The course focuses on the foundation of Numerical Linear Algebra. The aim therefore is to give the student knowedge of the main algorithms for the execution of matrix operations on the computer, in particular for the solution of linear systems and the eigenvalues-eigenvectors computation. Problems of this kind arise in many computer simulations based on mathematical models, e.g., in engineering, physics, astronomy, biomathematics, finance, and informatics. Lectures will take place side by side with lab classes at the Computer Lab of the Mathematics Department.

Lectures. Exercises. Labs.

Written and oral exam. Lab report.

First year "Algebra lineare" course.

1) Error analysis.
Error propagation. Conditioning of a problem.

2) Direct methods for the solution of linear systems.
Triangular systems. Gauss elimination. LU factorization. Pivoting. Other factorizations, Choleski factorization. Banded, block and sparse matrices. Condition number. Forward and backward a priori analysis. Stability of LU factorization. Over-constrained systems; QR factorization; modified Gram-Schmidt algorithm and Householder matrices.

3) Iterative methods for the solution of linear systems.
Splitting methods: Jacobi and Gauss-Seidel methods. Iteration matrix and spectral radius. JOR and SOR methods. Convergence study and stopping criteria. Richardson-like methods; analysis of stationary Richardson method. Gradient method (steepest descent). Conjugate gradient method; preconditioned conjugate gradient method. Preconditioners.

4) Eigenvalues and eigenvectors approximation.
Conditioning of eigenproblems and eigenvalue localisation. Power method. Inverse power method. Shifting. Deflation. Similarity methods; QR method.

Lecture notes;
Lloyd N. Trefethen, David Bau III. Numerical Linear Algebra. SIAM.