This paper is concerned with the parameterized least squares inverse eigenvalue problems for the case that the number of parameters to be constructed is greater than the number of prescribed realizable eigenvalues. Intrinsically, this is a specific problem of finding a zero of an under-determined nonlinear map defined between a Riemannian product manifold and a matrix space. To solve this problem, we propose a Riemannian under-determined BFGS algorithm with a specialized update formula for iterative linear operators, and an Armijo type line search is used. Global convergence properties of this algorithm are established under some mild assumptions. In addition, we also generalize a Riemannian inexact Newton method for solving this problem. Specially, the explicit form of the inverse of the linear operator corresponding to the perturbed normal Riemannian Newton equation is obtained, which improves the efficiency of Riemannian inexact Newton method. Numerical experiments are provided to illustrate the efficiency of the proposed method.

对于要构造的参数数量大于规定的可实现特征值数量的情况，本文涉及参数化的最小二乘反特征值问题。本质上，这是找到在黎曼乘积流形和矩阵空间之间定义的欠定非线性映射的零的特定问题。为了解决这个问题，我们提出了一种黎曼欠定BFGS算法，该算法具有针对迭代线性算子的专门更新公式，并使用了Armijo型线搜索。该算法的全局收敛性是在一些温和的假设下建立的。此外，我们还推广了黎曼不精确牛顿法来解决该问题。特别，得到与扰动的正常黎曼牛顿方程相对应的线性算子逆的显式形式，提高了黎曼不精确牛顿法的效率。数值实验表明了该方法的有效性。