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.

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 exceptional). 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.

**Prerequisites**

Good knowledge of linear algebra and some experience in programming in MATLAB.

The text "Preliminaries" collects the Linear Algebra prerequisites that are needed for this course. The material is presented in the form of exercises and can be used to “refresh” Linear Algebra knowledge and skills. Some issues in this collection may not belong to a standard Linear Algebra course. These less-current issues will be introduced and discussed in the course when needed.

On the course page you find a link to a Matlab Tutorial and also to a text with some simple ‘Matlab exercises’ (with some code) that may help you to familiarize you with Matlab and some of its peculiarities.

- Docent: Gerard Sleijpen