This course will be offered during period 2-3 every second year. The next start is autumn 2018.

This course is primarily intended for (but not restricted to) PhD students in the graduate program Applied and Computational Mathematics at KTH. PhD students in other programmes with a sufficient interest in mathematics and computation are also welcome.

The contents of the course SF2524 is a subset of the contents of this course. In addition to the material in SF2524, the course will involve

- extendend material and exercises related to:
- the QR-method (double shifts)
- Krylov methods (Krylov-Schur restarting, rational Krylov)
- Newton-type methods and Jacobi-Davidson method (for linear and nonlinear eigenvalue problems)
- methods for linear systems of equations (IDR, BiCGstab)
- applications of matrix functions (exponential integrators)

- numerical methods for matrix equations (such as the Lyapunov equation and the Sylvester equation).
- an individual project (related to the material of the course and/or the research of the PhD student) with oral and written presentation.

- Correctly solving the homeworks of SF2524 including additional questions. The homeworks need to be done in the julia programming language.
- A written exam or a take-home exam or an oral exam. A decision of which one will be made after a discussion with the students.
- Written and oral presentation of project

The lectures of SF2524 is a subset of the lectures in this course, including the first lecture. Note that the time of the first lecture was rescheduled some time ago. See schedule of SF2524.

Further lectures and exercises:
The additional lectures will take place in Dec-Jan 2018. Further information t.b.a.

The course will involve an individual project, which should be presented in February 2018 (preliminarily).

- Golub and Van Loan, Matrix computations, 2013 (4th edition)
- Nicholas Higham, Functions of matrices, 2008
- G.W. Stewart, Matrix algorithms, Volume 2: Eigensystems, 2001
- D. Watkins, Fundamentals of Matrix Computations, (3rd edition), 2010
- J. W. Demmel, Applied numerical linear algebra, 1997
- Å Björck, Numerical Methods in Matrix Computations, 2015

- Lecture notes / hand-outs: QR-methods (preliminary version), convergence of Arnoldi method, matrix equations (will be added at a later stage)
- Lecture notes / hand-outs: Matrix equations (preliminary version)
- Lecture notes / hand-outs: convergence of Arnoldi method,
- Article: A new iterative method for solving large-scale Lyapunov matrix equations, V. Simoncini, SIAM Journal on Scientific Computing, 29 (3), 1268-1288
- Article: A Jacobi-Davidson iteration method for linear eigenvalue problems, G.L.G. Sleijpen and H.A. van der Vorst, SIAM J. Matrix Anal. Appl. 17 (2): 401-425
- Other articles

- Rational Krylov
- Krylov-Schur restarting
- Bartel-Stewart for Sylvester equations
- Harmonic eigenvalue extraction
- Preconditioning of a specific PDE
- GMRES for non-symmetric matrices which are symmetric with respect to a non-standard inner product form
- Specialized matrix function algorithms for, e.g., matrix square root,
- Algorithms for the Ricatti matrix equation
- Model reduction with balanced trunction and Lyapunov equations

- An enthusiastic TED-talk about the beauty of matrices: TED-talk by Margot Gerritsen (Stanford).
- This course should not be confused with the course (with the same name) offered by SeSE described here.