The l parameterized generalized eigenvalue problems for the nonsquare matrix pencils, proposed by Chu et al.in 2006, can be formulated as an optimization problem on a corresponding complex product Stiefel manifold. In this paper, an effective and efficient algorithm based on the Riemannian Newton’s method is established to solve the underlying problem. Under our proposed framework, to solve the corresponding Newton’s equation, it can be converted to solve a standard real symmetric linear system with a dimension reduction. By combining the Riemannian curvilinear search method with Barzilai–Borwein steps, a hybrid algorithm with both global and quadratic convergence is obtained. Numerical experiments are provided to illustrate the efficiency of the proposed method. Detailed comparisons with some latest methods are also provided to show the merits of the proposed approach.

该升Chu等人在2006年提出的非平方矩阵铅笔的参数化广义特征值问题，可以表示为对应复杂产品Stiefel流形上的优化问题。本文建立了一种基于黎曼牛顿法的高效算法来解决潜在的问题。在我们提出的框架下，为了求解相应的牛顿方程，可以将其转换为求解尺寸减小的标准实对称线性系统。通过将黎曼曲线搜索方法与Barzilai–Borwein步骤结合，获得了具有全局收敛性和二次收敛性的混合算法。数值实验表明了该方法的有效性。