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 solely for the computation of correction part, whence we propose a fixed-point iteration and a structure-preserving doubling algorithm. Additionally, 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$A$是一种可以表示为半无限Toeplitz矩阵与修正矩阵之和的无限$M$-矩阵。本文研究可逆拟Toeplitz $M$-矩阵平方根的计算问题，该平方根保持拟Toeplitz结构。我们证明，可通过在$m$次单位根上的求值/插值轻松计算平方根的Toeplitz部分。这一优势使得我们能够仅针对修正部分的计算提出算法，进而提出不动点迭代和保结构加倍算法。此外，我们展示了修正部分可以通过求解有限大小系数的非线性矩阵方程，然后将解扩展至无穷来近似。数值实验验证了所提出算法的有效性。