Backward error analysis of specified Eigenpairs and its application in solving inverse Eigenvalue problems

Abstract

KEYWORDS: Matrix pencil, matrix polynomial, two-parameter matrix system, back ward error, semisimple eigenvalue, sparsity, Hankel matrix, symmetric Toeplitz matrix, perturbation theory, generalized inverse eigenvalue problem, quadratic inverse eigenvalue problem, Frobenius norm. This thesis deals with the structured and unstructured backward errors, and pertur bation analysis of one or more specified eigenpairs for matrix pencils, matrix polynomials and two-parameter matrix systems. For a given set of specified eigenpairs, we develop a general framework on backward error analysis in such a way that various types of inverse eigenvalue problems, viz., matrix inverse eigenvalue problems, generalized inverse eigen value problems, and polynomial inverse eigenvalue problems are solved using our obtained backward error results. We raise the following two questions throughout the thesis with respect to matrix pencils, matrix polynomials, and two-parameter matrix systems. The first question is, what is the cumulative backward error of one or more approximate eigenpairs ? And the second question is, what is the nearest matrix pencil for which given approximate eigenpairs become the exact eigenpairs ? To answer the above-raised questions, first, we develop a general framework of the structured and unstructured backward error analysis of two specified eigenpairs of a double-semisimple eigenvalue for matrix pencils. We establish relationships between the unstructured backward error of a single eigenpair, structured backward error of two eigen pairs of a double semisimple eigenvalue, and the structured backward error of a single eigenpair. Next, we move towards the answers of the above-raised questions in a more general sense, i.e., the number of specified eigenpairs can be more than two and eigenval ues can be distinct. We further use the developed backward error results for solving the different inverse eigenvalue problems; for example, we solve real symmetric quadratic in verse eigenvalue problem and the symmetric generalized inverse eigenvalue problem with submatrix constraints. After then, we discuss the backward error analysis of symmetric-Toeplitz and Hankel matrix pencils. These two structured matrix pencils are particular types of a symmetric matrix pencil. We present the backward error analysis of these matrix pencils in such a way that the solutions of the symmetric-Toeplitz inverse eigenvalue problem and Hankel inverse eigenvalue problem are a consequence of it. Next, we discuss the backward error analysis for structured and unstructured ma trix polynomials and answer the above-raised questions. An n-by-n matrix polynomial of degree l have ln eigenvalues (finite or infinite) and the corresponding ln eigenvec tors. Hence for each structured matrix polynomial, we provide the upper bound on the maximum number of approximate eigenpairs whose backward error analysis can be done simultaneously. This challenge has not arisen during the backward error analysis of a single eigenpair. Further, we use the developed backward error results in solving different kinds of quadratic inverse eigenvalue problems. In particular, we solve symmetric and palindromic quadratic inverse eigenvalue problems. Finally, the backward error analysis has been developed for two-parameter matrix systems. We classify the two-parameter matrix systems based on the normal rank def inition. For two-parameter matrix systems, we obtain the structured and unstructured backward error results of two approximate eigenpairs provided eigenvalue is semisimple. Throughout the thesis, we answer the above-raised questions with respect to Frobenius norm.