A quasi-Toeplitz $M$-matrix $A$ is an infinite $M$-matrix that can be written as the sum of a semi-infinite Toeplitz matrix and a correction matrix. This paper is concerned with computing the square root of invertible quasi-Toeplitz $M$-matrices which preserves the quasi-Toeplitz structure. We show that the Toeplitz part of the square root can be easily computed through evaluation/interpolation at the $m$ roots of unity. This advantage allows to propose algorithms only for the computation of correction part, whence we propose a fixed-point iteration and a structure-preserving doubling algorithm. Moreover, we show that the correction part can be approximated by solving a nonlinear matrix equation with coefficients of finite size followed by extending the solution to infinity. Numerical experiments showing the efficiency of the proposed algorithms are performed.

翻译：拟Toeplitz $M$-矩阵$A$是一类可表示为半无限Toeplitz矩阵与修正矩阵之和的无限$M$-矩阵。本文研究保持拟Toeplitz结构的可逆拟Toeplitz $M$-矩阵平方根的计算问题。我们证明通过单位根$m$点处的求值/插值可轻松计算平方根的Toeplitz部分，该优势使得仅需针对修正部分设计算法，据此提出不动点迭代与保结构加倍算法。进一步地，我们证明修正部分可通过求解有限尺寸系数矩阵的非线性矩阵方程并延拓解至无穷域来逼近。数值实验验证了所提算法的有效性。