# Numerics II

## News

- The results of the make-up exam can be found here.
- The results of the exam can be found here.
- The morning lecture on February 7 is skipped in light of the written exam on the same day.
- Section 5.4.3 Generalized minimal residual method (GMRes) has been revised.
- The 12th problem set is online.
- The exam will take place on Monday,
**7 February**at**14:15**. There will be no lecture in the morning. - The 11th problem set is online.
- The 10th problem set is online.
- The nineth problem set is online.
- Section 1.9 Extrapolation Methods has been revised (special thanks to Vincent Dallmer).
- New reference added: P. Kunkel and V. Mehrmann: Differential-algebraic equations
- The eigth problem set is online.
- The seventh problem set is online.
- The sixth problem set is online.
- Due to the rapidly increasing number of Covid-19 infections, we came to the conclusion that switching to a non-virtual environment is not responsible at this point. Therefore, all lectures and tutorials will remain
**online**for the time being. - The fifth problem set is online.
- The fourth problem set is online.
- The third problem set is online.
- The second problem set is online.
- The first problem set is online.
- The introductory notes from the first lectures can be found here
~~Starting from~~**December**, we plan to have lectures and the tutorial sessions as regular (non-online) meetings at the institute.~~The usual measures will apply (Proof of vaccination, proof of recovery, or recent negative test).~~

## Dates

Lecture | Monday, 10-12 | WebEx | Prof. Dr. Ralf Kornhuber |

Monday, 14-16 | WebEx | ||

Tutorial | Tuesday, 14-16 | WebEx | Lasse Hinrichsen-Bischoff |

Exam | Monday, 7 Feb. 2022, 14:15 | online | |

Second exam | Monday, 28 March 2022, 14:15 | online |

## General Information

### Description

Extending basic knowledge on odes from Numerics I, we first concentrate on one-step methods for stiff and differential-algebraic systems and then discuss Hamiltonian systems. In the second part of the lecture we consider the iterative solution of large linear systems.

### Target Audience

Students of Bachelor and Master courses in Mathematics and of BMS.

### Prerequisites

Basics of calculus (Analysis I, II), linear algebra (Lineare Algebra I, II) and numerical analysis (Numerik I).

## Registration

All participants should register at the Whiteboard. In addition, depending on your program of study, you have to register in the Campus Management.

## Exercises and Criteria for a Certificate

### Tutorial & Exercises

- The tutorial offers the possibility to discuss and better understand the presented material and exercises. Furthermore solutions of exercises are presented by the students.
- Each week a sheet with exercises will be made available electronically on this web page (see below).
- The exercises are intended to be solved by teams of three members.
- The exercises consist of theoretical and numerical problems. The latter should be solved using Python 3 (or Matlab, if you must). Both types of exercises are rated seperately.
- The solutions have to be finished before the tutorial a week after they were handed out.
- The solutions of the numerical problems should be delivered by email to l.hinrichsen@fu-berlin.de. Note that a complete solution for a numerical experiment consists of a running code, a program executing the required test runs, and protocols of the execution of these test runs. Delivering a correct and running code without knowing what is going on in the code will be regarded and rated as attempt of deception.
- The solutions of the theoretical problems have to be presented in the tutorial on request (without regard to preference and presence) by a member of a given group.

### Criteria for the Certificate

- Passing the exam.
- Regular participation in the tutorials, i.e. 85% attendance at the tutorials (not checked).
- Active participation: at least as much success as failure in the presentation of the solution of theoretical problems and 50% of the maximal number of points for numerical problems.
- The grade of this course is based only on the result of the exam.

## Exercises

Here you'll find problem sets which will help you to understand the material.

- 1st problem set
- 2nd problem set
- 3rd problem set
- 4th problem set
- 5th problem set
- 6th problem set
- 7th problem set
- 8th problem set
- 9th problem set
- 10th problem set (Note that you can do the programming part in Python, of course).
- 11th problem set
- 12th problem set (Fixed due date)

## Accompanying Material

### Lecture Notes

The lecture notes are available here. Lecture notes for Numerik I are also available here (only in German).

### Papers

- P. Kaps, P. Rentrop: Generalized Runge-Kutta Methods of Order Four with Stepsize Control for Stiff Ordinary Differential Equations Numer. Math. 33(1), pp. 55-68 (1979).
- E. Hairer, Ch. Lubich: Asymptotic Expansions of the Global Error of Fixed-Stepsize Methods Numer. Math. 45(3), pp. 345-360 (1984)

### Python

- Think Python 2nd Ed. (Free introduction into programming in general and Python in particular).
- Numpy for Matlab users (Helps you to transit from Matlab to Python (or rather, Numpy).

## Literature

The following selection of textbooks can be found in the library.

- Deuflhard, Peter: Newton Methods for Nonlinear Problems. Springer, Berlin, 2004.
- Deuflhard, Peter und Folkmar Bornemann: Numerische Mathematik II - Gewöhnliche Differentialgleichungen. Walter de Gruyter, Berlin, 2002.
- Deuflhard, Peter und Folkmar Bornemann: Scientific computing with ordinary differential equations. Springer, Berlin, 2002.
- Deuflhard, Peter und Andreas Hohmann: Numerische Mathematik I - Eine algorithmisch orientierte Einführung. Walter de Gruyter, Berlin, 2002.
- Golub, Gene und Charles Van Loan: Matrix computations. Johns-Hopkins-University Press, Baltimore, 1993.
- Hairer, Ernst, Syvert Paul Nørsett und Gerhard Wanner: Solving Ordinary Differential Equations I - Nonstiff Problems, Band 8 der Reihe Springer Series in Computational Mathematics. Springer, Berlin, Heidelberg, New York, 1987.
- Hairer, Ernst und Gerhard Wanner: Solving Ordinary Differential Equations II - Stiff and Differential-Algebraic Problems, Band 14 der Reihe Springer Series in Computational Mathematics. Springer, Berlin, Heidelberg, New York, 1991.
- Kunkel, Peter, and Mehrmann, Volker. Differential-algebraic equations: analysis and numerical solution. Vol. 2. European Mathematical Society, 2006.
- Meister, Andreas: Numerik linearer Gleichungssysteme. Vieweg, Braunschweig, 1999.
- Ortega, James und Werner Rheinboldt: Iterative solution of nonlinear equations in several variables. Academic Press, New York, 1972.
- Quarteroni, Alfio, Riccardo Sacco und Fausto Saleri: Numerische Mathematik 1. Springer, Berlin, 2002.
- Quarteroni, Alfio, Riccardo Sacco und Fausto Saleri: Numerische Mathematik 2. Springer, Berlin, 2002.
- Stoer, Josef und Roland Bulirsch: Numerische Mathematik - eine Einführung, Band 1. Springer, Berlin, 2005. aus dem FU-Netz auch Online verfügbar: Springer Link
- Stoer, Josef und Roland Bulirsch: Numerische Mathematik - eine Einführung, Band 2. Springer, Berlin, 2005. aus dem FU-Netz auch Online verfügbar: Springer Link
- Walter, Wolfgang: Gewöhnliche Differentialgleichungen - eine Einführung. Springer, Berlin, 1996.
- Werner, Dirk: Funktionalanalysis. Springer, Berlin, 2000.

Further literature on numerics is deposited under the shelf marks H.1.0 and H.1.1.

Introductory literature in computer science can be found under XA.1, introductions in Unix operating system under XD.4.0.

## Contact

Prof. Dr. Ralf Kornhuber | ralf.kornhuber@fu-berlin.de | Arnimallee 6, Room 130 open consultation hours: tba |

Lasse Hinrichsen-Bischoff (Assistent) | l.hinrichsen@fu-berlin.de | Arnimallee 6, Room 122 consultation hours: on appointment |