Minimal decompositions and iterative methods

Abstract

Summary. We study how singular values and approximation numbers by algebraic operators of a bounded linear operator A on a separable Hilbert space H can be related to the speed of convergence of iterative methods. In infinite dimensional H these two quantities lead us to consider those A that can be split as $A=N+K$ with N normal andK compact. Then the spectrum of N, via a classical polynomial approximation problem, bounds below the speed of convergence of GMRES. In finite dimensional H we combine these two approximation numbers to approximation numbers by normal operators of a matrix A that lead us to consider splittings ofA as $A=N+F$ with N normal andF of smallest possible rank. We show, analogously to infinite dimensions, that the eigenvalues of N can be used, via a new polynomial approximation problem, in assessing the speed of convergence of GMRES. For upper bounds we obtain estimates that can be combined with the results of [28].