Errors and finite arithmetic; perturbation analysis and stability. Polynomial and spline interpolation and approximation. Newton-Cotes quadrature rules and composite rules; adaptive quadrature. Nonlinear equations: bisection, Newton's and quasi-Newton's methods. Fixed point iterations. Linear systems: LU factorization, variant with pivoting; Jacobi and Gauss-Seidel methods; QR factorization of rectangular matrices; overdetermined linear systems. Newton's method for nonlinear systems. MATLAB
- L. Brugnano, C. Magherini, A. Sestini, Calcolo Numerico, Master, Uni- versità e Professioni, Terza Edizione, Firenze 2014.
- Bini, Capovani, Menchi, Metodi Numerici per l'Algebra Lineare, Ed. Zanichelli, Bologna, 1993.
Learning Objectives
Knowledge acquired:
numerical methods for solving basic mathematical problems by using computers (linear and nonlinear equations and systems of equations, data and function approximation, definite integrals) with a particular attention devoted to implementation issues.
Competence acquired: finite arithmetic;
knowledge of classical numerical methods for solving the considered mathematical problems.
Skills acquired (at the end of the course):
Ability to convert methods in algorithms and to use the Matlab environment in order to solve the mathematical problems under study. Understanding of the performances of an algorithm and of the obtained numerical results.
Prerequisites
Required courses: Analysis I, Geometry I.
Teaching Methods
Lectures: Presentation of the theory described in the course program, with teacher-student direct interaction.
Training Sessions in the computer lab: practicing with numerical problem solving in Matlab environment.
The training sessions are conducted so to:
-- help the students develop skills to apply the theoretical knowledge;
-- encourage independent judgement in the students, particularly in the understanding of the results obtained in finite arithmetic.
Further information
Texts for the computer lab training sessions and possible integrative material will be given by the Moodle platform.
Type of Assessment
Practical Matlab test plus oral exam.
The practical Matlab test consists of two intermediate tests (or one final test): a selection of exercises is proposed (usually three for each intermediate test, four for the final one). Tests are designed to assess the ability of the students to apply their skills to problem solving. In the evaluation, special attention is paid to the correctness of the solution procedure, as well as to the ability of understanding the obtained numerical results.
Oral examination: some questions are posed. The oral examination is designed to evaluate the degree of understanding of the theory presented in the course. In the assessment, special attention is paid to communication skills, critical thinking and appropriate use of mathematical language.
Course program
Numerical methods and algorithms: definitions. Errors in scientific computing: floating-point representation, machine precision and arithmetic operations; analytic (convergence and discretization) errors and effects of finite precision; perturbation analysis and stability.
Polynomial and piecewise polynomial interpolation: Lagrange and Newton form of the interpolating 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; adaptive formulae.
Solution of nonlinear equations: conditioning of the problem; bisection, fixed point iteration, Newton's and quasi Newton's methods; convergence properties and implementation issues.
Direct methods for linear systems: LU factorization and variant with pivoting. Cholesky factorizations; error analysis. Householder reflections and QR factorization of a rectangular matrix. Overdetermined linear systems: normal equations; solution by QR factorization. Newton's method for nonlinear systems of equations.
Iterative methods for large linear systems: basics; convergence analysis; metodi di splitting (Jacobi and Gauss-Seidel iterations).
How to use MATLAB, an interactive system for scientific computations.