NUMERICAL LINEAR ALGEBRA

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.

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.