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

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  • Meng, Jie
  • Wang, Yuezhi

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.

Suggested Citation

  • Meng, Jie & Wang, Yuezhi, 2026. "On the matrix equation X + AX−1B = Q with semi-infinite quasi-Toeplitz coefficients," Applied Mathematics and Computation, Elsevier, vol. 509(C).
    • Handle: RePEc:eee:apmaco:v:509:y:2026:i:c:s0096300325003741
    DOI: 10.1016/j.amc.2025.129648
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    References listed on IDEAS

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    1. Gertsbakh, Ilya, 1984. "The shorter queue problem: A numerical study using the matrix-geometric solution," European Journal of Operational Research, Elsevier, vol. 15(3), pages 374-381, March.
    2. Engwerda, J.C., 1993. "On the existence of a positive definite solution of the matrix equation X = ATX-1A = I," Other publications TiSEM 9d762863-0dfe-4aeb-8a13-5, Tilburg University, School of Economics and Management.
    3. Engwerda, J.C. & Ran, A.C.M. & Rijkeboer, A.L., 1992. "Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X+A*X-1A=Q," Other publications TiSEM cbc6bc1e-3bbf-49a4-8222-d, Tilburg University, School of Economics and Management.
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