CODE  42927 

ACADEMIC YEAR  2023/2024 
CREDITS 

SCIENTIFIC DISCIPLINARY SECTOR  MAT/08 
LANGUAGE  Italian (English on demand) 
TEACHING LOCATION 

SEMESTER  2° Semester 
TEACHING MATERIALS  AULAWEB 
Numerical linear algebra deals with the study of problems related to the use of large and / or structured matrices. Many recent technological developments in the field of IT and data processing involve this kind of matrices. The aim of the course is to deepen the topics related to numerical linera algebra that were introduced during the bachelor degree.
The course aims to deepen the knowledge of numerical linear algebra, with particular reference to the numerical treatment of large matrices, favoring the understanding of the most efficient methods, both direct and iterative.
The course aims to provide students with the mathematical tools necessary to identify, understand and solve linear problems related to large and/or structured matrices that are present in most of the current application fields, such as, for example, page ranking on the Internet, image processing, tomography and nondestructive analysis in the civil and biomedical fields, machine learning.
At the end of the course the student will have acquired sufficient theoretical knowledge:
 to know and identify the main problems of a linear nature that require specially developed algorithms to be able to manage the large dimensions characterizing models and/or data, such as, for example, page ranking on the internet, image processing, tomography and nondestructive analysis in the civil and biomedical fields, machine learning from examples;
 to choose and apply numerical linear algebra tools to solve these problems via computer;
 to optimize the algorithms and the numerical code implemented for the numerical resolution of these problems;
 to implement these algebraic methodologies in a highlevel programming language.
The mathematical prerequisites are contained in the linear algebra and numerical analysis courses of the threeyear degree (I cycle). For an indepth understanding, however, it may be useful to have some rudiments concerning the analysis of functions of several variables, iterative methods for linear systems and measure theory. The topics of the "Numerical Calculus" course, an optional threeyear degree course, may be useful, although not necessary.
Lectures (44 hours)
Laboratory (6 hours)
Sparse matrices, structured matrices. Analysis of sparse matrices using graphs and permutation techniques. Connection and irreducibility. Centrality indices. PerronFrobenius theory for nonnegative matrices. Regular splitting.
Inverse of matrices with lowrank modifications, WoodburyShermanMorrison formula. Inverse of block partitioned matrices, Schur's complement and its applications.
QR method for sparse matrices. Separable matrices, Kronecker product, Kronecker sum and associated matrix equations. Spectral decomposition of Kronecker products. Integral equations, discretization and convolution. Structured matrices. Toeplitz matrices and SzegoTyrtyshnikov theorem. Circulant matrices. Fast Fourier Transform (FFT) and its applications to matrix algebra and polynomial algebra.
Nonlinear systems.
Insights on conjugate gradient, convergence analysis in relation to the spectrum of the matrix. Preconditioning techniques.
Office hours: By appointment via email.
Office hours: By appointment, to be booked via email.
CRISTINA CAMPI (President)
CLAUDIO ESTATICO (President)
ISABELLA FURCI (President)
All class schedules are posted on the EasyAcademy portal.
The exam consists in an oral test preceded by a brief discussion on the results obtained during the laboratory
The oral test focuses on the theoretical topics developed during the lessons. Discussions and intuitive justification of theoretical concepts will be also carried on during the exam.
The discussion of the laboratory outocomes focuses on the codes written by the students and the interpretation if the results.
Attendance, although not mandatory, is recommended.