A note on the numerical solution of complex Hamiltonian and skew-Hamiltonian eigenvalue problems
type: Item
instance of: scholarly article
title: A note on the numerical solution of complex Hamiltonian and skew-Hamiltonian eigenvalue problems
zbMATH Open document ID: 0936.65045
author: P. Benner
author: Volker Mehrmann
author: Hongguo Xu
publication date: 1999-09-19T00:00:00Z
full work available at URL: https://eudml.org/doc/120255
full work available at URL: pp115-126.html
review text: Computation of eigenvalues of a \((2n,2n)\) skew-Hamiltonian matrix \(N\) with complex elements is effected by transforming the \((4n,4n)\) block diagonal matrix with diagonal blocks \(N\), \(\overline N\) into a real skew-Hamiltonian matrix \({\mathcal N}\) whose eigenvalues agree with those of \(N\). The matrix \({\mathcal N}\) may be put into a skew-Hamiltonian Schur from \({\mathcal R}\) and the eigenvalues may be obtained by a method of \textit{C. F. Van Loan} [Linear Algebra Appl. 61, 233-251 (1984; Zbl 0565.65018)]. The results apply to complex Hamiltonian matrices as well since if \(N\) is skew-Hamiltonian then \(H= (-iN)\) is Hamiltonian. The eigenspaces associated with the positive and negative eigenvalues of \(H\), \(\text{Inv}_+H\), \(\text{Inv}_-H\), are computed from the Hamiltonian Schur form \((-i{\mathcal R})\) assuming that \(H\) has no purely imaginary eigenvalues. These eigenspaces are useful in certain control problems. A code ZHAEV has been written using the new numerical method and an error analysis of the method is presented. In two numerical tests ZHAEV is compared with ZGEEV, a general code for determining eigenvalues of nonsymmetric matrices. The results obtained show that the new code is faster and more accurate than the more general code.
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 93B36
Mathematics Subject Classification ID: 93B40
zbMATH DE Number: 1336577
zbMATH Keywords: skew-Hamiltonian matrix
zbMATH Keywords: algebraic Riccati equation
zbMATH Keywords: invariant subspace
zbMATH Keywords: eigenvalues
zbMATH Keywords: complex Hamiltonian matrices
zbMATH Keywords: eigenspaces
zbMATH Keywords: error analysis
zbMATH Keywords: numerical tests
reviewed by: John B. Butler jun.
describes a project that uses: LAPACK
MaRDI profile type: Publication
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published in: ETNA - Electronic Transactions on Numerical Analysis
description: scientific article; zbMATH DE number 1336577