Efficient Approaches for Enclosing the United Solution Set of the Interval Generalized Sylvester Matrix Equations

TL;DR

This paper introduces efficient methods for estimating the solution set of interval generalized Sylvester matrix equations, reducing computational complexity and leveraging spectral decompositions for improved enclosure accuracy.

Contribution

It presents a modified Krawczyk operator and an iterative technique for solution set enclosure, utilizing spectral properties and diagonalizability assumptions to enhance efficiency.

Findings

Reduced computational complexity to cubic order

Effective solution set enclosures demonstrated through numerical experiments

Spectral decomposition-based methods improve accuracy and efficiency

Abstract

In this work, we investigate the interval generalized Sylvester matrix equation ${\bf{A}}X{\bf{B}}+{\bf{C}}X{\bf{D}}={\bf{F}}$ and develop some techniques for obtaining outer estimations for the so-called united solution set of this interval system. First, we propose a modified variant of the Krawczyk operator which causes reducing computational complexity to cubic, compared to Kronecker product form. We then propose an iterative technique for enclosing the solution set. These approaches are based on spectral decompositions of the midpoints of ${\bf{A}}$, ${\bf{B}}$, ${\bf{C}}$ and ${\bf{D}}$ and in both of them we suppose that the midpoints of ${\bf{A}}$ and ${\bf{C}}$ are simultaneously diagonalizable as well as for the midpoints of the matrices ${\bf{B}}$ and ${\bf{D}}$. Some numerical experiments are given to illustrate the performance of the proposed methods.