OVERVIEW

The course Numerical Methods for Linear Algebra is devoted to stregthen the knowledges of numerical linear algebra which have been introduced in the first degree studies. The most recent technological developments require the resolution of problems associated with large and/or structured matrices. These problems and their numerical solution will be covered during the course.

AIMS AND CONTENT

LEARNING OUTCOMES

Deepening knowledge of numerical linear algebra, with particular reference to the numerical treatment of large matrices. Understanding of the most efficient methods, both direct and iterative, and their use in Matlab.

TEACHING METHODS

Teaching style: In presence.

SYLLABUS/CONTENT

Numerical methods for large scale matrices: sparse matrices, structured matrices. Graph theory and permutation techniques.

Matrix inverse by low rank corrections. Matrix inverse by block partitioning. Schur complements and Woodbury-Sherman-Morrison formula.

Structured matrices. Kronecker product. Lyapunov-Sylvester matrix equation and Kronecker sum. Spectral decompositions of Kronecker product.

QR factorization of sparse matrices.

Integral equations, discretization and convolution. Structured matrices.Toeplitz matrices, generating function, spectrum, equidistribution and Szegö-Tyrtysnikov Theorem. Fast Fourier Transform (FFT) and its applications in matrix algebra and polynomial.

Convergence theory for stationary iterative methods for linear systems. Splitting and block-splitting methods.Perron-Frobenius theory for nonnegative matrices. Regular splitting. 
Spectral radius and localization of eigenvalues.

Iterative methods of minimization for the solution of linear systems.  Non-stationary methods. Methods with optimum step length. Method of steepest descent. Conjugate gradient. Convergence via matrix spectrum distribution. Preconditioning techniques.

Matlab laboratory exercises.

RECOMMENDED READING/BIBLIOGRAPHY

 D. Bini, M. Capovani, O. Menchi, Metodi Numerici per l'Algebra Lineare. Zanichelli, Bologna, 1988. 

More information will be given during the course.

TEACHERS AND EXAM BOARD

Exam Board

CLAUDIO ESTATICO (President)

PAOLA BRIANZI

FABIO DI BENEDETTO

LESSONS

LESSONS START

September 26, 2016

Class schedule

NUMERICAL LINEAR ALGEBRA

EXAMS

EXAM DESCRIPTION

Orale exam.

ASSESSMENT METHODS

Oral exam, and possible preliminary laboratory test in Matlab. No in-itinere exams.

Exam schedule

FURTHER INFORMATION

Prerequisites: Basic tools of linear algebra and calculus of functions of several variables, in addition to the foundations of numerical analysys.

Attendance: Recommended