NUMERICAL LINEAR ALGEBRA
Anno accademico 2021/2022 - 3° anno - Curriculum APPLICATIVOCrediti: 6
SSD: MAT/08 - Analisi numerica
Organizzazione didattica: 150 ore d'impegno totale, 108 di studio individuale, 42 di lezione frontale
Semestre: 2°
Obiettivi formativi
Description
The module covers the design and analysis of numerical algorithms to solve or accurately approximate problems from linear algebra, such as linear systems and eigenvalue problems.
The module also aims at providing solid implementation skills by developing small software programs of the different numerical algorithms, with applications to real-world problems.
Contenuti del corso
Linear systems Reminders on some special matrices and their properties
Direct methods for full matrices linear systems: LU and QR factorization, Gaussian elimination, pivoting, Doolittle and Crout. Thomas algorithm. Preconditioning techniques: ILU, ILU(p), Incomplete Cholesky preconditioning.
Iterative methods for sparse matrices linear systems: Jacobi, Gauss-Seidel, SOR, SSOR, Krylov methods, Arnoldi orthogonalization, FOM and GMRES, Multigrid methods.
Eigenvalues
Direct methods
Non symmetric matrices: Power and inverse power methods. Similarity transformations: Householder and Givens. Simultaneous iteration, QR algorithm without and with shift.
Symmetric matrices: Tridiagonal QR iteration, Rayleigh ratio iteration, Divide & Conquer, Jacobi, Bisection, Sturm sequencies.
Iterative methods
Arnoldi method for non symmetric matrices and Lanczos method for symmetric matrices.