Numerical analysis

5.00 credits

30.0 h + 22.5 h

Teacher(s)

Remacle Jean-François;

Language

French

Prerequisites

First cycle level in numerical calculus and programming (LEPL1104) and in linear algebra (LEPL1101).

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

Learning outcomes

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

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

Learning outcomes

Additionnal module in Mathematics

APPMATH

Minor in Applied Mathematics

LMINOMAP

Specialization track in Applied Mathematics

FILMAP