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 $ \varPhi $ 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^{-1}C\ge 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(\varPhi)\le 1 $ for this QME. In this paper, under the same condition, we slightly improve their result and prove that $ \rho(\varPhi) < 1 $, which is important for the quadratic convergence of the structure-preserving 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.