A structure-preserving doubling algorithm for solving a class of quadratic matrix equation with M-matrix

Consider the problem of finding the maximal nonpositive solvent Phi of the quadratic matrix equation (qme) X-2 + BX + C = 0 with B being a nonsingular M-matrix and C an M-matrix such that B-1C >= 0. Such qme arises from an overdamped vibrating system. Recently, under the condition that B - C - I is a nonsingular M-matrix, Yu et al. (Appl. Math. Comput., 218 (2011): 3303-3310) proved that rho(Phi) <= 1 for this qme. In this paper, under the same condition, we slightly improve their result and prove that rho(Phi) < 1, which is important for the quadratic convergence of the structurepreserving doubling algorithm. Then, a new globally monotonically and quadratically convergent structure-preserving doubling algorithm for solving the qme is developed. Numerical examples are presented to demonstrate the feasibility and effectiveness of our method.