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. This advantage allows us 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.

拟托普利茨M矩阵A是无限M矩阵，可以写为半无限托普利茨矩阵和校正矩阵的和。本文涉及计算保留拟托普利茨结构的可逆拟托普利茨M矩阵的平方根。我们表明，平方根的托普利茨部分可以通过评估/插值轻松计算。这一优点使我们能够提出仅用于计算校正部分的算法，由此我们提出了定点迭代和结构保留加倍算法。此外，我们还表明，可以通过求解具有有限大小系数的非线性矩阵方程，然后将解扩展到无穷大来近似校正部分。数值实验显示了所提出算法的效率。