Abstract  The course gives an introduction into fundamental techniques and algorithms of numerical mathematics which play a central role in numerical simulations in science and technology. The course focuses on fundamental ideas and algorithmic aspects of numerical methods. The exercises involve actual implementation of numerical methods in C++. 
Objective  * Knowledge of the fundamental algorithms in numerical mathematics * Knowledge of the essential terms in numerical mathematics and the techniques used for the analysis of numerical algorithms * Ability to choose the appropriate numerical method for concrete problems * Ability to interpret numerical results * Ability to implement numerical algorithms afficiently 
Content  First two weeks: A gentle introduction to C++
1. Computing with Matrices and Vectors 1.1 Fundamentals 1.2 Software and Libraries 1.4 Computational Effort 1.5 Machine Arithmetic and Consequences
2. Direct Methods for (Square) Linear Systems of Equations 2.1 Introduction: Linear Systems of Equations 2.3 Gaussian Elimination 2.6 Exploiting Structure when Solving Linear Systems 2.7 Sparse Linear Systems
3. Direct Methods for Linear Least Squares Problems 3.1 Least Squares Solution Concepts 3.2 Normal Equation Methods 3.3 Orthogonal Transformation Methods 3.3.1 Transformation Idea 3.3.2 Orthogonal/Unitary Matrices 3.3.3 QRDecomposition 3.3.4 QRBased Solver for Linear Least Squares Problems 3.4 Singular Value Decomposition
4. Filtering Algorithms 4.1 Filters and Convolutions 4.2 Discrete Fourier Transform (DFT) 4.3 Fast Fourier Transform (FFT)
5. Machine Learning of OneDimensional Data (Data Interpolation and Data Fitting in 1D) 5.1 Abstract Interpolation (AI) 5.2 Global Polynomial Interpolation
8. Iterative Methods for NonLinear Systems of Equations 8.1 Introduction 8.2 Iterative Methods 8.3 FixedPoint Iterations 8.4 Finding Zeros of Scalar Functions 8.5 Newton’s Method in Rn 8.6. QuasiNewton Method 
Lecture notes  Lecture materials (PDF documents and codes) will be made available to the participants through the course web page and online repositories. Access information will be communicated in the beginning of the course. 
Literature  U. ASCHER AND C. GREIF, A First Course in Numerical Methods, SIAM, Philadelphia, 2011.
A. QUARTERONI, R. SACCO, AND F. SALERI, Numerical mathematics, vol. 37 of Texts in Applied Mathematics, Springer, New York, 2000.
W. Dahmen, A. Reusken "Numerik für Ingenieure und Naturwissenschaftler", Springer 2006.
W. Gander, M.J. Gander, and F. Kwok "Scientific Computing", Springer 2014.
M. HankeBourgeois "Grundlagen der Numerischen Mathematik und des wissenschaftlichen Rechnens", BG Teubner, 2002
P. Deuflhard and A. Hohmann, "Numerische Mathematik I", DeGruyter, 2002 
Prerequisites / Notice  The course will be accompanied by programming exercises in C++ relying on the template library EIGEN. Familiarity with C++, object oriented and generic programming is an advantage. Participants of the course are expected to learn C++ by themselves, in case they do not know it already. 
Competencies  Subjectspecific Competencies  Concepts and Theories  assessed   Techniques and Technologies  assessed  Methodspecific Competencies  Analytical Competencies  assessed   Decisionmaking  fostered   Problemsolving  assessed   Project Management  fostered 
