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

• Published in: Applied mathematics and computation 2021-10, Vol.406, p.126288, Article 126288
• Main Authors: Ahmad, Sk. Safique, Kanhya, Prince
• Format: Article
• Language: English
• Subjects: Backward error; Generalized inverse eigenvalue problem; Hankel generalized eigenvalue problems; Matrix pencil; Symmetric-Toeplitz generalized eigenvalue problem
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2021.126288

•Backward error analysis of one or more specied eigenpairs for Hankel and symmetricToeplitz matrices/ matrix pencils is discussed in detail.•For both the structures, minimum norm perturbed matrix pencils are obtained such that given eigenpairs become exact, which also preserve the sparsity.•Backward error results are developed using Frobenius norm.•Inverse eigenvalue problems and generalized inverse eigenvalue problems are solved for Hankel and symmetric-Toeplitz structures.•Solutions of inverse eigenvalue problems using backward error analysis show that backward error analysis and inverse eigenvalue problems are interconnected. 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 entrywise 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.