A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix
type: Item
instance of: scholarly article
title: A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix
zbMATH Open document ID: 0565.65018
DOI: 10.1016/0024-3795(84)90034-X
author: Charles F. Van Loan
published in: Linear Algebra and its Applications
publication date: 1984-01-01T00:00:00Z
review text: The paper presents a fast method for finding the eigenvalues of a Hamiltonian matrix M, that is of the partitioned form \(\left[ \begin{matrix} A^ T\\ H\end{matrix} \begin{matrix} G\\ -A\end{matrix} \right].\) It does this by squaring M and showing that using a special form of orthogonal matrix \(M^ 2\) can be reduced to the form \(\left[ \begin{matrix} H\\ 0\end{matrix} \begin{matrix} R\\ H^ T\end{matrix} \right].\) The standard QR algorithm is then used to calculate the eigenvalues of H which are the squares of the eigenvalues of M. Algorithms are given explicitly which minimize the work involved and which reduce the storage required. Examples are given and an error analysis is provided.
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 3901911
zbMATH Keywords: symplectic method
zbMATH Keywords: Hamiltonian matrix
zbMATH Keywords: QR algorithm
zbMATH Keywords: eigenvalues
zbMATH Keywords: error analysis
reviewed by: Brian L. Burrows
describes a project that uses: EISPACK
describes a project that uses: LINPACK
MaRDI profile type: Publication
cites work: Matrix eigensystem routines - EISPACK guide
cites work: Q4401802
cites work: A Schur method for solving algebraic Riccati equations
cites work: Q4057472
cites work: Q3850185
cites work: Q5674306
full work available at URL: 0024-3795(84)90034-x
OpenAlex ID: W2328218167
description: scientific article; zbMATH DE number 3901911