Numerical linear algebra
Numerical linear algebra SF3580 - PhD level course
This course will be offered during period 2-3 every second year, every even year, 2018,2020,...
This course is primarily intended for (but not restricted to) PhD students in
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
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.
Further information will
be provided at a later stage.
This is a pass/fail course. Course requirements:
- 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
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).
The literature of SF2524
is a subset of the literature in this course. We
will additionally use some chapters
from the books:
- 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
The topic of the project should connect with a specific technique
or method of the course. Examples (references available):
- 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