This course covers the basic tools of numerical analysis by relying on a complete theory (definitions and properties) and various examples (in the theoretical lectures and more actively in the exercise sessions).
This course aims to familiarise students with the approach to solving mathematical problems using numerical methods and to develop the critical thinking linked to this approach (error analysis, quality of the numerical solution, etc.).
The first part of this course gives an introduction to the concepts of numerical analysis (error analysis, numerical algorithm). The second and main part of the course deals successively with the numerical approximation of functions in the sense of Tchebycheff and least squares, interpolation and quadrature.
Part I: Introduction to numerical analysis
A. Error analysis
1) Approximation error
2) Rounding error
B. Numerical algorithms
1) Definition and characteristics (complexity, etc.)
2) Stability of an algorithm
3) Libraries and software
Part II: Approximation
A. Finite-dimensional linear approximation
1) Tchebycheff approximation
2) Least squares approximation
3) Orthogonal polynomials
4) Tchebycheff series
B. Interpolation
1) Lagrange interpolation
2) Hermit interpolation
3) Interpolation error
4) Optimal choice of Lagrange interpolation nodes
5) Piecewise polynomial interpolation -- Splines
C. Quadrature
1) Choice of weights -- Newton-Cotes quadrature
2) Choice of nodes -- Gauss quadrature
Exercise sessions are given for 1h30 per week.
Formula: Two exams per session: the first one on the theory is an oral exam and the second one for the exercises is either a written exam on computer or a work to be presented orally.
Modality: The teaching unit (TU) includes two learning activity assessments (LAA) per session: one on the theory covered in the course, the other on exercises. The TU will be considered as passed if the arithmetic average of the two marks obtained for each A.A. reaches at least 10/20. During the same academic year, the student is exempted from repeating the assessment of one of the two A.A. if it is passed (10/20) and provided that he/she presented both parts the first time.
Slides provided before the course.
| Training | Study programme | Block | Credits | Mandatory |
|---|---|---|---|---|
| Bachelier en sciences mathématiques | Standard | 0 | 5 | |
| Bachelier en sciences mathématiques | Standard | 3 | 5 |