NUMERICAL ANALYSIS

The course provides a brief introduction to numerical methods for solving linear systems, interpolation and approximation of functions, solving nonlinear equations, and calculating integrals. The implementation aspects of these methods will be demonstrated during the course using Matlab software.

Knowledge and understanding: The course aims to foster an active and critical understanding of the subject, not limited to the mere learning of methodologies, but also, and above all, to a deep comprehension of the fundamental concepts.

Applying knowledge and understanding: The computational tools learned during the course should be mastered and applied to real-world problems.

Making judgments: Students should be able to compare the different methods learned during the course and determine which is the most suitable for solving a particular problem, considering both the characteristics of the problem itself and the available resources (the issue of computational efficiency).

Communication skills: Students are encouraged to clearly present the topics covered during the oral exam and explicitly illustrate the various steps involved in solving problems.

Learning skills: Learning is stimulated during lectures through direct questions from the instructor.

Primarily lectures by the instructor, exercises on the blackboard (or on a graphic tablet if the lessons are conducted online), and coding in Matlab.

Information for students with disabilities and / or SLD

Introduction to the use of the computer for numerical calculations.

Use of matlab.

loating point representation. Machine numbers. Truncation and rounding. Machine operations. Digital cancellation. Order of accuracy.

Numerical linear algebra: vectors, matrices, determinants, inverse matrix. Vector and matrix norms. Natural norms and their representation. Eigenvalues. Spectral radius and its properties. Some special matrices. Direct methods for solving linear systems: triangular systems, Gaussian elimination, pivoting. Factorization A = LU and PA = LU. Choleski factorization and their implementation. Conditioning of a linear system. Condition number. Sparse matrices and their representation. Iterative methods for solving linear systems: Jacobi method, Gauss-Seidel method, SOR method. Stopping criteria. . Localization of the eigenvalues: the theorems Gershgorin-Hadamard. Eigenvalues calculation: direct and inverse power methods.

Approximation of functions and data: Polynomial interpolation. Lagrange form. Linear interpolation operator. Calculation of the interpolation polynomials. Newton's formula of divided differences. The error formula (Lagrange form). Chebyshev polynomials, the property of minimal norm. The problem of the convergence of a sequence of interpolators schemes. Interpolation by piecewise-polynomials. Splines. Calculation of cubic splines. Method of least squares and applications. Normal equations and their geometric interpretation.

Solution of nonlinear equations: General concepts. Methods of bisection, secant and Newton. General theory of iterative methods for nonlinear equations and fixed point problems. Convergence order. Stopping criteria.

Quadrature formulas: Weighted integrals. General form of a quadrature formula. Polynomial order (or degree of precision) of a quadrature formula. Interpolatory formulas. Convergence theorem. Newton-Cotes formulas. Gaussian formulas. Composite formulas: trapezoid and Simpson formulae. Romberg method. Adaptive quadrature.

Overview of Numerical Methods for Solving Ordinary Differential Equations.
Runge-Kutta methods and multistep methods.

G.Naldi, L.Pareschi, G.Russo, Introduzione al calcolo scientifico, McGraw-Hill, 2001.

V.Comincioli, Analisi Numerica: metodi, modelli, applicazioni, McGraw-Hill, Milano, 1990.G. 

Monegato, Calcolo Numerico, Levrotto e Bella, Torino, 1985. 

A. Quarteroni, R. Sacco, F. Saleri, Matematica Numerica, Springer Italia, Milano, 1998.