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

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Abstract

Let A be a positive definite, symmetric matrix. We wish to determine the largest eigenvalue, λ1. We consider the power method, i.e. that of choosing a vector v0 and setting vk = Akv0; then the Rayleigh quotients Rk = (Avk, vk)/(vk, vk) usually converge to λ1 as k → ∞ (here (u, v) denotes their inner product). In this paper we give two methods for determining how close Rk is to λ1. They are both based on a bound on λ1 - Rk involving the difference of two consecutive Rayleigh quotients and a quantity ωk. While we do not know how to directly calculate ωk, 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 λ1 - Rk which is proportional to (λ2/λ1)2k, which holds with a prescribed probability (the prescribed probability being an arbitrary δ > 0, with the upper bound depending on δ). © 1998 Elsevier Science Inc. All rights reserved.

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Friedman, J. (1998). Error bounds on the power method for determining the largest eigenvalue of a symmetric, positive definite matrix. Linear Algebra and Its Applications, 280(2–3), 199–216. https://doi.org/10.1016/S0024-3795(98)10020-4

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