Error bounds on the power method for determining the largest eigenvalue of a symmetric, positive definite matrix

Let A be a positive definite, symmetric matrix. We wish to determine the largest eigenvalue, lambda(1). We consider the power method, i.e. that of choosing a vector upsilon(0) and setting upsilon(k) = A(k)upsilon(0); then the Rayleigh quotients R-k = (A upsilon(k), upsilon(k))/(upsilon(k), upsilon(k)) usually converge to lambda(1) as k --> infinity (here (u, upsilon) denotes their inner product). In this paper we give two methods for determining how close R-k is to lambda(1). They are both based on a bound on lambda(1) - R-k involving the difference of two consecutive Rayleigh quotients and a quantity omega(k). While we do not know how to directly calculate omega(k), we can give an algorithm for giving a good upper bound on it, at least with high probability. This leads to an upper bound for lambda(1) - R-k which is proportional to (lambda(2)/lambda(1))(2k), which holds with a prescribed probability (the prescribed probability being an arbitrary delta > 0, with the upper bound depending on delta). (C) 1998 Elsevier Science Inc. All rights reserved.