A Riemannian Inexact Newton-CG Method for Nonnegative Inverse Eigenvalue Problems: Nonsymmetric Case

TL;DR

This paper introduces a Riemannian inexact Newton-CG method to solve the nonnegative inverse eigenvalue problem for nonsymmetric matrices, demonstrating its efficiency and convergence properties through numerical experiments.

Contribution

It develops a novel Riemannian Newton-CG algorithm for the nonsymmetric nonnegative inverse eigenvalue problem, including extensions and convergence analysis.

Findings

Method achieves quadratic convergence.

Numerical experiments confirm efficiency.

Extension to prescribed entries case.

Abstract

This paper is concerned with the nonnegative inverse eigenvalue problem of finding a nonnegative matrix such that its spectrum is the prescribed self-conjugate set of complex numbers. We first reformulate the nonnegative inverse eigenvalue problem as an under-determined constrained nonlinear matrix equation over several matrix manifolds. Then we propose a Riemannian inexact Newton-CG method for solving the nonlinear matrix equation. The global and quadratic convergence of the proposed method is established under some mild conditions. We also extend the proposed method to the case of prescribed entries. Finally, numerical experiments are reported to illustrate the efficiency of the proposed method.