CODE 42927 ACADEMIC YEAR 2021/2022 CREDITS 5 cfu anno 2 MATEMATICA 9011 (LM-40) - GENOVA 5 cfu anno 3 MATEMATICA 8760 (L-35) - GENOVA 5 cfu anno 1 MATEMATICA 9011 (LM-40) - GENOVA SCIENTIFIC DISCIPLINARY SECTOR MAT/08 TEACHING LOCATION GENOVA SEMESTER 2° Semester TEACHING MATERIALS AULAWEB OVERVIEW 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. AIMS AND CONTENT LEARNING OUTCOMES Deepening of the knowledge of numerical linear algebra, with particular attention to the numerical treatment of large matrices. Study of computationally efficient methods, both direct and iterative, and their use in Matlab. AIMS AND LEARNING OUTCOMES The course aims to equip students with a set of tools and knowledge necessary to identify, understand and solve linear problems related to large and / or structured matrices. These matrices are present in many applications, such as, for example, page ranking on the internet, image processing, tomography and non-destructive analysis in the civil and biomedical fields, machine learning. PREREQUISITES The mathematical prerequisites are contained in the linear algebra and numerical analysis courses of the three-year degree (I cycle). For an in-depth 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 three-year degree course, may be useful, although not necessary. TEACHING METHODS Lectures (40 hours) SYLLABUS/CONTENT Sparse matrices, structured matrices. Analysis of sparse matrices using graphs and permutation techniques. Connection and irreducibility. Perron-Frobenius theory for non-negative matrices. Regular splitting. Inverse of matrices with low-rank modifications, Woodbury-Sherman-Morrison 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 Szego-Tyrtyshnikov theorem. Circulant matrices. Fast Fourier Transform (FFT) and its applications to matrix algebra and polynomial algebra. Insights on conjugate gradient, convergence analysis in relation to the spectrum of the matrix. Preconditioning techniques. RECOMMENDED READING/BIBLIOGRAPHY D. Bini, M. Capovani, O. Menchi, Metodi Numerici per l'Algebra Lineare (Zanichelli, Bologna, 1988); C. Estatico, Gradiente coniugato e regolarizzazione di problemi mal posti (Quaderni del Gruppo Nazionale per l’Informatica Matematica, C.N.R., I.N.d.A.M., 1996). Notes provided during lessons TEACHERS AND EXAM BOARD CLAUDIO ESTATICO Ricevimento: By appointment via email. CRISTINA CAMPI Ricevimento: By appointment via email. FABIO DI BENEDETTO Ricevimento: Reception hours: 13-14 on lesson days, prior to email confirmation. Exam Board CRISTINA CAMPI (President) FABIO DI BENEDETTO CLAUDIO ESTATICO (President Substitute) LESSONS LESSONS START Lessons will start according to the academic calendar. Class schedule The timetable for this course is available here: Portale EasyAcademy EXAMS EXAM DESCRIPTION The exam consists in an oral test ASSESSMENT METHODS 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. FURTHER INFORMATION Attendance, although not mandatory, is recommended.
