## A geometric theory for preconditioned inverse iteration III: A short and sharp convergence estimate for generalized eigenvalue problems (2003)

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### BibTeX

@MISC{Knyazev03ageometric,

author = {Andrew V. Knyazev and Klaus Neymeyr},

title = {A geometric theory for preconditioned inverse iteration III: A short and sharp convergence estimate for generalized eigenvalue problems},

year = {2003}

}

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### Abstract

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.

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Citation Context ...sh [19] and later by Petryshyn [18]. Estimates on the convergence rate were given by Godunov et. al. [9] and D ′ yakonov et. al. [4, 8]. See Knyazev for a recent survey on preconditioned eigensolvers =-=[13]-=-. The viewpoint of preconditioned gradient methods for the convergence analysis of PINVIT seems to be less than optimal, since the convergence estimates are not sharp and some assumptions on the Rayle... |

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Citation Context ... given. One such simplification is that in equation (1.10): instead of the matrix Θ, a constant diagonal matrix is200 KLAUS NEYMEYR considered in order to make the convergence analysis feasible (see =-=[5, 6, 7]-=-). Usually the main difficulty for the convergence analysis of (1.10) is seen in the dependence of the iteration operator (which acts in (1.10) on the columns of V )ontheRitz values θi. Here we do not... |

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Citation Context ...et al. [1]). SPINVIT can also be embedded in an adaptive multigrid algorithm to solve eigenproblems for elliptic operators. Such a method and a posteriori error estimation for SPINVIT is the topic of =-=[14]-=-. Appendix A. Some auxiliary lemmata The next lemma provides a crude estimate from above for the sharp convergence estimate (2.5) of PINVIT (see [8]). Lemma A.1. Let λ ∈]λi,λi+1[ and γ ∈ [0, 1]. Then ... |

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Citation Context ...h quotient. Preconditioned gradient methods for the eigenvalue problem were first studied by Samokish [19] and later by Petryshyn [18]. Estimates on the convergence rate were given by Godunov et. al. =-=[9]-=- and D ′ yakonov et. al. [4, 8]. See Knyazev for a recent survey on preconditioned eigensolvers [13]. The viewpoint of preconditioned gradient methods for the convergence analysis of PINVIT seems to b... |

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