NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. (2011) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nla.803 Moments of a linear operator, with applications to the trace of the inverse of matrices and the solution of equations Claude Brezinski 1, * ,† , Paraskevi Fika 2 and Marilena Mitrouli 2 1 Laboratoire Paul Painlevé, UMR CNRS 8524, UFR de Mathématiques Pures et Appliquées, Université des Sciences et Technologies de Lille, 59655 Villeneuve d’Ascq cedex, France 2 University of Athens Department of Mathematics, Panepistimiopolis, 15784 Athens, Greece SUMMARY Let H be a real finite dimensional Hilbert space and A an invertible linear operator on it. In this paper, we are interested in obtaining estimations of Tr.A 1 / and of the norm of the error when solving the equation Ax D f 2 H . These estimates are obtained by extrapolation of the moments of A. Numerical results are given, and applications are discussed. Copyright © 2011 John Wiley & Sons, Ltd. Received 5 November 2010; Revised 16 July 2011; Accepted 17 July 2011 1. INTRODUCTION–MOTIVATION OF THE PROBLEM Let H be a real (to simplify) finite dimensional Hilbert space with the inner product denoted by h, i. Let A be a linear operator from H to H . We assume that A 1 exists. Our aim is to build estimates of Tr.A 1 / (the trace will be defined in Section 2) and of hx  y , x  y i where y is any approximation of x, the exact solution of the operator equation Ax D f 2 H . These estimates are obtained by extrapolation of the moments of A. Under appropriate conditions, these estimates could be extended to infinite dimensional Hilbert spaces. However, because most of the applications we are considering deal with the matrix case, we will focus on this case except for the definitions of the mathematical notions that we will be using. The case of an operator on a Hilbert space will be shortly discussed in the last section. The computation of the trace of the inverse of a matrix has received much attention (see [1, 2] and the references given therein). Methods dealing with estimates of Tr.A 1 / for symmetric pos- itive definite matrices by Gaussian quadratures [1, 3], modified moments [1, 4], and Monte Carlo techniques [5] have been derived. Such estimates have many applications in mathematics, statistics, and physics.  Statistics: weighing design problem. In weighing designs, the design matrix e X , must be specified such that the trace of the inverse of the symmetric positive definite matrix e X T e X is minimized (A-optimality criterion). Methods for constructing such A-optimal chemical balance weighing designs are described, for example, in [6, 7].  Physics: calculation of quark loops in lattice quantum chromodynamics. Many physical systems, classical or mechanical, require to compute the trace of a matrix M 1 when the dimension of the matrix M grows fast with the physical variables of the problem [8]. The quark matrix in lattice gauge calculations of quantum chromodynamics falls into this *Correspondence to: Claude Brezinski, Laboratoire Paul Painlevé, UMR CNRS 8524, UFR de Mathématiques Pures et Appliquées, Université des Sciences et Technologies de Lille, 59655 Villeneuve d’Ascq cedex, France. † E-mail: Claude.Brezinski@univ-lille1.fr Copyright © 2011 John Wiley & Sons, Ltd.