CALCOLO SCIENTIFICO

Anno accademico 2020-2021

CALCOLO SCIENTIFICO

Docenti:
Rossana Vermiglio
Davide Liessi
Anno di corso: 2
Totale crediti: 6
Tipologia: Affine/Integrativa
Periodo didattico: Secondo Periodo
Prerequisiti. The knowledge provides by the courses of “Matematica Discreta (Discrete Mathematics)” and “Analisi Matematica (Mathematical Analysis)” and the following concepts are mandatory:

-linear algebra (vector spaces, linear applications, matrix computing, Euclidean spaces)

-real numbers, functions anf their graphs, sequences, function’s limits, derivatives. Absolute value and norm of a vector.

It is also highly recommended to have skills on basic tools of programming.

Metodi didattici. 48 hours of frontal lectures and exercises on blackboard and by using slides. In order to support the practical lessons, a tutorial activity (8 hours) is scheduled for further exercises, to strengthen the concepts acquired in class.
Modalità di verifica. The final exam consists in verifying the achievement of the formative objectives previously stated. The final exam consists in a written test to verify the acquisition and the comprehension of the main topics of the course. The available time is 2 hours. Students will not be allowed to consult texts. The evaluation of the answers is based on the correctness, on the clarity and on the completeness of the script.

A score is assigned to each question. If the score of the written test is greater than or equal to 18/30, the oral exam is optional (compulsory in car of distance teaching) The student, absent to the oral test, which has a positive score in the written test can refuse it by using the esse3 procedure

OBIETTIVI FORMATIVI

Scientific Computing is concerned with the development, the analysis and the implementation of algorithms for solving mathematical problems involving continuous variables by using the computer. The aim of the course is to present the basic theoretical aspects for solving the main problems of scientific computing and to analyze the convergence,

Objectives

stability and complexity of the related numerical methods. In this introductive course the students learn about

-the sources of errors due to the use of the computer (rounding errors and their propagation, well/ill-conditioned problems; stability of algorithms);

-the numerical methods for solving nonlinear equations and linear systems, to approximate data and functions, together with the relative analysis of convergence, stability and complexity.

The theoretical knowledge is complemented with the solution of exercises and the study of some test examples to learn how critically analyze the results of numerical simulations.

The learned skills allow the interested students to continue the study of the discipline at advanced level and, to encompass scientific computing problems arising not only in computer science but also throughout the natural sciences, social sciences, engineering, medicine, and business. -critically analyze the results of the numerical simulations.

CONTENUTI

Computer arithmetic: finite number representation and related operations, condition of a problem and stability of an algorithm; error propagation;

– Numerical methods for solving nonlinear equations.

– Linear systems: triangular systems, direct methods (LU factorization, Choleski factorization), perturbation analysis and condition number. Overdetermined systems of linear equations.

-Approximation of data and functions via interpolation: polynomial interpolation (Lagrange polynomial, Newton polynomial, Hermite polynomial; interpolation error, Chebyshev points); interpolation by piecewise polynomials and spline functions; B-splines, parametric interpolation; Bezier and B-spline curves; trigonometric interpolation and FFT.

-Approximation of data and functions via least squares method.

All the topics also include the solution of exercises and the presentation of case-studies in MATLAB.

TESTI DI RIFERIMENTO

– Teacher’s notes, slides, exercises available on e-learning.

– A. Quarteroni, F. Saleri “Introduzione al calcolo scientifico”. Springer Verlag 2002

– C. Moler “Numerical computing with MATLAB”. SIAM 2005