1. PREREQUISITES:
Students already 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, to which they should moreover have access.
[Why MATLAB? Because all examples and demonstrations will take place in MATLAB and the computational assessments are formulated in MATLAB]
This requires that apart from a first-year Bachelor course in Linear Algebra, student has followed an advanced course in Linear Algebra or a Bachelor course in Numerical Linear Algebra or Representation Theory. Moreover, they have successfully passed a course in Numerical Mathematics that includes the mathematical definitions of conditioning of a mathematical problem and of backward stability of an algorithm to solve that problem. Furthermore, student is able to work efficiently in MATLAB.
Students already know how to use and compute LU-, QR-, and Cholesky factorizations. They have worked with plane rotations (Givens) and with (Householder) reflectors in hyperplanes and know how these generate the (special) orthogonal and unitary groups. Students understand the Schur matrix factorization and is able to prove the Spectral Theorems for Hermitian, normal and unitary linear transformations. They have worked with basic iterative methods for linear systems (such as the Jacobi method) and for eigenvalue problems (such as the power method).
These prerequisites and assumed prior knowledge can for example be obtained from the combination of both:
[1] G.W. Strang (2023). Introduction to Linear Algebra (sixth edition). Wellesley-Cambridge Press, USA.
[2] L.N. Trefethen and D. Bau (1997). Numerical Linear Algebra, SIAM Society for Industrial and Applied Mathematics.
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2. 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 smallerdimensions. The smaller problem yields an approximate solution of the original problem and 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 implement 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 great detail, from defining mathematical principles via algorithms to their efficient and stable implementation.
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3 RULES ABOUT HOMEWORK AND EXAMS
The final grade for the course is composed from the following components:
10 points: For free.
30 points: Two assignments (0-15 points each), individual or in pairs, that combine theory, implementation, and doing/discussing numerical experiments, and that are handed in as a Zoom-recorded presentation of the assignment.
60 points: An individual written Exam (0-60 points) at the end of the course.
To pass:
1) The Final Exam should score at least 30 points.
2) The total should be at least 55 out of 100.
REPAIR OPTIONS
If you did not score 30 for the Final Exam, or if you did not score 55 out of 100 in total, you will need to do the resit for the exam, which can again give you 60 points.
The score for the resit should be at least 30 points and will be added to the scores of the two assignments and the ten free points. The total should be at least 55 points.
A repair option for an assignment may only be given if the Study Adviser of student's university will declare that due to special circumstances, student was not able to do the assignment optimally or hand it is in time. The lecturer should be informed of this by the Study Adviser as soon as possible.
If a repair option for an assignment is given, then this will only be done after the resit of the exam, and only if it is necessary and possible to pass the course.
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4. COURSE LITERATURE:
The course will be based mostly on the material provided by the lecturer, such as short videos, slides, quizzes, photographs of blackboards, notes, exercise sets, and the assignment texts.
In the two classics [3] [4] by the renowned expert Yousef Saad, much more details can be found. Both books [3] and [4] are available via SIAM and bol.com but updated pdf files are available as well directly from the web-page of author Yousef Saad at http://www-users.cs.umn.edu/~saad/books.html
[3] Yousef Saad (2003).
Iterative Methods for Sparse Linear Systems.
SIAM Society for Industrial and Applied Matematics, 2nd revised edition.
[4] Yousef Saad (2011).
Numerical Methods for Large Eigenvalue Problems.
SIAM Society for Industrial and Applied Matematics, 2nd revised edition.
- Docent: Jan Brandts