Errors and finite precision arithmetic; perturbation analysis and stability. Polynomial and piecewise polynomial interpolation and approximation. Interpolatory quadrature rules and composite rules; Richardson extrapolation. Nonlinear equations: fixed point iteration, bisection, secant and Newton's method. Linear systems: Gaussian elimination with partial pivoting; LU, Cholesky and QR factorizations; Jacobi and Gauss-Seidel iterations; linear least squares problems. Introduction to MATLAB.
Bini, Capovani, Menchi, “Metodi Numerici per l’Algebra Lineare”, Ed. Zanichelli, Bologna, 1993.
Gasparo, Morandi, “Elementi di calcolo numerico: metodi e algoritmi, McGraw-Hill, Milano, 2008.
Monegato, "Fondamenti di Calcolo Numerico", Edizioni C.L.U.T., Torino, 1998.
Learning Objectives
Knowledge acquired:
The course deals with the definition and study of methods for solving mathematical problems by using computers.
Purpose of the course is to present the basic methodologies of numerical analysis for data and functions polynomial approximation, for computing definite integrals, for solving linear systems of equations, and for finding roots of nonlinear equations, with a particular attention devoted to implementation issues.
Competence acquired:
Knowledge of classical numerical methods for finding roots of nonlinear equations, for solving linear systems, for polynomial interpolation, and for computing definite integrals.
Skills acquired (at the end of the course):
Ability to develop simple programs and to use the Matlab environment in order to solve the mathematical problems under study. Understanding of the obtained numerical results.
Prerequisites
Courses required: Mathematical Analysis I, Geometry I.
Teaching Methods
CFU: 9
Total hours of the course (including the time spent in attending lectures, seminars, private study, examinations, etc...): 225
Hours reserved to private study and other indivual formative activities: 138
Frequency of lectures, practice and lab: Recommended
Teaching Tools: Matlab lecture notes, : textbook , UniFi E-Learning: http://e-l.unifi.it
Office hours:
Prof. Papini
Monday, from 3:00 to 5:00 p.m. or by appointment. Department of Industrial Engineering, viale Morgagni 40, 50134 – Firenze
E-mail: alessandra.papini@unifi.it
Tel. 055 4796716 Fax 055 4796744
Prof. Bellavia
By appointment.
Department of Industrial Engineering, viale Morgagni 40, 50134 – Firenze
E-mail: stefania.bellavia@unifi.it
Tel. 055 4796703 Fax 055 4796744
Type of Assessment
Practical Matlab test plus oral exam.
Course program
Numerical methods and algorithms: definitions. Errors in scientific computing: floating-point representation, machine precision and arithmetic operations; discretization error and effects of finite precision; perturbation analysis and stability.
Polynomial and piecewise polynomial interpolation: Lagrange and Newton form of the interpolant polynomial, interpolation error, conditioning of the problem, Chebyshev's abscissae; spline functions, cubic spline interpolants. Polynomial least squares approximation.
Numerical integration: Newton-Cotes formulas; composite quadrature rules; error analysis and conditioning; Richardson extrapolation; adaptive formulae.
Solution of nonlinear equations: conditioning of the problem; bisection, fixed point iteration, secant and Newton's method; convergence properties and implementation issues. Newton’s method for nonlinear systems of equations.
Direct methods for linear systems: Gaussian elimination; LU and Cholesky factorizations; Householder reflections and QR factorization; pivoting strategies; error analysis. Linear least squares problems: normal equations; solution by QR factorization.
Stationary iterative methods for large linear systems: basics; convergence analysis; metodi di splitting (Jacobi and Gauss-Seidel iterations); Richardson method.
How to use MATLAB, an interactive system for scientific computations.