TY - JOUR

T1 - Network Flows That Solve Sylvester Matrix Equations

AU - Deng, Wen

AU - Hong, Yiguang

AU - Anderson, Brian D.O.

AU - Shi, Guodong

N1 - Publisher Copyright:
© 1963-2012 IEEE.

PY - 2022/12/1

Y1 - 2022/12/1

N2 - In this article, we study methods to solve a Sylvester equation in the form of A {X}+ {X} {B}={C} for given matrices {A}, {B}, {C} R n× n}, inspired by the distributed linear equation flows. The entries of {A}, {B}, and {C} are separately partitioned into a number of pieces (or sometimes we permit these pieces to overlap), which are allocated to nodes in a network. Nodes hold a dynamic state shared among their neighbors defined from the network structure. Natural partial or full row/column partitions and block partitions of the data {A}, {B}, and {C} are formulated by use of the vectorized matrix equation. We show that existing network flows for distributed linear algebraic equations can be extended to solve this special form of matrix equations over networks. A 'consensus + projection + symmetrization' flow is also developed for equations with symmetry constraints on the matrix variables. We prove the convergence of these flows and obtain the fastest convergence rates that these flows can achieve regardless of the choices of node interaction strengths and network structures.

AB - In this article, we study methods to solve a Sylvester equation in the form of A {X}+ {X} {B}={C} for given matrices {A}, {B}, {C} R n× n}, inspired by the distributed linear equation flows. The entries of {A}, {B}, and {C} are separately partitioned into a number of pieces (or sometimes we permit these pieces to overlap), which are allocated to nodes in a network. Nodes hold a dynamic state shared among their neighbors defined from the network structure. Natural partial or full row/column partitions and block partitions of the data {A}, {B}, and {C} are formulated by use of the vectorized matrix equation. We show that existing network flows for distributed linear algebraic equations can be extended to solve this special form of matrix equations over networks. A 'consensus + projection + symmetrization' flow is also developed for equations with symmetry constraints on the matrix variables. We prove the convergence of these flows and obtain the fastest convergence rates that these flows can achieve regardless of the choices of node interaction strengths and network structures.

KW - Convergence rate

KW - Sylvester equation

KW - distributed algorithms

KW - network flows

UR - http://www.scopus.com/inward/record.url?scp=85120573350&partnerID=8YFLogxK

U2 - 10.1109/TAC.2021.3130877

DO - 10.1109/TAC.2021.3130877

M3 - Article

SN - 0018-9286

VL - 67

SP - 6731

EP - 6738

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

IS - 12

ER -