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Numerical Linear Algebra (Numeric Math.)
| Credits | 8 credit points |
| Instructors | Sleijpen, G.L.G. (Universiteit Utrecht) |
| G.L.G.Sleijpen@math.uu.nl | |
| Aim | To provide theoretical insight and to develop practical skills for solving numerically large scale linear algebra problems. Particular emphasis lies on large-scale linear systems and on eigenvalue problems. |
| Description | New developments in many applications, such as weather forecasting, airplane design, tomographic problems, analysis of the stability of structures, design of chips and other electrical circuits, etc, rely on numerical simulations. Such simulations require the numerical solution of linear systems or of eigenvalue problems. The matrices involved are sparse and high dimensional (1 billion is not acceptional). The solution of these linear problems are normally by far the most time-consuming part of the whole simulation. Therefore, the development of new solution algorithms is extremely important and forms a very active area of research. The course will give an overview of the modern solution algorithms for linear systems and eigenvalue problems. Modern approaches rely on schemes that improve approximate solutions iteratively. The course will start with a review of basic concepts from linear algebra, after which solution methods for dense systems (LU, QR and Choleski decomposition) will be discussed. Next, the basic ideas for iterative solution methods of sparse systems will be explained, which will lead to the main topic of the course: modern Krylov subspace methods. The main ideas of these methods will be explained and how they lead to efficient solvers. Solution algorithms for linear systems that will be discussed include CG, GMRES, CGS, Bi-CGSTAB, Bi-CGSTAB(l) and IDR(s). Furthermore several preconditioning and deflation techniques will be explained. For large scale eigenvalue problems the Lanczos methods, Arnoldi's method and the Jacobi-Davidson method will be treated. |
| Organization | Fourteen lectures, each consisting of instruction and theoretical and practical assignments. The practical assignments require programming in MATLAB. |
| Examination | Quiz, homework assignments and a final project assignment. |
| Prerequisites | Good knowledge of linear algebra and some experience in programming in MATLAB. |
| Remarks | Course homepage: http://www.math.uu.nl/people/sleijpen/Opgaven/NumLinAlg NOTE: The course starts in week 37 (September 12). |