HMDR and FMDR algorithms for the generalized eigenvalue problem
type: Item
instance of: scholarly article
title: HMDR and FMDR algorithms for the generalized eigenvalue problem
zbMATH Open document ID: 0664.65034
DOI: 10.1016/0024-3795(89)90593-4
author: D. C. Dzeng
author: Wen-Wei Lin
published in: Linear Algebra and its Applications
publication date: 1989-01-01T00:00:00Z
review text: The authors present a new algorithm for the solution of the generalized eigenproblem \(Ax=\lambda Bx\), with \(A,B\in {\mathbb{R}}^{n\times n}\). The matrix B may be reduced to a positive semidefinite diagonal matrix D by the transformation \(\pi BQ^ T=_ R\), where \(\pi\) is a permutation matrix and Q is orthogonal; this implies that the standard form may be written in the form \(Ax=\lambda Dx.\) The MDR transformation is essentially two-dimensional and is designed to annihilate an element of A in an efficient well-conditioned manner [cf. \textit{A. Bunse-Gerstner}, ibid. 58, 43-68 (1984; Zbl 0575.65028)]. The current paper presents two variants of this method; the HMDR is based on Householder transformations and the FMDR is based on a fast MDR which is nearly as well conditioned but requires less computational work. To overcome problems arising in the presence of infinite eigenvalues a new `preprocessing' procedure is described. A comparison of the number of `flops' required by the new method as against the commonly used QZ algorithm shows a 60\% improvement over the latter. The stability of the new method, which in principle resembles the QR algorithm, is similar to that of the MDR method.
Mathematics Subject Classification ID: 65F15
zbMATH DE Number: 4086881
zbMATH Keywords: Hessenberg matrices
zbMATH Keywords: preprocessing procedure
zbMATH Keywords: generalized eigenproblem
zbMATH Keywords: Householder transformations
zbMATH Keywords: QZ algorithm
zbMATH Keywords: stability
zbMATH Keywords: QR algorithm
zbMATH Keywords: MDR method
MaRDI profile type: Publication
full work available at URL: 0024-3795(89)90593-4
OpenAlex ID: W2005758608
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description: scientific article; zbMATH DE number 4086881