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.

翻译：本文关注从可变频谱构建对称非负式矩阵的逆向问题。 我们重新将反向问题作为里曼尼亚产品方块上一个不确定的非线性矩阵方程式。 为了解决这个问题, 我们开发了里曼尼人不确定的牛顿狗腿方法, 以解决里曼尼人方块和欧几里德空间之间定义的普通非线性方程式。 所提议方法的全球和四面形趋同是在一些温和假设下确定的。 然后我们通过将拟议方法应用于等同的非线性矩阵方程式来解决反向问题, 同时也为周遭的里曼人牛顿方程式搭建了先决条件。 数字测试显示了解决反问题的拟议方法的效率 。