Riemannian Newton-CG Methods for Constructing a Positive Doubly Stochastic Matrix From Spectral Data

TL;DR

This paper develops Riemannian Newton-CG algorithms to solve the inverse eigenvalue problem for positive doubly stochastic matrices, ensuring convergence and providing practical numerical solutions.

Contribution

It introduces novel Riemannian inexact Newton-CG methods for constructing positive doubly stochastic matrices from spectral data, with proven convergence properties.

Findings

Methods achieve global and quadratic convergence.

Numerical tests demonstrate effectiveness in practical applications.

Application to digraphs shows real-world utility.

Abstract

In this paper, we consider the inverse eigenvalue problem for the positive doubly stochastic matrices, which aims to construct a positive doubly stochastic matrix from the prescribed realizable spectral data. By using the real Schur decomposition, the inverse problem is written as a nonlinear matrix equation on a matrix product manifold. We propose monotone and nonmonotone Riemannian inexact Newton-CG methods for solving the nonlinear matrix equation. The global and quadratic convergence of the proposed methods is established under some assumptions. We also provide invariant subspaces of the constructed solution to the inverse problem based on the computed real Schur decomposition. Finally, we report some numerical tests, including an application in digraph, to illustrate the effectiveness of the proposed methods.