Approximated least-squares solutions of a generalized Sylvester-transpose matrix equation via gradient-descent iterative algorithm

Adisorn Kittisopaporn,Pattrawut Chansangiam

doi:10.1186/s13662-021-03427-4

Approximated least-squares solutions of a generalized Sylvester-transpose matrix equation via gradient-descent iterative algorithm

Adisorn Kittisopaporn, Pattrawut Chansangiam

Open Access

https://doi.org/10.1186/s13662-021-03427-4

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Abstract

This paper proposes an effective gradient-descent iterative algorithm for solving a generalized Sylvester-transpose equation with rectangular matrix coefficients. The algorithm is applicable for the equation and its interesting special cases when the associated matrix has full column-rank. The main idea of the algorithm is to have a minimum error at each iteration. The algorithm produces a sequence of approximated solutions converging to either the unique solution, or the unique least-squares solution when the problem has no solution. The convergence analysis points out that the algorithm converges fast for a small condition number of the associated matrix. Numerical examples demonstrate the efficiency and effectiveness of the algorithm compared to renowned and recent iterative methods.

Highlights

In differential equations and control engineering, there has been much attention for the following linear matrix equations: AXB = C, (1)AX + XAT = B : Lyapunov equation, (2)AX + XB = C : Sylvester equation, (3)AXB + CXD = E : a generalized Sylvester equation, (4)AXB + CXT D = E : a generalized Sylvester-transpose equation, (5)X + AXB = C : Stein equation
A group of methods, called gradient-based iterative methods, aim to construct a sequence of approximated solutions that converges to the exact solution for any given initial matrices
The following two methods were proposed to produce the sequence X(k) of approximated solutions converging to the exact solution X∗ of Eq (8)

Summary

Introduction

A group of methods, called gradient-based iterative methods, aim to construct a sequence of approximated solutions that converges to the exact solution for any given initial matrices. The following two methods were proposed to produce the sequence X(k) of approximated solutions converging to the exact solution X∗ of Eq (8). Our varied step sizes are the optimal convergence factors that guarantee the algorithm to have a minimum error at each iteration. 2 Exact and least-squares solutions of the matrix equation by the Kronecker linearization

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