B006819 - NUMERICAL ANALYSIS

Italian

Errors and finite precision arithmetic; numerical methods for solving a nonlinear equation; Numerical solution of linear and nonlinear systems of equations; Approximation of functions and of definite integrals.

L.Brugnano, C.Magherini, A.Sestini. Calcolo numerico, quinta edizione. Master, Universita' e Professioni, Firenze, last available edition.

Knowledge: the course aims at giving the basic instruments for scientific computing, with a major emphasis to their computer implementation.

Skills: basic numerical methods for scientific computing; ability to solve numerical problems on a computer.

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

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

Credits: 9
Oral exam,
Teaching hours: 88
Text-book:
L.Brugnano, C.Magherini, A.Sestini. Calcolo Numerico, quinta edizione. Master Books, Firenze.

Intermediate evaluations for students attending the lectures.

Additional informations at the web pages:

http://web.math.unifi.it/users/brugnano/Corsi/index.htm

For reception see the web page:

http://web.math.unifi.it/users/brugnano/Corsi/ORARIO%20RICEVIMENTO.htm

Address:

Viale Morgagni, 67/a - 50134 Firenze
Tel: 055 2751-421

E-mail: luigi.brugnano@unifi.it
Web: http://web.math.unifi.it/users/brugnano/

The exam consists in an oral dissertation, concerning the methodological aspects of the course, plus a a written report concerning the efficient Matlab implementation of the methods. The written report is done by the student before the exam, alone or in a group of 2-3 people. The final score is obtained as the combination of the oral dissertation (with weight 2/3) and of the written report (with weight 1/3) scores.

There are intermediate written exams for the attending students, with open answers concerning the subject of each chapter of the text-book, which substitute the oral dissertation.

Errors and finite precision arithmetic: errors of discretization, convergence errors, round-off errors, conditioning of a problem. The language Matlab. Roots 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, adaptive formulas.