Digital Analysis

The aim of this module is to introduce the basic tools of numerical analysis: interpolation, quadrature (numerical integration) and direct methods for solving linear systems.
Outline:
1°) Interpolation:

- Lagrange interpolation: Lagrange polynomials, split differences, interpolation remainder, Runge phenomenon

- Hermite interpolation: basic polynomials, generalization of divided differences. 

2°) Quadrature:    

- Principle and definitions: elementary quadrature formula, compound formula, degree of accuracy, order  

- Classical quadrature methods: rectangles, trapezoids, Simpson order, error increase results

- Gauss method: obtaining the optimal order formula (Gauss-Legendre), generalizations: formulas with constraints (e.g. Gauss-Lobatto) or formulas for other scalar products (e.g. Gauss-Laguerre). 

 3°) Direct methods for solving linear systems:

- LU decomposition: principle, algorithm, cost, variant with permutations 

- Cholesky decomposition: algorithm, cost, interest