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Consider the problem of finding the maximal nonpositive solvent $\Phi$ of thequadratic 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 $B - C - I$ anonsingular $M$-matrix. Such QME arises from an overdamped vibrating system.Recently, Yu et al. ({\em Appl. Math. Comput.}, 218: 3303--3310, 2011) provedthat $\rho(\Phi)\le 1$ for this QME. In this paper, we slightly improve theirresult and prove $\rho(\Phi)< 1$, which is important for the quadraticconvergence of the structure-preserving doubling algorithm. Then, a newglobally monotonically and quadratically convergent structure-preservingdoubling algorithm to solve the QME is developed. Numerical examples arepresented to demonstrate the feasibility and effectiveness of our method.

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