ON THE SQUARE ROOT OF QUASI-TOEPLITZ M-MATRICES

초록

A quasi-Toeplitz M-matrix A is an infinite size M-matrix with an almost Toeplitz structure, that is, A = alpha I - T(a) E with T(a) + E >= 0 and rho(T(a)+E) <= alpha, where T(a) is a semi-infinite Toeplitz matrix associated with the function a(z) = Sigma(i is an element of z)alpha(i)z(i )in the sense that T(a)(i,j), and E = (e(i-j))(i, j is an element of z)+ is a correction matrix such that lim(i Sigma j=1)|e(i-j)| = 0. In this paper, we first show some theoretical applications of properties of invertible quesi-Toeplita M-matrices. In particular, we deduce properties of function s(z) such that s(2)(2) = a(z), where a(z) is the function associated with the Toeplitz part of a quasi-Toeplitz M-matrix. Then, we pay our attention to the computation of square root of invertible quasi-Toeplitz M-matrices. We show that the square root can be approximated by computing square root of a finite M-matrix via Schulz method, followed by extending the computed square root to infinity. Finally, we show by numerical experiments the efficiency of the proposed iterative method.

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