The course deals with the following subjects:
1 Error Analysis
Representation of numbers, Floating-point-numbers, Rounding errors, Error Propagation, Error propagation in arithmetic operations, Condition numbers
2 Nonlinear equations
The method of Bisection, The secant method, Newton‘s method, Fixed point iteration, Polynomial equations, Systems of nonlinear equations, Newton‘s method for systems
3 Systems of Linear Equations
The LR and Cholesky Decomposition, The LR-Decomposition, The Cholesky Decomposition, Gauss Elimination and Back-Substitution, Pivoting strategies, The QR Decomposition, Data fitting; Least square problems, lterative solutions, Jacobi Iteration (total-step-method), Gauss-Seidel-Iteration (single-step-method), Convergence properties
4 Finding Eigenvalues
The Power method, Localizing eigenvalues, The QR-method, Hessenberg matrices
5 Ordinary Differential Equations
Basic analytic methods, Separation of variables, Linear differential equations, One-step-methods, Euler‘s Method, Midpoint Euler, Two-stage-models, Runge-Kutta-methods
6 Polynomial Interpolation
Lagrange form of Interpolation Polynomial, Interpolation Error, Divided Differences, Spline Interpolation
7 Numerical Integration
The students should learn, to solve typical problems in engineering-mathematics by numerical methods, among others: Linear and nonlinear systems, eigenvalues, interpolation, differential equations and integration. They should learn to implement general methods into a practical computation and to evaluate them with respect to accuracy and efficiency.
·1 Gautschi, W. Numerical Analysis, Birkhäuser,1997.
·2 Hammerlin und Hoffmann. Numerische Mathematik, Springer,1994.
·3 Householder. A.S. Principles of Numerical Analysis, Dover Publications,1974.
·4 Kincaid,D. and Cheney, W. Numerical Analysis, Brooks/Cole Publishing,1991.
·5 Locher. Numerische Mathematik für Informatiker,1993.
·6 Philipps,C. and Cornelius, B. Computional Numerical Methods, Ellis Hoorwood.
·7 Stoer, J. and Burlisch, R. Introduction to numerical Analysis,2005.