Numerical analysis [SMATB303]

Learning outcomes

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).

Objectives

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.).

Content

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.

Table of contents

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

Exercises description

Exercise sessions are given for 1h30 per week.