Prerequisites
Students should have linear algebraic capabilities that surpass the mere ability
to perform linear algebraic computations 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.
This requires that apart from a first-year BSc course in Linear Algebra, student
has followed an advanced course in Lineair Algebra or a BSc course in Numerical
Linear Algebra or even Representation Theory. Moreover, they have successfully
passed a course in Numerical Mathematics that includes the formal definitions of
conditioning of a mathematical problem and backward stability of an algorithm
to solve that problem.
Students know how to use and compute LU- Cholesky, and QR-factorizations. They
have worked with plane rotations (Givens) and reflectors in hyperplanes (House-
holder) and know how these generate the (special) orthogonal and unitary groups.
Knowledge of other matrix Lie groups is a plus but not strictly necessary.
Students know the Spectral Theorems for selfadjoint, normal and unitary linear
transformations. They know and understand the Schur-, Jordan-, and Singular
Value factorizations and know how to compute them by hand for small matrices.
They know the Power Method, the Rayleigh Quotient Iteration, and the QR-iteration
for the approximation of eigenpairs.
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.
OBTAINING AND TESTING YOUR ASSUMED PRIOR KNOWLEDGE:
The first two lectures will be spent on reviewing this material. Note that reviewing
is not the same as explaining in detail. If you have not seen the material before it
may be hard to absorb everything in just these two weeks.
Long before the course starts, a yes/no quiz will be placed on the elo-website of
the course with 30 easy questions about the prerequisites. If they do not seem easy,
please reconsider taking this course, or work hard to get to the required level.
Aim of the course
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 aim is to teach students how to approximate solutions of large scale linear
algebra problems by cleverly designed small scale linear algebra problems, how
to analyse the approximation properties mathematically, and how to iplement the
corresponding methods in MatLab. Students are taught how to perform experiments
in MatLab and how to discuss their outcomes.
The focus will be on mathematical theorems and proofs. Instead of covering a
large number of algorithms, we study a smaller number of central algorithms in
greater detail, from defining mathematical principles via algorithms to their
efficient and stable implementation.
This course is part of Master Programmes in Mathematics and can be of added
value in the other MasterMath courses Parallel Algorithms, Systems and Control,
and Numerical Bifurcation Analysis of Large-scale systems. It also supplements
Numerical Methods for PDEs (stationary or time-dependent).
Lecturer
Jan Brandts, Korteweg-de Vries Institute for Mathematics, UvA
- Docent: Jan Brandts