A divide-and-conquer method for the tridiagonal generalized eigenvalue problem
type: Item
instance of: scholarly article
title: A divide-and-conquer method for the tridiagonal generalized eigenvalue problem
zbMATH Open document ID: 1007.65024
DOI: 10.1016/S0377-0427(97)00152-0
author: A. Fasse
author: Ludwig Elsner
author: Edwin Langmann
published in: Journal of Computational and Applied Mathematics
publication date: 2003-03-09T00:00:00Z
review text: This paper presents a new method to solve the generalized eigenvalue problem \[ Ax= \lambda Bx,\quad \tag{1} \] where \(A\) and \(B\) are real symmetric tridiagonal matrices, with \(B\) positive definite. In the first phase, by means of a convenient rank one modification, the problem is reduced to solving two similar problems whose dimensions can be arbitrarily chosen so that their sum is the dimension of problem (1). With the solutions of these smaller problems, (1) is transformed into a different generalized eigenvalue problem. The core of the paper is devoted to show that this modified problem can be nicely solved. As pointed out by the authors, the method thus proposed is an extension of the one analysed by \textit{J. J. Cuppen} [Numer. Math. 36, 177-195 (1981; Zbl 0431.65022)] to solve (1) when \(B=I\). Also some aspects of the method are discussed in relation to the one proposed by \textit{C. F. Borges} and \textit{W. B. Gragg} [Proc. Conf. Numer. Linear Algebra Sci. Comput. 1992, 11-29 (1993; Zbl 0799.65043)] for the same problem, which show the interest of its further examination.
Mathematics Subject Classification ID: 65F15
zbMATH DE Number: 1115929
zbMATH Keywords: divide-and-conquer method
zbMATH Keywords: generalized eigenvalue problem
zbMATH Keywords: tridiagonal symmetric matrices
reviewed by: Juan Pedro Milaszewicz
MaRDI profile type: Publication
cites work: Q4296886
cites work: Rank-one modification of the symmetric eigenproblem
cites work: A Divide and Conquer method for the symmetric tridiagonal eigenproblem
cites work: A Fully Parallel Algorithm for the Symmetric Eigenvalue Problem
cites work: A Stable and Efficient Algorithm for the Rank-One Modification of the Symmetric Eigenproblem
description: scientific article; zbMATH DE number 1115929