Advanced Numerical Analysis

1. General considerations on matrices

Matrices:definitions and properties; norm of matrices
The condition number of a matrix
Sparse matrices and sparse formats (sparsity, structure, functionals)
The role of the PDE discretization (e.g., parameter dependence)

2.a Direct methods for general linear systems

Factorizations: definitions and properties
Factorization algorithms
Cost and numerical stability

2.b Direct methods for sparse linear systems

Factorizations of banded matrices
Ordering strategies to minimize the fill-in of a matrix
Solution of sparse triangular systems
Sparse matrices in Matlab: memorization and handling
Predefined functions for the direct solution of systems

3. Numerical solution of large-scale linear systems

Krylov subspace methods (CG, MINRES, GMRES, IDR family)
Structured problems
Algebraic multigrid methods (hints)
Numerical experiments with Matlab and the IFISS package

4. Numerical solution of eigenvalue problems

Standard and generalized eigenproblems
Typical numerical methods
Equation of motion in structural dynamics: quadratic eigenproblems