**Prerequisites and Assumed Prior Knowledge**

Students should have linear algebraic capabilities that surpass the mere ability to perform linear algebraic computions and that include geometric intuition in normed spaces and inner product spaces. They should be acquainted with the basic principles of numerical mathematics and have programming skills that allow them to work in MatLab or to learn it quickly.

This usually requires that apart from a first-year BSc course in Linear Algebra, the student has followed an advanced course in lineair algebra and/or a course in numerical linear algebra and/or even a course in representation theory. They have also followed an introductory course in Numerical Mathematics, preferablyone that includes the concepts of:

1) Finite Precision Arithmetic

2) Conditioning of a problem, stability of an algorithm

in the context of LU- and QR-factorization. They know elementary unitary maps like plane rotations and reflectors in hyperplanes and their matrices, know the spectral theorems for selfadjoint, normal and unitary linear transformations and the Schur-, Jordan-, and Singular Value Decomposition.

These prerequisites and assumed prior knowledge can for example be obtained from:

[1] L.N. Trefethen and D. Bau (1997). Numerical Linear Algebra, SIAM Society for Industrial and Applied Matematics. Lectures 1-31.

[2] A. Quarteroni, R. Sacco and F. Saleri (2006). Numerical Mathematics. Springer Verlag, 2nd edition. Chapters 1-5.

**Aim**

This course is a first introduction into the main aspects of iterative methods to approximate the solutions of finite- but high-dimensional linear equations, eigenvalue-, and singular value problems. Many of these methods are based on the clever reduction of the problem to an approximating problem of much smaller

dimensions. The smaller problem yields an approximate solution of the original problem but simultaneously provides information how to set up the next reduced problem whose corresponding approximation is better than the previous one. This leads to a sequence of smaller problems that need to be solved in order to get increasingly better approximations of the solution of the original problem.

The course includes the following topics: iterative, subspace, and in particular Krylov methods, such as CG, GMRES, Lanczos, Arnoldi and Jacobi-Davidson methods. Abstract embedding. Analysis of stability and convergence.

The course can be part of an MSc program in Applied Mathematics and will be of great added value in the other MasterMath courses Parallel Algorithms, Systems and Control, and Numerical Bifurcation Analysis of Large-scale systems. It also supplements Aplied Finite Elements and Numerical Methods for Time-dependent PDEs

- Docent: Jan Brandts