An Ulm-like algorithm for generalized inverse eigenvalue problems

Abstract

In this paper, we study the numerical solutions of the generalized inverse eigenvalue problem (for short, GIEP). Motivated by Ulm’s method for solving general nonlinear equations and the algorithm of Aishima (J. Comput. Appl. Math. 367, 112485 2020) for the GIEP, we propose here an Ulm-like algorithm for the GIEP. Compared with other existing methods for the GIEP, the proposed algorithm avoids solving the (approximate) Jacobian equations and so it seems more stable. Assuming that the relative generalized Jacobian matrices at a solution are nonsingular, we prove the quadratic convergence property of the proposed algorithm. Incidentally, we extend the work of Luo et al. (J. Nonlinear Convex Anal. 24, 2309–2328 2023) for the inverse eigenvalue problem (for short, IEP) to the GIEP. Some numerical examples are provided and comparisons with other algorithms are made.