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arXiv · 2303.15019

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

Hongjia Chen·Hyun-MIn Kim·Jie Meng
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Abstract

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.

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Hongjia Chen, Hyun-MIn Kim, Jie Meng. 2023-03-27. Algorithms for square root of semi-infinite quasi-Toeplitz $M$-matrices. https://arxiv.org/abs/2303.15019

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