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 $B^{-1}C\ge 0$, 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 $\rho(\Phi)\le 1$ for this QME. In this paper, we slightly improve their result and prove $\rho(\Phi)< 1$, 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 to solve the QME is developed. Numerical examples are presented to demonstrate the feasibility and effectiveness of our method.
翻译:考虑找到四边矩阵方程式的最大非阳性溶剂$\Phi$(QME) $X%2 + BX + C =0美元,其中B$为非单币美元和1美元美元,例如,B ⁇ -1}C\ge 0美元和B-C美元-C美元是非单币美元。这种QME产生于一个过份的振动系统。最近,Yu 等人(them Appl. Math.comput.}),218:3303-3310,2011年)证明,$\rho(\\\\le 1美元,用于这一QME。在本文件中,我们略微改进了结果,并证明$\rho(\Phi) < 1美元,这对于结构-保值双倍算法的四边汇合十分重要。随后,开发了一个新的全球单调和二次趋同结构-保持双倍算法来解决QME。