Subject: NUMERICAL ANALYSIS AND STATISTICS (A.A. 2023/2024)
Unit Analisi Numerica e Statistica
Mathematics, Information Technology and Statistics (lesson)
The course of Numerical Analysis and Statistics aims to train students with basic knowledge of probability theory and parametric statistics and able to analyze from a numerical point of view basic mathematical problems (such as linear algebra, roots finding or data/functions approximation), selecting the most appropriate algorithm according to the peculiarities of the specific problem. The implementation of the analyzed methods in Matlab language will allow the student on one hand to put into practice the theoretical knowledge and on the other hand to acquire knowledge and practice of a programming language that leads the world in scientific computing.
To better understand the objectives of the course, please refer to the section devoted to the expected results that students should have acquired at the end of this course.
- Differential calculus for real functions of real variables.
- Integral calculus for real functions of one real variable.
- Basics of linear algebra.
- Basics of computer programming.
The scanning of the contents for CFU is to be understood as purely indicative. In fact, it may undergo changes during the course of teaching in light of the feedback from students.
1 CFU (8 ore). Computer representation of numbers: rounding errors and floating-point arithmetic.
1.5 CFU (12 ore). Systems of linear equations: Gaussian elimination, pivoting in Gaussian elimination, LU decomposition. Iterative methods: Jacobi method, Gauss-Seidel method, stopping rules.
1 CFU (8 ore). Solution of nonlinear equations: bisection method, Newton's method, convergence analysis for Newton's method.
1.5 CFU (12 ore). Data and functions approximation: basic functions for approximation, polynomial interpolation, interpolation by splines, least-squares approximation.
3 CFU (24 ore). Probability and elements of combinatorics. Introduction to parametric statistics, confidence intervals and tests, simple linear regression.
1 CFU (8 ore). Basic statements of the MATLAB computing environment: array features, subprograms, operations on files, advanced statements for matrix operations and graphics, character arrays, structured data.
The course is delivered through face-to-face lectures and practical exercises carried out with the aid of a blackboard and audiovisual tools. Attendance to face-to-face lessons is not compulsory. The course is delivered in Italian. Working students who cannot regularly attend classes must agree with the teacher specific support activities.
The exam will take place at the end of the course according to the official exam schedule. The exam is oral, lasting about 45 minutes. The candidate must demonstrate a thorough knowledge of: - the course content and teaching training, including both the institutional part, i.e. frontal lessons/classroom activities, and the laboratory practices; - the memorization and the operations of and between real numbers on the computer; - the main techniques for the solution of a linear system or a nonlinear equation, and the computation of eigenvalues and eigenvectors of a matrix; - the main notions on the data fitting problem, as the difference between interpolation and regression, the numerical strategies to design and compute an optimal model and the application to the numerical computation of the integral of a function; - the MATLAB syntax for the implementation of an elementar algorithm; - basic notions of combinatorics, probability and statistics. The verification is integral with respect to the course's content; it is also verified the student's ability to relate specific subject content with the knowledge listed as pre-requisite. The oral exam consists in the implementation of one of the algorithms analyzed during the class in a Matlab environment and in the in-depth analysis of some topics treated during the lectures. The score of the oral exam, in a scale of thirty, is divided in: 5 points for the communicative skills; 5 points for multidisciplinary skills; 20 points for the knowledge of the contents. Basic knowledge of the topics and partial ability to apply this knowledge are necessary to obtain the minimum grade (18/30). Full knowledge of all topics and excellent ability to apply knowledge are necessary to obtain the maximum grade (30/30 cum laude). The graduation of the intermediate grades is done on the basis of the achievement of the expected learning outcomes, including the transversal ones, demonstrated during the oral exam. The result will be communicated to the individual student at the end of the oral exam.
Knowledge and understanding:
At the end of the course it is hoped that the students will know basic tools of probability and statistics and fundamental methods in numerical analysis, and will be able to implement these methods within the Matlab software environment and to analyse their performance in terms of convergence properties and computational complexity.
Applying knowledge and understanding:
At the end of the course it is hoped that the students
a) will have adequate knowledge to face some problems in scientific computing arising from real applications;
b) will be able to find the statistical and numerical methods suitable for the considered problems and to develop their Matlab codes.
At the end of the course it is hoped that the students will be able to independently choose the appropriate methods for a specific problem in scientific computing.
At the end of the course it is hoped that the students will be able to describe the considered statistical and numerical methods in a clear and rigorous way and to discuss their efficiency.
At the end of the course it is hoped that the students will be able to go deep by himself into the main aspects of the subjects proposed in the course.
E' fornito materiale didattico preparato dal docente sulla piattaforma moodle.unimore.it sia per la parte di teoria che per la parte di esercitazioni numeriche.
Eventuali testi consigliati per approfondire le tematiche sviluppate nel corso sono i seguenti:
A. Mazzia: Laboratorio di calcolo numerico. Applicazioni con Matlab e Octave, Pearson, 2014.
G. Naldi, L. Pareschi, G. Russo: Introduzione al Calcolo Scientifico - Metodi e applicazioni con Matlab, McGraw-Hill, Milano 2001.
A. Quarteroni, R. Sacco, F. Saleri: Matematica Numerica (3a edizione), Springer, 2008.
A. Quarteroni, F. Saleri, P. Gervasio: Scientific Computing with MATLAB and Octave, Springer, 2010.
W. Navidi, I. Negri: Probabilità e statistica per l'ingegneria e le scienze, McGraw-Hill, 2006.