Spectrum and pseudspectrum of \(D\)-stable matrices of economy models

Volume 38, Issue 3, pp 298--312 https://dx.doi.org/10.22436/jmcs.038.03.02

Publication Date: December 16, 2024 Submission Date: October 02, 2024 Revision Date: October 17, 2024 Accteptance Date: November 03, 2024

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Authors

M.-U. Rehman - Center for Research and Innovation, Asia International University, Yangiobod MFY, G'ijduvon street, House 74, Bukhara, Uzbekistan. S. H. Alshabhi - Department of Mathematics, Applied college at Khulis, University of Jeddah, Jeddah, Saudi Arabia. A. O. Mustafa - College of Business at Khulis, University of Jeddah, Jeddah, Saudi Arabia. M. M. Mohammed - Department of Mathematics, Applied college at Khulis, University of Jeddah, Jeddah, Saudi Arabia. S. Aljohani - Department of Mathematics and Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia. H. S. Mahgoub - College of Business at Khulis, University of Jeddah, Jeddah, Saudi Arabia. R. A. A. Bashir - Department of Mathematics, Applied college at Khulis, University of Jeddah, Jeddah, Saudi Arabia. A. A. Bakery - Department of Mathematics, Applied college at Khulis, University of Jeddah, Jeddah, Saudi Arabia.

Abstract

The computation and analysis of structured singular values and \(D\)-stable structured matrices have an important and crucial role in system theory to study the stability and \(D\)-stability of dynamical systems. In this study, novel theoretical results are obtained to analyze interconnections between \(D\)-stability, \(D(\alpha)\)-stability, \(H\)-stability, a rank-1 perturbation to \(D\)-semi-stable matrices, and the computation of the bounds of structured singular values for structured and unstructured matrices. The proposed methodology is based on various tools from linear algebra, matrix analysis and system theory. The pseudo-spectrum of structured matrices appearing in economic models provides insights to analyze and characterize stability and instability analysis and sensitivity of linear dynamical models. The numerical experimentation's on the computation and comparison of lower bounds of structured singular values show the effectiveness of the proposed methodology. The Matlab EigTool is used on the computation of pseudo-spectrum for structured matrices arising from economy models.

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Keywords

• Singular values
• structured singular values
• \(D\)-stable matrix
• \(H\)-stable matrix

MSC

•  15A18
•  15A16
•  15A23

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