Numerische Lineare Algebra - Lineare Gleichungssysteme und Matrixgleichungen SS 2017
Wir diskutieren die numerische Loesung von Matrixgleichungen und linearen Gleichungssystemen
| Termine: | Montags, | 7:30 bis 9:00 | G05-208 | ||
| Freitags, | 13:00 bis 15:00 | G02-210 | |||
| Übung: | Jede 2. Woche bzw. nach Ankündigung | ||||
| Nächste Termine: | 30.6.,3.7. (V), 7.7. (Ü) | ||||
| Sprache/Language: | English, | ||||
Content
- Linear Systems:
- Direct solvers
- Stationary iterations
- Krylov methods (CG, MINRES,GMRES)
- Preconditioning (basic algorithms, saddle point examples)
- Multigrid
- Matrix equations:
- Introduction to matrix equations
- Direct solvers for linear matrix equations
- Low-rank solver for large-scale linear matrix equations
- Methods for algebraic Riccati equations
Certificate of attendance (Teilnahmeschein)
Criteria for getting a course assessment:
- at least 50% of all possible points from the homework
- at least 50% of the points in one homework ( on n-1 of n homework sheets)
- demonstrate and explain at least one exercise in the tutorial
Even if you don't need a certificate, we warmly encourage you to work on the homework assignments. It will be invaluable for understanding.
Literature
Linear Systems:
- Trefethen, Bau, III.: Numerical Linear Algebra, SIAM, Philadelphia, 1997.
- Elman, Silverster, Wathen: Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, Oxford University Press, 2005.
- Saad: Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, 1996.
- Greenbaum: Iterative methods for solving linear systems, SIAM, Philadelphia, 1987.
- Meister: Numerik linearer Gleichungssysteme, Vieweg+Teubner, 2011.
Matrix Equations:
- B.N. Datta: Numerical Methods for linear Control Systems, Academic Press, 2003.
- A.C. Antoulas: Approximation of Large-scale Dynamical Systems, SIAM, 2005.
- Y. Zhou: Numerical Methods for Large-scale Matrix Equations with Applications in LTI System Model Reduktion, PhD Thesis, Rice University 2002.
- J. Saak: Efficient Numerical Solution of Large-scale Matrix Equations in PDE Control and Model Order Reduction, PhD Thesis, TU Chemnitz, 2009.
- G. Golub, C. Van Loan: Matrix Computations, 3. Aufl., The John Hopkins University Press, 1996.
- Petersen & Pederson: The Matrix Cookbook.
- K. Schäcke: On the Kronecker Product, 2004.
- M. Marcus & H. Minc: A Survey of Matrix Theory and Matrix inequalities, Allyn and Bacon, 1964.
- D. A. Bini, B, Iannazzo, B. Meini: Numerical Solution of Algebraic Riccati Equations, SIAM, 2012.
Useful Links
- Matlab introductions:
- Short intro by M. Pester (in german)
- Practical introduction by M. Gockenbach
- Matlab introduction by MIT
- Matlab primer
- More on GMRES:
- Original paper: Saad/Schultz: GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems.
- Seminal work on GMRES convergence: Greenbaum/Pták/Strakoš: Any Nonincreasing Convergence Curve is Possible for GMRES.