Describe the main numerical methods used in computer codes for solving linear systems. Sensitize students to problems of rounding errors and conditioning. Study of polynomial interpolation and numerical integration.
Upon completion of this course, students should:
a. be able to have a good practice: the study and resolution of major systems.
b. interpolate functions of a real variable and computing integrals approached with a study of the quadrature error.
c. control the Matlab environment. In particular, they will be able to manipulate matrices, using some basic functions (computing, display, plot, ...) to find and implement their own, news, features for future uses.
1. Introduction, causing problems of numerical analysis matrix.
2. Recalls linear algebra matrix statement of the theorem of Jordan.
3. Direct methods for solving linear systems:
• Methods for Gaussian elimination with and without spindles, factorization of a matrix by Gauss method A = LU.
• Theorem and Cholesky method.
• Matrix and Householder algorithm.
4. Spectrum and standards matrix.
• Theorem Gershgorin - Hadamard.
• Matrix multiplicative norms, norm of Frobenus-Schur.
• Subordinate norms. Calculation of particular norms: ||A||1, ||A||2, ||A||∞
• Equivalence ρ (A) <1 there exists a matrix norm such that (A) <1.
• Effect of rounding errors - the study of a posteriori error - method of permutations.
5. Iterative methods for solving linear systems:
• Classical methods (Jacobi, Gauss Seidel relaxation). Convergence of these methods.
• Hermitian matrices ρ (M-1N) <1 A is symmetric positive definite.
6. Linear Interpolation: Polynomial interpolation, Lagrange, Hermite, error estimation of Hermite interpolation. Aitken's method of divided differences. Newton interpolation, error, choice of medium. Chebychev points.
7. Numerical integration (quadrature methods, rectangle, midpoint, trapezoid, Newton-Cotes, composite formulas, undetermined coefficients, Gaussian quadrature formulas using orthogonal polynomials, error estimate).