The shift-inverted \(J\)-Lanczos algorithm for the numerical solutions of large sparse algebraic Riccati equations
type: Item
instance of: scholarly article
title: The shift-inverted \(J\)-Lanczos algorithm for the numerical solutions of large sparse algebraic Riccati equations
zbMATH Open document ID: 0889.65034
DOI: 10.1016/S0898-1221(97)00074-6
author: Wen-Wei Lin
author: Chern-Shuh Wang
author: William R. Ferng
publication date: 1998-06-02T00:00:00Z
review text: A shift invert Lanczos algorithm applicable to compute the eigenvalues of a large and sparse Hamiltonian matrix is described. It computes a symplectic basis two columns at a time, in which the matrix is represented by a 2 by 2 block matrix with 3 diagonal and one tridiagonal block. It is shown that convergence occurs, and symplecticity gets lost, when a quantity computed from the eigenvectors of the small block matrix and its last nondiagonal elements gets small, in a way very similar to the symmetric case. A numerical example is described, and different shift and invert transformations are discussed.
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 1067178
zbMATH Keywords: large sparse algebraic Riccati equations
zbMATH Keywords: shift invert Lanczos algorithm
zbMATH Keywords: eigenvalues
zbMATH Keywords: large and sparse Hamiltonian matrix
zbMATH Keywords: symplectic basis
zbMATH Keywords: convergence
zbMATH Keywords: numerical example
reviewed by: Axel Ruhe
MaRDI profile type: Publication
cites work: A Schur method for solving algebraic Riccati equations
cites work: A symplectic QR like algorithm for the solution of the real algebraic Riccati equation
cites work: A Chart of Numerical Methods for Structured Eigenvalue Problems
cites work: A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix
cites work: On Hamiltonian and symplectic Hessenberg forms
cites work: Computation of Stable Invariant Subspaces of Hamiltonian Matrices
cites work: Q5284141
cites work: A Convergence Analysis for Nonsymmetric Lanczos Algorithms
cites work: Eigenvalues of perturbed Hermitian matrices
cites work: On some algebraic problems in connection with general eigenvalue algorithms
cites work: Residual Bounds on Approximate Eigensystems of Nonnormal Matrices
full work available at URL: s0898-1221(97)00074-6
OpenAlex ID: W2147129570
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 1067178