B006819 - NUMERICAL ANALYSIS

Italian

Errors and finite arithmetic. Conditioning of a problem. The Matlab language. Basic methods for finding the roots of an equation. Methods for solving linear systems, overdetermined linear systems. Basic numerical methods for solving nonlinear systems. Polynomial interpolation, spline functions, least squares polynomial approximation. Quadrature formulas for solving definite integrals. Basic numerical methods for computing the eigenvalues of a matrix.

L. Brugnano, C. Magherini, A. Sestini, Calcolo Numerico, seconda edizione, Master, Università & Professioni, 2010, Firenze.Materiale di esercitazione (a cura della Dott.ssa Sestini) http://web.math.unifi.it/users/sestini/

Knowledge: The course aims to provide the basic numerical methods used in scientific computing, with particular emphasis on aspects related to their efficient implementation on a computer. Gained expertise: The student acquires the knowledge about the basic numerical methods, together with the notions of finite arithmetic used in a computer, and the conditioning analysis of a problem.
Acquired abilities (at the end of the course): At the end of the course the student will be able to locate a suitable numerical method to solve certain classes of problems and to translate this method in a coputer algorithm.

Courses required: Analysis I: Integral and Differential Calculus, Linear Algebra, Programming.
Recommended Courses: Analysis I: Integral and Differential Calculus, Linear Algebra, Programming.

Frequency of lessons and exercises: Recommended
Tools to support teaching UniFi E-Learning: http://el.unifi.it
Office hours:
- Prof. Brugnano
http://web.math.unifi.it/users/brugnano/Corsi/ORARIO% 20RICEVIMENTO.htm
Agree or via e-mail (@ luigi.brugnano unifi.it)
Department of Mathematics "U. Dini ", ground floor, room 22.

- Dr. Sestini
http://web.math.unifi.it/users/sestini/ricevimento.html
or by appointment.
Department of Mathematics "Ulisse Dini", Viale Morgagni, 67 / a 50134 - Florence (FI) Tel: 055 4237481 Fax: 055 4237165
E-Mail: alessandra.sestini @ unifi.it

Oral exam with report

And finite arithmetic errors: errors of discretization, convergence errors, round-off errors, conditioning of a problem. The language Matlab.Radici of an equation: the bisection method, stopping criteria and conditioning of the problem, order of convergence, Newton's method, local convergence, the case of multiple roots, quasi-Newton methods. Solution of linear systems: simple cases, the LU factorization of a matrix, computational cost, diagonally dominant matrix, symmetric matrices and positive definite, LDL ^ T factorization, pivoting, conditioning of the problem, QR factorization and overdetermined linear systems. Basic iterative methods for solving linear systems: motivation, the Jacobi method, the Gauss-Seidel, splitting regular matrices. Outline of the basic methods for solving systems of nonlinear equations. Approximation of functions: polynomial interpolation, Lagrange shape and form of Newton interpolation error, conditioning of the problem, the Chebyshev abscissas, spline interpolation, cubic spline, polynomial approximation to the minimum quadrati.Formule quadrature: Newton-Cotes formulas , error and composite formulas, formulas adattative.Metodi for research of the eigenvalues of a matrix: the power method, applied to the calculation of "Google pagerank".