Catalogo dei Corsi di studio

The goal of the course is to teach students how to use numerical methods to solve some engineering problems for which no analytical solution can be found. The course focuses on understanding the theoretical issues behind the numerical methods studied and implementing them using a programming language. This approach is essential for learning how to choose a numerical method correctly, while considering its limitations.

Programme

The course unit consists of 90 hours duration, with lectures and exercises, for a total of 9 ECTS. Arguments of the teaching modules are the following.

Finite precision and accumulation of round-off errors; truncation errors; conditioning of a mathematical problem; numerical stability (2 hours).

Introduction to the solution of non linear equations and non linear systems of equations by iterative methods (14 hours).

Numerical linear algebra: direct and iterative methods for linear systems (8 hours).

Introduction to approximation theory: polynomial interpolation, Lagrange formula, interpolation error; least squares approximation by algebraic and trigonometric polynomials; splines, linear interpolating splines (14 hours).

Numerical integration: elementary and composite Newton-Cotes formulas, error and convergence, method of trapezes, Simpson’s composite rule (4 hours).

Numerical methods for initial value problems: discretization error, consistency, stability; one-step methods, Euler method, Heun method, classical Runge-Kutta method, convergence. Finite difference methods for boundary value problems (20 hours).

Introduction to Matlab (28 hours).

Adopted texts

L. Gori, Calcolo Numerico, Ed. Kappa, 2006

L. Gori, M.L. Lo Cascio, F. Pitolli, Esercizi di Calcolo Numerico, Ed. Kappa, 2007

Course Slides (to be downloaded)

Bibliography

V. Comincioli, Analisi numerica: metodi, modelli, applicazioni, Mcgraw-Hill Libri Italia s.r.l., Milano, 1990 A. Quarteroni, R.Sacco, F, Saleri. Matematica numerica. Springer, Milano, 2008 F. Fontanella, A. Pasquali, Calcolo numerico: Metodi e algoritmi, Vol. 1, Pitagora Editrice, Bologna. F. Fontanella, A. Pasquali, Calcolo numerico: Metodi e algoritmi, Vol. 2, Pitagora Editrice, Bologna.

Prerequisites

Knowledge of fundamentals of calculus, geometry and linear algebra

Study modes

The course includes both lectures and lab exercises. During the lectures, the teacher will outline and discuss the main features of the numerical methods listed in the program. During lab exercises, firstly the teacher will give an introduction to programming in Matlab, then the teacher will show how to code algorithms. During the course the teacher will also provide guided exercises on numerical methods and programming and will assign homeworks to students.

Frequency modes

Attending of the course is warmly recommended.

Exam modes

Assessment is based on two components:
• solution of exercises (55%): students should identify the numerical method suitable to solve a given problem and discuss numerical issues (accuracy, convergence, stability);
• computer implementation of numerical algorithms (45%): students should implement a numerical algorithm on the computer, realize numerical tests and critically analyze the results.
The oral part is not compulsory.