A Riemannian Inexact Newton Dogleg Method for Constructing a Symmetric Nonnegative Matrix with Prescribed Spectrum

TL;DR

This paper introduces a Riemannian inexact Newton dogleg method to efficiently construct symmetric nonnegative matrices with a given spectrum, addressing an inverse problem in matrix theory.

Contribution

It develops a novel Riemannian underdetermined inexact Newton dogleg algorithm with proven convergence for solving the inverse matrix spectrum problem.

Findings

Method demonstrates quadratic convergence.

Numerical tests confirm efficiency and robustness.

Preconditioner improves solution accuracy.

Abstract

This paper is concerned with the inverse problem of constructing a symmetric nonnegative matrix from realizable spectrum. We reformulate the inverse problem as an underdetermined nonlinear matrix equation over a Riemannian product manifold. To solve it, we develop a Riemannian underdetermined inexact Newton dogleg method for solving a general underdetermined nonlinear equation defined between Riemannian manifolds and Euclidean spaces. The global and quadratic convergence of the proposed method is established under some mild assumptions. Then we solve the inverse problem by applying the proposed method to its equivalent nonlinear matrix equation and a preconditioner for the perturbed normal Riemannian Newton equation is also constructed. Numerical tests show the efficiency of the proposed method for solving the inverse problem.