Reduction to condensed form for the eigenvalue problem on distributed memory architectures
type: Item
instance of: scholarly article
title: Reduction to condensed form for the eigenvalue problem on distributed memory architectures
zbMATH Open document ID: 0758.65040
DOI: 10.1016/0167-8191(92)90011-U
author: Jack Dongarra
author: Robert A. van de Geijn
published in: Parallel Computing
publication date: 1993-01-17T00:00:00Z
review text: The authors discuss a parallel implementation of the LAPACK routines for reduction of a general matrix to Hessenberg form (and a symmetric matrix to tridiagonal form). The LAPACK project is designed to update the classical sequential codes for shared memory machines and this implementation is for running on the Intel Touchstone Delta. It is assumed the multicomputer has \(p\) nodes \(P_ 0,\dots,P_{p-1}\) connected by some network. If \(A\in \mathbb{R}^{n\times n}\) and the panelwidth \(m\) is such that \(n=r*m\), the partition \(A^{(k)}=(A_ 1^{(k)}A_ 2^{(k)}\dots A_ r^{(k)})\), where \(A_ j^{(k)}\in\mathbb{R}^{n\times m}\) is a panel of width \(m\). A panelwrapped scheme assigns \(A^{(k)}_ j\) to node \(P_{(j-1)\mod p}\), so that \(A_{i+1},A_{i+p+1},\dots\) are assigned to \(P_ i\). The authors then describe both sequential and parallel implementations of the reduction of Hessenberg form (and tridiagonal form in the symmetric case) using Householder transformations and follow with a description of the blocked form of these. The paper concludes with results of extensive numerical experiments with the parallel routines on matrices up to \(n=8000\) in size.
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 65Y05
zbMATH DE Number: 97047
zbMATH Keywords: eigenvalue problem
zbMATH Keywords: distributed memory architecture
zbMATH Keywords: parallel computation
zbMATH Keywords: reduction
zbMATH Keywords: Hessenberg form
zbMATH Keywords: tridiagonal form
zbMATH Keywords: LAPACK
zbMATH Keywords: shared memory machines
zbMATH Keywords: multicomputer
zbMATH Keywords: Householder transformations
zbMATH Keywords: numerical experiments
reviewed by: Adrian Swift
describes a project that uses: BLAS
describes a project that uses: LAPACK
MaRDI profile type: Publication
full work available at URL: 0167-8191(92)90011-u
OpenAlex ID: W2071237893
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description: scientific article; zbMATH DE number 97047