Teaching plan for the course unit

 

 
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General information

 

Course unit name: Numerical Linear Algebra

Course unit code: 572661

Academic year: 2021-2022

Coordinator: Arturo Vieiro Yanes

Department: Faculty of Mathematics and Computer Science

Credits: 6

Single program: S

 

 

Estimated learning time

Total number of hours 150

 

Face-to-face and/or online activities

60

(face-to-face and online)
 

-  Lecture with practical component

Face-to-face

  

30
 

(face-to-face and online)
 

-  IT-based class

Face-to-face

  

30
 

(face-to-face and online)

Supervised project

40

Independent learning

50

 

 

Teaching blocks

 

1. Basic definitions

1.1. Vectors and matrices. Rank and null-space. Norms.

1.2. Linear systems.

1.3. Stability. Condition number. Complexity of an algorithm.

1.4. Structured matrices. Blocking algorithms.

2. Linear systems

2.1. Gaussian methods, LU, Cholesky.

2.2. Orthogonalization methods: QR factorization (Gram-Schmidt, Householder). Application to the least squares problem (LSP).

2.3. Iterative methods: Jacobi, Gauss-Seidel, successive over-relaxation method (SOR).

2.4. Introduction to Krylov methods (Lanczos, Arnoldi, GMRES,...)

3. Eigenvalues and eigenvectors

3.1. Power method

3.2. LR and QR iteration

4. Singular value decomposition (SVD).

4.1. Singular values and vectors.

4.2. Applications to LSP and graphic compression.

4.3. Principal component analysis (PCA).

 

 

Teaching methods and general organization

 

The teaching methodology  consists of:

  •      Two hours of master classes per week
  •      Two hours of computer laboratory per week
  •      Supervised personal work on the projects to solve
  •      Autonomous independent learning


During the lectures, the lecturer will explain the definitions and main results of the syllabus, which will be illustrated with examples. Several practical situations, exercises and implementation tricks will be discussed.

During the semester some short projects will be stated. The students should work around each project and implement the codes needed to solve the proposed exercises. A short summary of the methods used and the results obtained will be required for grading, as well as the codes implemented.

 

 

Official assessment of learning outcomes

 


To succeed in the assessment of the subject, students must show a good understanding of the foundations of the algorithms presented in the lectures, including coding details of the algorithms, and a good ability to solve concrete problems.

Continuous assessment is based on the completion of some projects (2 or 3) throughout the course, for which a report of the methodology as well as a short summary of the results must be handed in to the teacher together with the developed code necessary for their solution. To solve these problems student will receive advise to face the difficulties encountered, these being either theoretical, concerning the implementation process or the interpretation of results. The global mark for the projects comprises 50% of the final grade.

The marks awarded for the final exam make up the remaining 50%.

To approve the course, the realization of all the assigned projects is mandatory and a minimum mark of 3,5/10 in both the projects and the final exam is needed.

 

Examination-based assessment

Students who wish to opt for a single assessment must inform the Secretary by the date set in the Faculty calendar.


Single assessment consists of projects (50% of the final grade) and an on-site examination (50% of the final grade).

To approve the course with this modality, the realization of all the assigned projects is also mandatory and a minimum mark of 3,5/10 in both the projects and the final exam is needed.