Numerical solution of singular Sylvester equations

Eric K. W. Chu; Liangshao Hou; Daniel B. Szyld; Jieyong Zhou

doi:10.1016/j.cam.2023.115426

Numerical solution of singular Sylvester equations

Eric K. W. Chu, Liangshao Hou, Daniel B. Szyld, Jieyong Zhou^*

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

2 Citations (Scopus)

Abstract

We consider the numerical solution of the large scale singular Sylvester equation of the form AX - XD^T = E (or AXB^T - CXD^T = E), where the spectra Λ (A) and Λ (D) (or, Λ (A -λC) and Λ (D -λB)) have a nonempty intersection. Using appropriate invariant subspaces, the singular Sylvester equation is rewritten as four Sylvester equations, three of which are nonsingular and one singular. When the invariant subspaces are small, so are three of the equations (including the singular one) which can be solved efficiently. The fourth is large but nonsingular with structures and may be solved using the projection method with Krylov subspaces or techniques involving hierarchical matrices. Some numerical examples for the subspace method are provided.

Original language	English
Article number	115426
Journal	Journal of Computational and Applied Mathematics
Volume	436
Early online date	6 Jul 2023
DOIs	https://doi.org/10.1016/j.cam.2023.115426
Publication status	Published - 15 Jan 2024

Scopus Subject Areas

Computational Mathematics
Applied Mathematics

User-Defined Keywords

Krylov subspace
Least squares solution
Lyapunov equation
Projection method
Singular equation
Sylvester equation

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10.1016/j.cam.2023.115426

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@article{e261c33db24f4d49915c4322a33a9e12,

title = "Numerical solution of singular Sylvester equations",

abstract = "We consider the numerical solution of the large scale singular Sylvester equation of the form AX - XDT = E (or AXBT - CXDT = E), where the spectra Λ (A) and Λ (D) (or, Λ (A -λC) and Λ (D -λB)) have a nonempty intersection. Using appropriate invariant subspaces, the singular Sylvester equation is rewritten as four Sylvester equations, three of which are nonsingular and one singular. When the invariant subspaces are small, so are three of the equations (including the singular one) which can be solved efficiently. The fourth is large but nonsingular with structures and may be solved using the projection method with Krylov subspaces or techniques involving hierarchical matrices. Some numerical examples for the subspace method are provided.",

keywords = "Krylov subspace, Least squares solution, Lyapunov equation, Projection method, Singular equation, Sylvester equation",

author = "Chu, {Eric K. W.} and Liangshao Hou and Szyld, {Daniel B.} and Jieyong Zhou",

note = "Publisher copyright: Crown Copyright {\textcopyright} 2023 Published by Elsevier B.V. All rights reserved. Funding Information: Part of the work was carried out when the first and fourth authors respectively visited the School of Mathematical Sciences at Fudan University since 2018 and the Department of Mathematics at Temple University in 2016.",

year = "2024",

month = jan,

day = "15",

doi = "10.1016/j.cam.2023.115426",

language = "English",

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journal = "Journal of Computational and Applied Mathematics",

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TY - JOUR

T1 - Numerical solution of singular Sylvester equations

AU - Chu, Eric K. W.

AU - Hou, Liangshao

AU - Szyld, Daniel B.

AU - Zhou, Jieyong

N1 - Publisher copyright: Crown Copyright © 2023 Published by Elsevier B.V. All rights reserved. Funding Information: Part of the work was carried out when the first and fourth authors respectively visited the School of Mathematical Sciences at Fudan University since 2018 and the Department of Mathematics at Temple University in 2016.

PY - 2024/1/15

Y1 - 2024/1/15

N2 - We consider the numerical solution of the large scale singular Sylvester equation of the form AX - XDT = E (or AXBT - CXDT = E), where the spectra Λ (A) and Λ (D) (or, Λ (A -λC) and Λ (D -λB)) have a nonempty intersection. Using appropriate invariant subspaces, the singular Sylvester equation is rewritten as four Sylvester equations, three of which are nonsingular and one singular. When the invariant subspaces are small, so are three of the equations (including the singular one) which can be solved efficiently. The fourth is large but nonsingular with structures and may be solved using the projection method with Krylov subspaces or techniques involving hierarchical matrices. Some numerical examples for the subspace method are provided.

AB - We consider the numerical solution of the large scale singular Sylvester equation of the form AX - XDT = E (or AXBT - CXDT = E), where the spectra Λ (A) and Λ (D) (or, Λ (A -λC) and Λ (D -λB)) have a nonempty intersection. Using appropriate invariant subspaces, the singular Sylvester equation is rewritten as four Sylvester equations, three of which are nonsingular and one singular. When the invariant subspaces are small, so are three of the equations (including the singular one) which can be solved efficiently. The fourth is large but nonsingular with structures and may be solved using the projection method with Krylov subspaces or techniques involving hierarchical matrices. Some numerical examples for the subspace method are provided.

KW - Krylov subspace

KW - Least squares solution

KW - Lyapunov equation

KW - Projection method

KW - Singular equation

KW - Sylvester equation

UR - https://www.sciencedirect.com/journal/journal-of-computational-and-applied-mathematics/vol/436/suppl/C

UR - http://www.scopus.com/inward/record.url?scp=85164730305&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2023.115426

DO - 10.1016/j.cam.2023.115426

M3 - Journal article

SN - 0377-0427

VL - 436

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

M1 - 115426

ER -

Numerical solution of singular Sylvester equations

Abstract

Scopus Subject Areas

User-Defined Keywords

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