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 $\mathrm{\Phi }$ 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 $B^{-1}C\ge 0$, and $B-C-I$ a nonsingular $M$-matrix. Such QME arises from an overdamped vibrating system. Recently, Yu et al. ({\em Appl. Math. Comput.}, 218: 3303–3310, 2011) proved that $\rho (\mathrm{\Phi })\le 1$ for this QME. In this paper, we slightly improve their result and prove $\rho (\mathrm{\Phi })<1$, 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 to solve the QME is developed. Numerical examples are presented to demonstrate the feasibility and effectiveness of our method.