Program of The Rome-Moscow school of Matrix Methods and Applied Linear Algebra

Lessons of Professor D Bertaccini

1. The unifying approach of projection methods
2. Krylov projection methods
3. Matrix representation
4. A taxonomy for projection methods
5. From the Arnoldi algorithm to conjugate gradients
6. From the Arnoldi algorithm to GMRES
7. Notes on a formal implementation of GMRES
8. A convergence bound of conjugate gratients
9. Complex Chebyshev polynomials
10. A convergence bound for GMRES based on eigenvalues and eigenvectors
11. Convergence bounds for preconditioned iterations with clustered spectra
12. Other tools: pseudoeigenvalues and the field of values
13. Applying the convergence bounds to some model problems

References
D. Bertaccini, G. H. Golub and S. Serra-Capizzano ``Spectral analysis of a preconditioned iterative method for the convection-diffusion equation''
SIAM J. Matr. Anal. Appl. 29-1, pp. 260--278, 2007

Lecture notes on GMRES convergence in presence of clustered spectra+2 examples (link) (link to slides)

Notes given by the lecturer during the lectures
Y. Saad, Iterative Methods for Sparse Linear Systems, PWS, 1996 (link to a part)