Numerical Algorithm For Solving The Non-negative Inverse Eigenpair Problem

The nonnegative inverse eigenvalue problem aims to find a non-negative matrix based on some of its eigenvalues or eigenvalue vectors which are given in advance.Nowadays,the problem attracts some attentions due to its wide applications in many fields,such as random analysis,application of physical control theory,and data processing and so on.The study of the problem mainly includes two important issues: one is about the existence and the uniqueness of the solutions of the problem;the other is the research of effective numerical algorithms for finding a solution.We focus on the later,more precisely,the effective numerical algorithms for solving the nonnegative matrix inverse eigenvalue problem with partial eigendata.Firstly,we propose the alternating projection algorithm for solving the general nonnegative inverse eigenvalue problem with partial eigendata by reformulating it as a convex feasibility problem with two convex sets.The linear convergence of the algorithm is established.At last,some numerical experiments are provided to illustrate the convergence effectiveness of the alternating projection algorithm and the non-smooth Newton method in [Bai,Z.J.,Stefano,S.C.,Zhao,Z.,Numer.Math.,387–431,2012].The numerical results show that the alternating projection algorithm performances sharply better than the non-smooth Newton method in two senses.Firstly,the alternating projection algorithm always converges to a solution of the problem in all the cases thanks to its global convergence property,while the non-smooth Newton method fails in some cases since it owes only local convergence property(under some assumptions made for the initial point).All of the numerical results show that the alternating projection algorithm is sharply more effective than that the non-smooth Newton method measured in terms of the time costs.Finally,we consider two special kinds of the nonnegative inverse eigenvalue problems with partial eigendata: the row stochastic one and a symmetric stochastic one.Both of them are reformulated as convex feasibility problems.The alternating projection algorithm and the projection algorithm are proposed for solving these two problems,respectively.The linear convergence of the alternating projection algorithm for solving the row stochastic inverse eigenvalue problems with the partial eigendata is established,and some numerical experiments are provided to show the convergence of both algorithms.