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<p>Theoretical and computational properties of semi-infinite quasi-Toeplitz <i>M</i>-matrices are investigated recently in [ Linear Algebra Appl., 653, 66–85, 2022]. In this paper, we propose a more general framework which allows to deal with semi-infinite quasi-Toeplitz <i>M</i>-matrix plus a rank-1 perturbation. Matrices of this kind arise in certain Markov processes involving restarts. We investigate properties of such <i>M</i>-matrices, and propose appropriate algorithms that take advantage of the matrix structure for the computational issues relating to the square root. In particular, a nonlinear equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(Vx-\mu _xx=b\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mi>x</mi> <mo>-</mo> <msub> <mi>μ</mi> <mi>x</mi> </msub> <mi>x</mi> <mo>=</mo> <mi>b</mi> </mrow> </math></EquationSource> </InlineEquation> where <i>V</i> is an invertible quasi-Toeplitz <i>M</i>-matrix and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mu _x=\Vert x\Vert _1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>μ</mi> <mi>x</mi> </msub> <mo>=</mo> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>x</mi> <mo stretchy="false">‖</mo> </mrow> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> is encountered. We propose algorithms that enable the numerical treatment of such nonlinear equations. Finally, we show by numerical examples the effectiveness of the proposed algorithms.</p>

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On a class of infinite M-matrices with an extended quasi-Toeplitz structure

  • 期刊论文

  • 发表日期：

    2025年4月22日

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  • Jie Meng

摘要

Theoretical and computational properties of semi-infinite quasi-Toeplitz M-matrices are investigated recently in [ Linear Algebra Appl., 653, 66–85, 2022]. In this paper, we propose a more general framework which allows to deal with semi-infinite quasi-Toeplitz M-matrix plus a rank-1 perturbation. Matrices of this kind arise in certain Markov processes involving restarts. We investigate properties of such M-matrices, and propose appropriate algorithms that take advantage of the matrix structure for the computational issues relating to the square root. In particular, a nonlinear equation \(Vx-\mu _xx=b\) V x - μ x x = b where V is an invertible quasi-Toeplitz M-matrix and \(\mu _x=\Vert x\Vert _1\) μ x = ‖ x ‖ 1 is encountered. We propose algorithms that enable the numerical treatment of such nonlinear equations. Finally, we show by numerical examples the effectiveness of the proposed algorithms.