## 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 APB-E001 (computer science building) / **FIRST LECTURE: OCT 8**

Exercises: Thursday, 3. DS (11:10-12:40) in APB-E006/U

**Format**

2 SWS lecture, 2 SWS exercise, self-study

5 credits

**Exam**

**Tuesday, February 19, 2019, 09:20-10:50h, APB/E006 (computer science building)**

If there are more than 10 registered students, the module examination consists of a written examination, with a duration of 90 minutes. If there are 10 or fewer registered students, it consists of an oral examination as an individual examination performance amounting to 30 minutes; this will be announced to the enrolled students at the end of the enrollment period.

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)

**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: Suryanarayana Maddu

**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, Solutions 01 PDF)
- Lecture 2 - linear systems of equations, LU decomposition of matrices, Gaussian elimination, iterative linear solvers, Jacobi method (Exercise 02 PDF, Solutions 02 PDF)
- Lecture 3 - Gauss-Seidel method, SOR method, Conjugate gradient methods, preconditioning schemes (Exercise 03 PDF, Solutions 03 PDF)
- Lecture 4 - Least-Squares methods, QR decomposition, singular value decomposition (Exercise 04 PDF, Solutions 04 PDF)
- Lecture 5 - Non-linear least squares, non-linear equations, Newton method, bisection method, secant method (Exercise 05 PDF, Solutions 05 PDF)
- Lecture 6 - Non-linear systems of equations, quasi-Newton method, rank-1 update, Broyden algorithm, Lagrange interpolation, barycentric interpolation (Exercise 06 PDF, Solutions 06 PDF)
- Lecture 7 - Interpolation algorithms: Aitken-Neville algorithm, Hermite and Spline interpolation (Exercise 07 PDF, Solutions 07 PDF)
- Lecture 8 - Trigonometric interpolation: Discrete Fourier transform and fast Fourier transform algorithms (Exercise 08 PDF, Solutions 08 PDF)
- Lecture 9 - Numerical integration (quadrature): trapezoidal rule, Simpson rule, Romberg extrapolation, Gauss quadrature (Exercise 09 PDF, Solutions 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)
- Lecture 11 - second-order schemes, Higher-order schemes, Runge-Kutta schemes
- Lecture 12 - Butcher tables, variable step size control, implicit schemes
- Lecture 13 - Systems of ODEs, Adams-Bashforth formula, numerical stability, stiff problems
- Lecture 14 - Partial differential equations 1, method of lines, parabolic problems (diffusion), hyperbolic problems (waves)
- Lecture 15 - Partial differential equations 2, elliptic problems (Poisson equation), finite difference schemes, stencil methods, Courant-Friedrichs-Levy condition