This course is divided into two parts, the first part is dedicated to the digital resolution of systems linear. At the end of the first two chapters, the engineering student will know some methods direct and iteratives to solve a linear system. The second part deals with the theory of optimization without constraints and with constraints in denied dimension. Numerical optimization methods are also discussed for both optimization problems. The purpose of non lineair numerical analysis is to enable the student to acquire and implement the different methods used in numerical approximation. These are methods for resolving non-linear systems, polynomial interpolation and approximation, numerical integration, as well as methods of approximating solutions of ordinary dierential equations
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Numerical analysis
Mathematics is as much an aspect of culture as it is a collection of algorithms, CARL BOYER.
Content of the module
I) 1. Direct methods for solving linear systems
(a) Reminder on the Gauss removal method
(b) LU and Cholesky factorization of a matrix: living conditions and unicity,
(c) Algorithm and number of operations: band matrix case
2. Iterative methods for solving linear systems
(a) Matrix standards and packaging
(b) Jacobi and Gauss-Seidel methods: algorithms and convergence for matrices
a diagonal strictly dominant
(c) Relaxation methods
(d) Cases of symmetrical denies positive matrices
(e) Convergence rates: case of tridiagonal matrices
3. Optimization without constraints
(a) Existence of a minimum
(b) Problems of convex and unicite optimizations
(c) Optimal condition
(d) Problems of lesser square
(e) Downhill methods and algorithms for pitch gradient, pitch gradient and optimal pitch gradient
and the gradient combines Application and convergence for a quadratic problem
4. Optimization with constraints
(a) Existence and uniqueness of an optimization problem with constraints
(b) Euler's optimal condition and inequation
(c) Projection on a convex
(d) Application to a quadratic problem with linear constraints: Conditions of
Kuhn and Tucker
(e) Case of a quadratic problem
(f) Projected gradient method
(g) Uzuwa method
II) 1. Resolution of non-linear equations
(a) Introduction: local convergence, global convergence, order of convergence
(b) Bisection method
(c) Point method xe : convergence theorem, convergence order
(d) Newton's method
(e) Secante method
(f) Non-linear system
2. Polynomial interpolation
(a) Lagrange Interpolation
(b) Newton form and divided dierences. Dierences nis
(c) Interpolation error
(d) Hermit interpolation and interpolation error
3. Digital integration
(a) Introduction: quadrature formula, degree of a formula
(b) Newton-Odds formulas: error and degree
(c) Gauss quadrature formula: orthogonal polyn^omes, error and degree of a
Gauss quadrature formula
4. Resolution of ordinary dierential equations
(a) Cauchy Lipchitz problem
(b) Principle of a digital schema, Euler method
(c) Consistency of a digital schema, stability, convergence
(d) Runge-Kutta methods of order 2 and order 4
Matrix Numerical Analysis
Non linear Numerical analysis
Problems & Exams