The Stability and D-Stability Analysis of Toeplitz Linear Systems

The stability and D-stability analysis of Toeplitz linear systems is important and interesting problem in applied mathematics. Toeplitz matrices often appear in control theory, as well as across the computational mathematics. In this paper, we study and analyze Toeplitz system of linear equations of the form Tx=b, where T is a Toeplitz matrix, and provide some new results on its stability and D-stability. The D-stability is a stronger type of stability and it ensures that a given Toeplitz matrix remains stable under diagonal scaling. We develop and proposed a theoretical approach for the computation of spectrum and the relationship between D-stability and structured singular values for Toeplitz matrices. We make use of EigTool to visualize the pseudo-spectrum of higher dimensional Toeplitz matrices from Toeplitz system of linear equations. The behavior of the spectrum, pseudo-spectrum, singular values, and structured singular values of Hermitian Toeplitz matrices for an optimal mass transportation is demonstrated through various numerical experimentation.

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