On the convergence of Krylov methods for solving non symmetric linear systems

Today Krylov methods are the most popular iterative methods for solving linear systems. In this talk we consider their residual norms and their decrease. We will recall what are the known results for GMRES (Generalized Minimum Residual method) and we will show how to obtain exact expressions for the residual norms for diagonalizable matrices. They involve the eigenvalues and eigenvectors of the matrix as well as the right-hand side. To a certain extent they allow to explain the good or bad convergence of the method. Then we will study how to extend some of these results to other Krylov methods like the biconjugate gradient or the QMR method.