Newton's method for the parameterized generalized eigenvalue problem with nonsquare matrix pencils

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.

Automated summary

In this paper, an efficient and effective algorithm based on Riemannian Newton's method is established to solve the nonsquare matrix penciled l-parameterized generalized eigenvalue problems. By combining different optimization methods, a hybrid algorithm with both global and quadratic convergence is obtained, and its efficiency is demonstrated through numerical experiments.