Analyzing the convergence factor of residual inverse iteration
type: Item
instance of: scholarly article
title: Analyzing the convergence factor of residual inverse iteration
zbMATH Open document ID: 1247.65070
author: Elias Jarlebring
author: Wim Michiels
published in: BIT
publication date: 2012-01-10T00:00:00Z
full work available at URL: https://lirias.kuleuven.be/handle/123456789/278609
review text: A formula for the convergence factor of the method called \textit{residual inverse iteration} for nonlinear eigenvalue problems and generalization of the well-known \textit{inverse iteration} is established. This formula is explicit and involves quantities associated with the eigenvalue to which the iteration converges, in particular the eigenvalue and eigenvector. Residual inverse iteration allows the choice of a vector \(w_k\) and the formula may be used for the convergence factor so as to analyze the dependence on the choice of \(w_k\). The formula is also used to illustrate the convergence when the shift is close to the eigenvalue. The slow convergence for double eigenvalues is explained by showing that under generic conditions the convergence factor is one, unless the eigenvalue is semisimple. Convergence similar to the simple case is expected when the eigenvalue is semisimple.
Mathematics Subject Classification ID: 65H17
zbMATH DE Number: 5996276
zbMATH Keywords: nonlinear eigenvalue problems
zbMATH Keywords: residual inverse iteration
zbMATH Keywords: convergence factors
zbMATH Keywords: double eigenvalues
describes a project that uses: NLEVP
MaRDI profile type: Publication
OpenAlex ID: W2071189202
cites work: The solution of characteristic value-vector problems by Newton's method
cites work: A GSVD formulation of a domain decomposition method forplanar eigenvalue problems
cites work: Stationary Schrödinger equations governing electronic states of quantum dots in the presence of spin-orbit splitting.
cites work: NLEVP
cites work: Invariance properties in the root sensitivity of time-delay systems with double imaginary roots
cites work: A block Newton method for nonlinear eigenvalue problems
cites work: Nonlinear Rayleigh-Ritz iterative method for solving large scale nonlinear eigenvalue problems
cites work: The Quadratic Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem
cites work: Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods
cites work: Residual Inverse Iteration for the Nonlinear Eigenvalue Problem
cites work: Inverse Iteration, Ill-Conditioned Equations and Newton’s Method
cites work: A minimax theory for overdamped systems
cites work: Algorithms for the Nonlinear Eigenvalue Problem
cites work: Q5390350
cites work: Q4337625
cites work: An Arnoldi method for nonlinear eigenvalue problems
cites work: Iterative projection methods for computing relevant energy states of a quantum dot
DOI: 10.1007/S10543-011-0336-2
description: scientific article; zbMATH DE number 5996276