 # Numerical analysis and optimisation

## Presentation

Programme (detailed contents):

Numerical Analysis :

−        Interpolation: Vand der Monde’s method, Lagrange’s method, Newton’s method and piece by piece interpolation,

−        Numerical integration: trapezoidal rule, Simpson’s method and Gauss’ methods,

−        The LU decomposition as well as the Cholesky factorization for solving linear systems,

−        The fixed point method and Newton’s method for solving nonlinear systems,

Optimization :

-          Introduction to numerical optimization, differential calculus

-          Definition of a local/global minimum/maximum, definition and characterization of convexity

-          Necessary optimality conditions of the first and the second order

-          Gradient methods (constant step and steepest descent), Newton method, (linear and nonlinear) least square methods

Organisation:

This module  is  organized in course and labworks..

Main difficulties for students: No difficulties for the ones regularly working.

## Objectives

 At the end of this module, the student will have understood and be able to explain (main concepts): Numerical Analysis : −        Some polynomial interpolation technics, −        Different methods for the numerical integration, −        Numerical errors and the problem of numerical stability through the condition number, −        The LU decomposition as well as the Cholesky factorization for solving linear systems, −        The fixed point method and Newton’s method for solving nonlinear systems,   Optimization : Introduction to unconstrained numerical optimization. The differentiable case. -          Concept of local extremum, inroduction to convexity -          Necessary optimality conditions -          Gradient methods, Newton method, least square problems       The student will be able to: Numerical Analysis : To be able to choose and to implement efficient numerical methods: to numerically compute an integral, to solve linear and nonlinear systems.   Optimization: To be able to choose and implement a suitable algorithm for solving a given unconstrained optimization problem.

## Needed prerequisite

- Precedent courses on the following subjects : linear algebra

- Differential Calculus from the 2nd year

## Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...

## Bibliography

Polycopié remis.

Trefethen & Bau, Linear Numerical Algebra, SIAM, ISBN 0-89871-361-1

Lascaux & Theodor, Analyse numérique matricielle appliquée à l'art de l'ingénieur-2, ISBN 2-225-84122-5