Description: |
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 Gaussian Quadrature |
Learning Targets: |
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. |
Literature: |
·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. |