Backward error analysis and inverse eigenvalue problems for Hankel and Symmetric-Toeplitz structures

This work deals with the study of structured backward error analysis of Hankel and symmetric-Toeplitz matrix pencils. These structured matrix pencils belong to the class of symmetric matrix pencils with some additional properties that a symmetric matrix pencil does not have in general. The perturbation analysis of these two structures is discussed one by one to depict the additional properties explicitly. Present work shows the entry wise structured perturbation of matrix pencils in Frobenius norm such that the specified eigenpairs become exact eigenpairs of an appropriately perturbed matrix pencil. The framework used here maintains the sparsity in the perturbation of the above-structured matrix pencils. Further, the backward error results help for solving a variety of inverse eigenvalue problems . (c) 2021 Elsevier Inc. All rights reserved. Backward error analysis and inverse eigenvalue problems are among the most important topics in numerical linear algebra. We discuss the backward error analysis of Hankel and symmetric-Toeplitz matrix pencils for one or more specified eigenpairs and its use in solving the inverse eigenvalue problem . There are many applications of structured matrix pencils; see, e.g., [1,7,17] , more specifically, for example, a Hankel matrix pencil arises in the shape reconstruction of the polygon from its moments [13] and a symmetric-Toeplitz matrix pencil appears in the estimation of sinusoidal signals in noise [21] . For more information regarding these two matrix pencils, see [6,11,12] . Computing eigenvalues and eigenvectors of a given matrix pencil is always a difficult task (see, [16,18,20,26] ). Therefore

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This work focuses on the structured backward error analysis of Hankel and symmetric-Toeplitz matrix pencils, showcasing their additional properties and applications in solving inverse eigenvalue problems. The perturbation analysis of these structured matrix pencils in Frobenius norm helps maintain sparsity while ensuring exact eigenpairs. This framework proves useful in various numerical linear algebra tasks, particularly in solving inverse eigenvalue problems.