Riemannian inexact Newton method for structured inverse eigenvalue and singular value problems

Abstract

Inverse eigenvalue and singular value problems have been widely discussed for decades. The well-known result is the Weyl-Horn condition, which presents the relations between the eigenvalues and singular values of an arbitrary matrix. This result by Weyl-Horn then leads to an interesting inverse problem, i.e., how to construct a matrix with desired eigenvalues and singular values. In this work, we do that and more. We propose an eclectic mix of techniques from differential geometry and the inexact Newton method for solving inverse eigenvalue and singular value problems as well as additional desired characteristics such as nonnegative entries, prescribed diagonal entries, and even predetermined entries. We show theoretically that our method converges globally and quadratically, and we provide numerical examples to demonstrate the robustness and accuracy of our proposed method. Having theoretical interest, we provide in the appendix a necessary and sufficient condition for the existence of a 2 × 2 real matrix, or even a nonnegative matrix, with prescribed eigenvalues, singular values, and main diagonal entries.Mathematics Subject Classification (2010) 15A29 · 65H17.Keywords Inverse eigenvalue and singular value problems, nonnegative matrices, Riemannian inexact Newton method.

An Extended Two-Step Method for Inverse Eigenvalue Problems with Multiple Eigenvalues

“…The following lemma presents the generalized Jacobian ∂ Q f(•) and the relative generalized Jacobian ∂ Q|S f(•), see for instance [4,26].…”

“…In this paper, we consider the formulation and local analysis of the numerical algorithm for solving the IEP (1.2). As described in [4,5,22], solving the IEP (1.2) is equivalent to solving the following nonlinear equation:…”

“…there exists a neighborhood of c * where the function f is analytic (cf. [4]). Thus, based on the equivalence of the IEP (1.2) and the nonlinear equation (1.3), Newton's method and various Newton-type methods were proposed to solve the IEP (1.2) in the distinct case [2,4,10,21,22].…”

“…[4]). Thus, based on the equivalence of the IEP (1.2) and the nonlinear equation (1.3), Newton's method and various Newton-type methods were proposed to solve the IEP (1.2) in the distinct case [2,4,10,21,22]. However, when multiple eigenvalues are present, solving the IEP (1.2) becomes more complicated in the sense that the eigenvalues are not in general differentiable at c * and that the eigenvectors of A(c * ) are not unique, and they cannot generally be defined to be continuous functions of c at c * .…”

On the globalization of Riemannian Newton method

“…Recent years have witnessed a growing interest in the development of numerical algorithms for nonlinear manifolds, as there are many numerical problems posed in manifolds arising in various natural contexts. For example, eigenvalue problems [14,34,46,50,51], low-rank matrix completion [49], loss minimization problem [42] and dextrous hand grasping problem [16,24,25]. For such problems, the solutions of a system of equations often have to be computed or the zeros of a vector field have to be found.…”