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

03/12/2021

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by   Cairong Chen, et al.

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Consider the problem of finding the maximal nonpositive solvent Φ 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^-1C≥ 0, and B - C - I a nonsingular M-matrix. Such QME arises from an overdamped vibrating system. Recently, Yu et al. (Appl. Math. Comput., 218: 3303–3310, 2011) proved that ρ(Φ)≤ 1 for this QME. In this paper, we slightly improve their result and prove ρ(Φ)< 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.