The Swiss engineering master's degree

After successful studying students are capable to solve selected practical mathematical problems by combining appropriate numerical methods with suitable computer algebra tools. Moreover, students know how to interprete and visualize computational outcomes resulting from numerical algorithms.

Requisiti

Linear Algebra

• Algebra with vectors and matrices
• Elementary solving linear systems of equations (Gauss Pivoting)
• Eigenvectors and Eigenvalues

Analysis

• Univariate and multivariate calculus (differentials, integrals)
• Knowing of simple numerical recipes for equations and integrals (e.g. Bi-Section, Newton, Trapezoidal-Rule, Simpson-Rule...)
• Ordinary differential equations including simple numerical recipes (e.g. Euler)

Basics in Computer Handling

• Operating system, software installation
• Elementary skills in procedural programming

Hardware and Software

• Notebook
• Mathematical software installed (e.g. Mathematica, Matlab, Maple ... according to preference and experience)

Obiettivi di apprendimento

Solving mathematical problems with practial relevance by

• capable handling a computer algebra system (CAS) or appropriate mathematical software
• mastering selected numerical methods

Knowing limits of computer based methods and comprehension of

• some internals of CAS (e.g. representations of numbers and functions)
• the problems of numerical stability, errors from rounding and discretization
• algorithmic complexity (e.g. convergence speed)

Combining analytical methods of CAS with efficient numerical software

Interpreting and visualizing computational results

Contenuti del modulo

Processing

• data from problems with practical relevance
• by tools from numerical mathematics and analytics
• up to interpretation and visualization of results

Based on a selection of methods listed below

• Solving systems of linear equations (LU-Decomposition, Cholesky Decomposition, Householder Transformations, QR Decomposition, sparse matrix strategies and Gauss-Seidel ...)
• Computations of zeroes and non-linear optimization
• Univariate and multivariate interpolation and approximation (Collocation, Osculation, Splining, Least-Squares Approximation, Chebyshev Approximation ...)
• Numerical differentiation and integration
• Initial and boundary value problems of ordinary differential equations

With consideration of

• Accuracy, efficiency and condition
• Problem identification and method selection
• Computeralgebra in order to establish analytical relations

Metodologie di insegnamento e apprendimento

• Derivation of mathematical facts in lectures
• Software demonstrations and visualizations by the lecturer during the lectures
• Teaching based on problems with practical relevance
• Software examples and additional materials on complimentary website (Zuerich)
• Hints to sources and literature on complimentary website (Zuerich)
• Self-studies based on sources and literature
• Doing homework as a preparation for dedicated exercise lessons

Bibliografia

• Schaum’s Outlines of Numerical Analysis, McGraw-Hill Professional, 2nd edition
• Schwarz, Hans R.; Köckler, Norbert; Numerische Mathematik, Vieweg & Teubner, 7. Auflage
• Bronstein et al., Taschenbuch der Mathematik, Harri Deutsch
• Bradie, Brian, A Friendly Introduction to Numerical Analysis, Prentice-Hall
• Alfio Quarteroni, Riccardo Sacco, Fausto Saleri, M´ethodes Num´eriques - Algorithmes, analyse et applications, Springer, 2007
• Jean-Philippe Grivet, Méthodes numériques appliqués, EDP sciences
• Koepf, Wolfram, Computeralgebra, Springer
• Moler Cleve, Numerical Computing with Matlab, www.mathworks.com/moler/chapters.html
• Erwin Kreyszig, Advanced Engineering Mathematics, Wiley
• Erwin Kreyszig, Advanced Engineering Mathematics – Students Solution Manual and Study Guide, Wiley
• Erwin Kreyszig/E.J. Norminton, Mathematica Computer Guide for Erwin Kreiszigs Advanced Engineering Mathematics, Wiley
• Michael Trott, The Mathematica Guide Book for Numerics, Springer