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A quasi-Toeplitz $M$-matrix $A$ is an infinite $M$-matrix that can be writtenas the sum of a semi-infinite Toeplitz matrix and a correction matrix. Thispaper is concerned with computing the square root of invertible quasi-Toeplitz$M$-matrices which preserves the quasi-Toeplitz structure. We show that theToeplitz part of the square root can be easily computed throughevaluation/interpolation at the $m$ roots of unity. This advantage allows topropose algorithms solely for the computation of correction part, whence wepropose a fixed-point iteration and a structure-preserving doubling algorithm.Additionally, we show that the correction part can be approximated by solving anonlinear matrix equation with coefficients of finite size followed byextending the solution to infinity. Numerical experiments showing theefficiency of the proposed algorithms are performed.

Algorithms for square root of semi-infinite quasi-Toeplitz $M$-matrices