2-phase eigenvalue computation

#### Thu, 01/26/2006:

(5.33) Hessenberg matrices
(5.34) QR-algorithm for Hessenberg matrix (Phase 2)
(5.35) Reduction to Hessenberg form (Phase 1)
(5.36) Stability of the 2-phase QR-algorithm

### §5.7 Bisection for selected eigenvalues

(5.37) Aim and prerequisites
(5.38) Theoretical basis: Sylvester's law of inertia
(5.39) Application of the law
(5.40) Bisection - algorithm, cost, and stability

### §5.8 How-To-Session: How to solve an eigenvalue problem

(5.41) The problem and properties - uniqueness, and characteristic polynomial
(5.42) Similarity transforms and normal forms
(5.43) Condition - condition of eigenvectors
(5.44) Selecting an algorithm
(5.45) Vector iteration
(5.46) QR-algorithm
(5.47) Bisection - not completed