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

TL;DR

This paper introduces a new structure-preserving doubling algorithm for solving a specific quadratic matrix equation with M-matrices, proving its quadratic convergence and demonstrating its effectiveness through numerical examples.

Contribution

The paper develops a globally convergent doubling algorithm for a class of quadratic matrix equations with M-matrices, improving convergence properties and providing numerical validation.

Findings

Proved that the spectral radius of the maximal nonpositive solvent is less than 1.

Established quadratic convergence of the proposed doubling algorithm.

Demonstrated the algorithm's effectiveness through numerical experiments.

Abstract

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.