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Contents

Chapter 1. Introduction

Tue, 10/25/2005:

§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)
(1.2) Example 2: GPS Navigation
(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 Pfeil)

Thu, 10/27/2005:

(1.8) Resolution and machine precision (eps)
(1.9) Roundoff (Examples, see also MATLAB script oldeps.m)
(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)
(Download hydromat.dat, hydrorhs.dat, and solvehydro.m)

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?


Tue, 11/22/2005:

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 Pfeil

Tue, 12/06/2005:

§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
(4.4) Solution of introductory example (download the MATLAB example, ZIP 4KB)

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:

§4.2 Numerical Integration (Quadrature)

(4.6) Introduction
(4.7) Structure of the problem
(4.8) Generic formulation of quadrature
(4.9) Interpolatory quadrature formulae
(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)

§4.3 Adaptive quadrature

(4.17) Adaptive principle

Tue, 01/10/2006:

(4.18) Adaptive refinement
(4.19) Error estimation
(4.20) Example: adaptive Simpson quadrature (see the slide, PDF, 18 KB)


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

(4.21) Structure of the problem
(4.22) Classical Newton-Cotes quadrature
(4.23) Approximation error
(4.24) Composite rules
(4.25) Adaptive quadrature


Thu, 01/12/2006:

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

Tue, 01/17/2006:

§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

Thu, 01/19/2006:

§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

Tue, 01/24/2006:

§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

Tue, 01/31/2006:


§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