On the matrix equation X + AX−1B = Q with semi-infinite quasi-Toeplitz coefficients

Jie Meng and Yuezhi Wang

Applied Mathematics and Computation, 2026, vol. 509, issue C

Abstract: This paper is concerned with computing the extremal solution of the matrix equation X+AX−1B=Q, where the coefficients A,B and Q are semi-infinite quasi-Toeplitz matrices. A quasi-Toeplitz matrix A is an infinite size matrix that can be written as the sum of a semi-infinite Toeplitz matrix and a correction matrix. We show that there is an extremely larger solution X+ of the nonlinear matrix equation under certain conditions. Moreover, the extremely larger solution preserves the quasi-Toeplitz structure and is an invertible quasi-Toeplitz M-matrix if Q is an invertible quasi-Toeplitz M-matrix. Fixed-point iterations, including a quadratically convergent algorithm based on the cyclic reduction, are analyzed for the computation of X+. Numerical experiments showing the efficiency of the proposed algorithms are performed.

Keywords: Matrix equation; Quasi-Toeplitz matrix; Infinite M-matrix; Fixed-point iteration; Cyclic reduction (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:509:y:2026:i:c:s0096300325003741

DOI: 10.1016/j.amc.2025.129648

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