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 $B^{-1}C\ge 0$. 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 $\rho (\Phi )\le 1$ 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 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.