Learning outcomes of the course unit
One of the main aims of the course is to provide students with the fundamentals that are the basis of the most common numerical methods used to solve various applicative problems, critically presenting the main algorithms and their properties such as convergence, stability, accuracy, complexity, using examples and counter-examples to illustrate the advantages and weaknesses of the aforementioned methods. During lectures, students will be able to experiment with the presented algorithms in a software environment widely used for scientific computing such as MATLAB. At the end of the course, the student will be able to use computational tools to understand, analyze and solve problems of moderate difficulty in different areas of Mathematics.
Basics of Calculus and Linear Algebra.
Course contents summary
Error analysis - Numerical methods for solving linear systems: direct methods and outline of iterative methods - Outline of numerical solution of nonlinear equations - Interpolation of data and functions by algebraic polynomials and splines - Numerical integration: simple and iterated Newton-Cotes formulas; extensions - Ordinary differential equations: discrete one-step methods - Introduction to Matlab
Error analysis: Representation of numbers in a computer. Rounding errors. Machine operations. Numerical cancellation. Conditioning of a problem. Stability of an algorithm.
Numerical Linear Algebra: Matrix norms - Conditioning of a linear system - Direct methods: the Gauss elimination method, Gauss decomposition and PA = LU factorization - Cholesky factorization - Inverse matrix computation - Direct methods for band matrices - Outline of iterative methods: Jacobi and Gauss-Seidel algorithms.
Outline of the numerical resolution of non-linear equations: bisection algorithm and Newton algorithm - Convergence results - Stop tests.
Interpolation of data and functions: interpolation by algebraic polynomials - Lagrange interpolation formula - Hermite interpolation formula - Newton's divided differences formula - Interpolation error theorems - Generalized interpolation - Interpolation by piecewise polynomial functions and spline functions - Linear and cubic splines - Convergence results.
Numerical integration: Interpolatory quadrature rules - Integration according to Newton-Cotes - Error estimates - Iterated formulas - Convergence theorems - Adaptive quadrature formulas - Applications to generalized integrals.
Numerical methods for ODEs: Explicit one-step methods - Series expansion methods - Runge-Kutta methods - Local truncation error - Stability and convergence of explicit one-step methods - Implicit one-step methods - Adaptive integration step - Absolute stability.
Introduction to Matlab: Matlab as matrix laboratory - Matlab as programming language: counter cycles, condition cycles, structured tests - Function files and script files - Matlab main numerical routines - Matlab for graphics.
A. Quarteroni, R. Sacco, F. Saleri, P. Gervasio: Matematica Numerica, Springer, 2014.
G. Naldi, L. Pareschi, G.Russo, Introduzione al Calcolo Scientifico. Metodi e applicazioni con Matlab, McGraw-Hill, 2003.
G. Monegato, Fondamenti di Calcolo Numerico, CLUT, 1998.
V. Comincioli: Analisi Numerica. Metodi Modelli Applicazioni, Mc Graw-Hill, 1995.
R. Bevilacqua, D. Bini, M. Capovani, O. Menchi: Metodi Numerici, Zanichelli, 1992.
D. Bini, M. Capovani, O. Menchi: Metodi Numerici per l’Algebra lineare, Zanichelli, 1988.
Classroom lectures and exercises. Numerical exercises with MATLAB in Computer Science lab. In the lab lessons, numerical and programming exercises will be assigned. The presentation of the solutions by the students will be taken into account for the final evaluation.
Assessment methods and criteria
Lab written test followed by an oral exam.