Inexact inverse iteration for generalized eigenvalue problems
type: Item
instance of: scholarly article
title: Inexact inverse iteration for generalized eigenvalue problems
zbMATH Open document ID: 0984.65032
DOI: 10.1023/A:1022388317839
author: Gene H. Golub
author: Qiang Ye
published in: BIT
publication date: 2002-04-30T00:00:00Z
review text: To solve the generalized eigenvalue problem \(Ax=\lambda Bx\), one can use the inverse iteration method where in each iteration a linear system of equation \(Az_{k+1}=Bx_k\), has to be solved. In recent years, it has been proposed to solve that system by iterative schemes leading to an inexact inverse iteration method. In this paper, convergence results in function of two stopping criteria are given. In particular, the stopping criteria used are \(\|q_k\|<\varepsilon \|Bx_k-Ay_k\|\) or \(\|q_k\|<\varepsilon_k \|y_{k+1}\|\), where \(y_{k+1}\) is an approximate to \(z_{k+1}\) and \(\varepsilon\) is a threshold. In the convergence analysis, under the hypotheses that the matrix \(B^{-1}A\) is diagonalizable and the two minimum eigenvalues have different absolute value, the authors prove that the convergence is linear under the natural assumption that \(\varepsilon_k\) decreases linearly with respect to a parameter \(\gamma\leq 1\). In particular, they prove that the convergence is linear at the rate of the maximum of \(\rho\) and the parameter \(\gamma\leq 1\), where \(\rho\) is the quotient of the absolute value of the two minimum eigenvalues of \(B^{-1}A\). When using the other stopping criterion with fixed \(\varepsilon\) the convergence is linear but not to zero. Further, the computational work is analyzed. More precisely, the effect of \(\gamma\) on the total amount of work is studied proving that the optimal choice of this parameter is greater than \(\rho\) and as close to it as possible. Finally, there are some numerical results for solving the eigenvalue problem of the convection diffusion equation. These numerical problems illustrate the theoretical results of.
reviewed by: R. Bru
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 65N25
Mathematics Subject Classification ID: 35P15
zbMATH DE Number: 1556488
zbMATH Keywords: inverse iteration method
zbMATH Keywords: shift-and-invert
zbMATH Keywords: inner-outer iterations
zbMATH Keywords: numerical examples
zbMATH Keywords: generalized eigenvalue problem
zbMATH Keywords: convergence
zbMATH Keywords: stopping criterion
zbMATH Keywords: numerical results
zbMATH Keywords: convection diffusion equation
MaRDI profile type: Publication
full work available at URL: a:1022388317839
OpenAlex ID: W62683955
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description: scientific article; zbMATH DE number 1556488