## Basic Numerical Methods

Upon completing this module, the students will acquire the basics of numerical mathematics and numerical simulation methods. This includes the theoretical understanding of how a computer calculates with finite floating-point numbers and what kind of errors and inaccuracies may arise from these, and how to reduce or control them same. They will be familiar with basic numerical methods for modelling and simulating statistical models, linear algebra models, and ordinary and partial differential equations. They will be able to estimate the approximation errors of the methods and determine the algorithmic intensity, and will be able to implement these methods themselves.

**Contents**

Floating point arithmetic, rounding errors, cancellation, numerical interpolation (Lagrange, Newton, Splines), Taylor developments, finite differences and their approximation errors, explicit and implicit time integrators, direct and iterative algorithms for matrix inversion, matrix decomposition (LU), solution for the Poisson equation.

**Program / Module**

M.Sc. Computational Modeling and Simulation

Module: CMS-COR-NUM - Basic Numerical Methods

**Time/Place**

**Winter Term**

Lecture: Mondays, 5. DS (14:50-16:20) in SCH-A316 (Georg-Schumann-Bau) / **FIRST LECTURE: OCT 14**

Exercises: Thursday, 3. DS (11:10-12:40) in HSZ/003 (Hörsaalzentrum). 17.10.2019: VMB/0E02/U (von-Mises-Bau)

**Format**

2 SWS lecture, 2 SWS exercise, self-study

5 credits

**Exam**

**Monday, February 10, 2020, 09:20-10:50h, M13/DÜLF/U (Dülfersaal, Mensa Mommsenstr. 13)**

**Since there are more than 10 registered students, the module examination consists of a written examination with a duration of 90 minutes.**

At the exam, the following may be used:

- 4 A4 sheets (8 pages if you print duplex) of hand-written summary. We recommend writing the summary by hand, but it can also be machine-written. In the latter case, the font size must be 8 points or larger throughout.
- A standard pocket calculator (devices with network or bluetooth access, as well as devices capable of storing and displaying documents are not allowed)

**Exam Review**

You can come and look at your exam, and ask questions about its correction and the answers given during the following times:

- T.B.A.

All exam check session are going to happen at the CSBD (Pfotenhauerstr. 108) in the rooms of the Professorship.

**Registration to the course**

For students of the Master program "Computational Modeling and Simulation: via CampusNet SELMA

For students of the Computer Science programs: via jExam

**Teachers**

Lecture: Prof. Ivo F. Sbalzarini

Exercises: Abhinav Singh

**Teaching language:** ENGLISH

Lecture notes are available as PDF here.

Below is the weekly syllabus and the exercise/solution handouts:

- Lecture 1 - finite-precision arithmetics, IEEE number representation, roundoff and extinction, error propagation, condition numbers, backward error analysis (Exercise 01 PDF, Solution 01 PDF,
**ATTENTION: EXERCISE SESSION IS IN ROOM VMB/0E02/U (von-Mises-Bau)**) - Lecture 2 - linear systems of equations, LU decomposition of matrices, Gaussian elimination, iterative linear solvers, Jacobi method (Exercise 02 PDF, Solution 02 PDF,
**ATTENTION: EXERCISE SESSION IS IN ROOM HSZ/003 (Hörsaalzentrum)**) - Lecture 3 - Gauss-Seidel method, SOR method, Conjugate gradient methods, preconditioning schemes (Exercise 03 PDF, Solution 03 PDF)
- Lecture 4 - Least-Squares methods, QR decomposition, singular value decomposition (Exercise 04 PDF, Solution 04 PDF)
- Lecture 5 - Non-linear least squares, non-linear equations, Newton method, bisection method, secant method (Exercise 05 PDF, Solution 05 PDF)
- Lecture 6 - Non-linear systems of equations, quasi-Newton method, rank-1 update, Broyden algorithm, Lagrange interpolation, barycentric interpolation (Exercise 06 PDF, Solution 06 PDF)
- Lecture 7 - Interpolation algorithms: Aitken-Neville algorithm, Hermite and Spline interpolation (Exercise 07 PDF, Solution 07 PDF)
- Lecture 8 - Trigonometric interpolation: Discrete Fourier transform and fast Fourier transform algorithms (Exercise 08 PDF, Solution 08 PDF)
- Lecture 9 - Numerical integration (quadrature): trapezoidal rule, Simpson rule, Romberg extrapolation, Gauss quadrature (Exercise 09 PDF, Solution 09 PDF)
- Lecture 10 - Numerical differentiation: finite difference methods, Romberg extrapolation, Initial value problems of ordinary differential equations, the explicit Euler scheme (Exercise 10 PDF, Solution 10 PDF)
- Lecture 11 - second-order methods, Heun's method, Runge-Kutta methods, variable step size control, embedded Runge-Kutta, Richardson extrapolation (Exercise 11 PDF, Solution 11 PDF)
- Lecture 12 - implicit methods, multistep methods, Systems of ODEs, higher-order ODEs (Exercise 12 PDF, Solution 12 PDF)
- Lecture 13 - numerical stability, stiff problems, partial differential equations introduction (Exercise 13 PDF)
- Lecture 14 - Partial differential equations 1: parabolic problems, elliptic problems
- Lecture 15 - Partial differential equations 2: hyperbolic problems, Courant-Friedrichs-Lewy condition