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

Preconditioning

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