Numerical analysis

Note from June 29, 2020
Although we do not yet know how long the social distancing related to the Covid-19 pandemic will last, and regardless of the changes that had to be made in the evaluation of the June 2020 session in relation to what is provided for in this learning unit description, new learnig unit evaluation methods may still be adopted by the teachers; details of these methods have been - or will be - communicated to the students by the teachers, as soon as possible.

5 credits

30.0 h + 22.5 h

Teacher(s)

Henrotte François (compensates Remacle Jean-François);

Language

French

Prerequisites

First cycle level in numerical calculus and programming (LFSAB1104) and in linear algebra (LFSAB1101).

Main themes

Numerical methods for solving non-linear equations
Numerical methods for solving linear systems : iterative methods
Numerical methods for solving eigenvalue and eigenvector problems
Numerical solution of ordinary differential equations : initial value problems

Aims

At the end of this learning unit, the student is able to :

With respect to the AA reference, this course contributes to the development, acquisition and evaluation of the following learning outcomes :

AA1.1, AA1.2, AA1.3
AA2.1, AA2.4
AA5.2, AA5.3, AA5.5

More precisely, after completing this course, the student will have the ability to :

Analyze in depth the various key methods and algorithms for the numerical solution of important classes of problems from science and industry, related to applied mathematics
Better understand the numerical behavior of the various numerical algorithms for the solution of linear as well as nonlinear problems
Implement these methods in a high level computer language and verify their numerical behavior on a practical problem

Transversal learning outcomes :

Collaborate in a small team to solve a mathematical problem using numerical methods

The contribution of this Teaching Unit to the development and command of the skills and learning outcomes of the programme(s) can be accessed at the end of this sheet, in the section entitled “Programmes/courses offering this Teaching Unit”.

Content

Reminder of the basic notions of linear algebra (linear spaces, vector and matrix norms, ...)
Floating point calculations.
Stability, precision and conditioning of algorithms.
QR and SVD factorizations.
Linear systems of equations : direct methods. LU, Choleski, Pivoting, Renumbering (RCMK), direct resolution of sparse systems, Fill-in.
Iterative methods (Krylov subspaces) : iteration of Arnoldi, conjugate gradients, GMRES, Lanczos.
Preconditioning of iterative methods, preconditioned conjugated gradients.
Computing eigenvalues, QR algorithm

Teaching methods

Classes organized following the EPL guidelines.
Homeworks done individually
A more detailed organization is specified each year in the course plan provided on Moodle.

Evaluation methods

Exam (50% of the grade) and homeworks (50% as well)

Online resources

https://moodleucl.uclouvain.be/course/view.php?id=10034

Bibliography

http://bookstore.siam.org/ot50/

Nous suivons relativement scrupuleusement l'excellent ouvrage :
Trefethen, L. N., & Bau III, D. Numerical linear algebra (Vol. 50). Siam.

Teaching materials

http://bookstore.siam.org/ot50/

Faculty or entity

MAP

Programmes / formations proposant cette unité d'enseignement (UE)

Title of the programme

Sigle

Credits

Prerequisites

Aims

Master [120] in Statistic: General

STAT2M

Additionnal module in Mathematics

LMATH100P

Minor in Engineering Sciences: Applied Mathematics (only available for reenrolment)

LMAP100I

Minor in Applied Mathematics

LFSA136I

Specialization track in Applied Mathematics

LMAP100P