# Numerical Analysis 2 - SS06

# Contents

## 6. Nonlinear systems

### 6.1 Introduction

- Motivation, problem formulation
- Existence of solutions
- Uniqueness
- Sensitivity
- Efficient computation

### 6.2 Bisection and fixed point iteration

- Bisection
- Contraction
- Fixed point theorem
- Implementation

### 6.3 Newton's method

- Geometric idea
- Example: %$f(x)=x^2-c$%
- Convergence theorem
- Implementation: convergence criterion, stopping criterion
- How to compute %$Df$%

### 6.4 Nonlinear least squares: Gauss-Newton method

- Introduction
- The Gauss-Newton method
- Convergence

## 7. Ordinary differential equations

### 7.1 Introduction, basics

- Examples: classical mechanics, population dynamics
- Initial value problems
- Existence theorem (Peano)
- Uniqueness theorem (Picard & Lindelöf)
- Evolution, flow

### 7.2 Condition of initial value problems

- Propagation matrix
- The variational equation
- Condition number & properties

### 7.3 Explicit one-step methods

- Idea of one-step methods
- Mesh, mesh function, discrete evolution
- Consistency, consistency order
- Convergence, order of convergence

### 7.4 Explicit Runge-Kutta methods

- Taylor methods
- Idea of Runge-Kutta methods, the Runge method
- The Butcher array
- Consistency of RK-methods
- Consistency and number of stages
- Invariance under autonomization
- On the construction of RK-methods
- RK-methods and quadrature

### 7.5 Step size control

- Step size formula
- Estimation of the local error
- Embedded RK-methods
- The idea of Fehlberg

### 7.6 Extrapolation methods in a nutshell

- Idea of extrapolation
- Extrapolation of the classical RK-method
- Richardson extrapolation
- Computational effort
- Theorem on the asymptotic expansion of the local error

## 8. Partial differential equations

### 8.1 Introduction

- What are PDEs and where do they occur?
- Classification
- The elliptic model problem: Poisson's equation
- Discretization principles

### 8.2 Finite difference methods for elliptic equations

- A one-dimensional example
- The two-dimensional case
- The discrete min-max principle
- Uniqueness of solutions
- Stability
- Convergence analysis

### 8.3 Finite element methods

# Literature

- P. Deuflhard, F. Bornemann (2002): Scientific Computing with Ordinary Differential Equations, Springer, Berlin.
- A. Hohmann, P. Deuflhard (2003): Numerical Analysis in Modern Scientific Computing, 2nd Edition, Springer, New York.
- J. Stoer, R. Bulirsch (2002): Introduction to Numerical Analysis, 3rd Edition, Springer, New York.
- C. Moler (2004): Numerical Computing with MATLAB
^{}. - E. Neuman: MATLAB Tutorials
^{} - J. Behrens and A. Iske: MATLAB: A friendly introduction (in German), (PDF, 200kB).

# Software

We will use MATLAB^{}for the programming examples throughout this course. MATLAB is installed on workstations in the mathematics and computer science departments. If you would like to use MATLAB on your private computer, you can either purchase a student version of MATLAB

^{}, or try to use the free software Octave

^{}. Octave, however, is not 100% compatible with MATLAB. Yet another possibility is to use the (completely incompatible but) free scientific software package Scilab

^{}that has been developed since 1990 first by French institutes INRIA and ENPC, and now by the Scilab Consortium. One can try to use the MATLAB to Scilab conversion tools provided by the editor in Scilab.