B005494 - NUMERICAL ANALYSIS I

Errors and finite precision arithmetic; perturbation analysis and stability. Polynomial and piecewise polynomial interpolation. Interpolatory quadrature rules and composite rules; Richardson extrapolation. Nonlinear equations: bisection, secant and Newton's method; Newton’s method for nonlinear systems of equations. Linear systems: Gaussian elimination with partial pivoting; LU and QR factorizations and their variants; Jacobi and Gauss-Seidel iterations. 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.

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 interpolation, 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.

Courses required: Mathematical Analysis I, Geometry I.

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

Lectures: 48 hours

Practical activities: 36 hours

Contact hours for: Laboratory-field/practice (hours): 0

Seminars (hours): 0

Stages: 0

Intermediate examinations: 3

Frequency of lectures, practice and lab: Recommended
Teaching Tools: textbook , Matlab lecture notes
Office hours:
Prof. Papini Wednsday, from 2:00 to 4:00 p.m. or by appointment. Dipartimento di Energetica “S.Stecco”, viale Morgagni 40, 50134 – Firenze E-mail: alessandra.papini@unifi.it Tel. 055 4796716 Fax 055 4796744
Prof. Bellavia By appointment. Dipartimento di Energetica “S.Stecco”, viale Morgagni 40, 50134 – Firenze E-mail: stefania.bellavia@unifi.it Tel. 055 4796703 Fax 055 4796744

Practical Matlab test plus oral exam.

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; Hermite interpolation; spline functions, cubic spline interpolants.
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, 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.
Stationary iterative methods for large linear systems: basics; convergence analysis; Jacobi and Gauss-Seidel iterations; regular splittings of matrices, the SOR method.
How to use MATLAB, an interactive system for scientific computations.