Fast verified computation for the solution of the T-congruence Sylvester equation
type: Item
instance of: scholarly article
title: Fast verified computation for the solution of the T-congruence Sylvester equation
zbMATH Open document ID: 1434.65047
author: Shinya Miyajima
published in: Japan Journal of Industrial and Applied Mathematics
publication date: 2018-12-21T00:00:00Z
review text: An algorithm is proposed to compute an interval matrix that contains the symmetric matrix $X$ that solves $AX+X^TB=C$ (all matrices are in $\mathbb{C}^{n\times n}$). It was shown in [\textit{M. Oozawa} et al., J. Comput. Appl. Math. 329, 51--56 (2018; Zbl 1373.15023)] how, under certain conditions, the problem can be transformed into a Lyapunov equation. Based on this result, conditions on the numerically computed generalized eigendecomposition of the pencil $(B,A^T)$ or $(A,B^T)$ are formulated to guarantee the existence of a unique solution. The algorithm finding the required interval matrix has complexity $O(n^3)$ assuming that the employed enclosure routines for products and inverses of matrices have complexity at most $O(n^3)$. The MATLAB code using INTLAB (interval arithmetic toolbox) is available online.
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 65F45
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 65G20
zbMATH DE Number: 6996761
zbMATH Keywords: T-congruence Sylvester equation
zbMATH Keywords: discrete Lyapunov equation
zbMATH Keywords: verified numerical computation
describes a project that uses: INTLAB
describes a project that uses: VERSOFT
MaRDI profile type: Publication
full work available at URL: s13160-018-0307-y
OpenAlex ID: W2801423590
cites work: An Algorithm for Computing Reducing Subspaces by Block Diagonalization
cites work: Structured Condition Numbers for Invariant Subspaces
cites work: Verified error bounds for solutions of Sylvester matrix equations
cites work: Q4917542
cites work: Q4303969
cites work: Implicit QR algorithms for palindromic and even eigenvalue problems
cites work: Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations
cites work: Numerical enclosure for each eigenvalue in generalized eigenvalue problem
cites work: Fast enclosure for solutions of Sylvester equations
cites work: Fast enclosure for solutions of generalized Sylvester equations
cites work: Fast Enclosure for All Eigenvalues and Invariant Subspaces in Generalized Eigenvalue Problems
cites work: On a relationship between the \(\operatorname{T}\)-congruence Sylvester equation and the Lyapunov equation
cites work: Q4949347
cites work: Q3165156
cites work: The solution of the equation \(AX + X^{\star}B =0\)
cites work: On the Kronecker products and their applications
DOI: 10.1007/S13160-018-0307-Y
description: scientific article; zbMATH DE number 6996761