In two previous papers by Neymeyr [Linear Algebra Appl. 322 (1--3) (2001) 61; 322 (1-- 3) (2001) 87], a sharp, but cumbersome, convergence rate estimate was proved for a simple preconditioned eigensolver, which computes the smallest eigenvalue together with the corresponding eigenvector of a symmetric positive definite matrix, using a preconditioned gradient minimization of the Rayleigh quotient. In the present paper, we discover and prove a much shorter and more elegant (but still sharp in decisive quantities) convergence rate estimate of the same method that also holds for a generalized symmetric definite eigenvalue problem. The new estimate is simple enough to stimulate a search for a more straightforward proof technique that could be helpful to investigate such a practically important method as the locally optimal block preconditioned conjugate gradient eigensolver.

ABSTRACT. In this paper we will discuss an abstract iteration scheme for the calculation of the smallest eigenvalue of an elliptic operator eigenvalue problem. The iteration is based on the preconditioned steepest descent of the Rayleigh quotient, also known as the preconditioned inverse iteration (PINVIT), and we will extend the convergence result obtained for algebraic eigenvalue problems in [D’yakonov and Orekhov, Math. Notes 27, 382-391 (1980)] to the case of operators. We show that the convergence is retained up to any tolerance if one uses only approximate applications of operators which leads to the perturbed preconditioned inverse iteration (PPINVIT). For wavelet discretization we briefly sketch how the iteration can be used to construct an adaptive convergent algorithm with quasi-optimal bases comparable to the Richardson iteration in elliptic operator equations [Cohen, Dahmen, and DeVore, Math. Comp. 70, 27-75 (2001)]. However a detailed discussion of such an optimal adaptive scheme is postponed to a forthcoming article. 1.

. Let's give the mesh discretization of an elliptic eigenvalue problem and the problem to determine the smallest eigenvalue together with an eigenvector. Gradient type methods solve this problem by consecutive correction steps, each in the direction of the negative gradient of the Rayleigh quotient. It is shown that the convergence rate of gradient methods, even with optimal scaling, converges to 1 if the mesh parameter tends to 0. In contrast to this, premultiplying the gradient vector by a preconditioner, which defines the preconditioned gradient method, leads to grid--independent convergence estimates, if the preconditioner is sufficiently accurate. Moreover, some suitable scaling strategy may lead to an improved convergence. For these reasons, preconditioning of gradient type methods is decisive to construct a reliable and efficient eigensolver for elliptic eigenvalue problems. 1. INTRODUCTION Let A be a symmetric positive definite matrix whose smallest eigenvalue 1 together with...

. The topic of this paper is a convergence analysis of preconditioned inverse iteration (PINVIT). A sharp estimate for the eigenvalue approximations (as given by the Rayleigh quotient of the iterates) is derived. The error of the eigenvector approximations is controlled by an upper bound for the residual vector. While in Part I the case of poorest convergence of PINVIT as a function of the choice of the preconditioner is analyzed, we determine here its dependence on the eigenvector expansion of the iteration vector. The analysis is mainly based on extremal properties of various quantities which define the geometry of PINVIT. 1. INTRODUCTION Let A 2 R m\Thetam be a symmetric positive definite matrix whose eigenvalues of arbitrary multiplicity are given by 0 ! 1 ! 2 ! : : : ! n . Preconditioned inverse iteration (PINVIT), as introduced in Part I, maps a given vector x with the Rayleigh quotient := (x) = (x; Ax) (x; x) (1.1) to the next iterate x 0 = x \Gamma B \Gamma1 (Ax \G...

of the discretization and iteration errors

We consider a large class of residuum based a posteriori eigenvalue/eigenvector estimates and present an abstract framework for proving their asymptotic exactness. Equivalence of the estimator and the error is also established. To demonstrate the strength of our abstract approach we present a detailed study of hierarchical error estimators for Laplace eigenvalue problems in planar polygonal regions. To this end we develop new error analysis for the Galerkin approximation which avoids the use of the strengthened Cauchy-Schwarz inequality and the saturation assumption, and gives reasonable and explicitly computable upper bounds on the discretization error. A brief discussion is also given concerning the design of estimators which are in the same spirit, but are based on different a posteriori techniques—notably, those of gradient recovery type.

Abstract. The so called Wilkinson’s Schur complement trick plays a prominent role in the theory of eigenvalue estimates for finite matrices. We apply this technique to the problem of obtaining Ritz value spectral estimates for a positive definite self-adjoint operator in a Hilbert space. New estimates have a form of a Temple–Kato like inequality and are particularly suited to a situation in which we are estimating a multiple eigenvalue of a positive definite operator by a cluster of Ritz values. An application of new estimates in a context of finite element computations will be discussed. 1.