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<p>We present two fast algorithms for finding the solution of the nonsingular lower Hessenberg quasi-Toeplitz linear system stem from Markov chain. And we confirm the complexity of these two algorithms is both O<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_6791_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">\((n\log n)\)</EquationSource> </InlineEquation> based on the fact that a lower Hessenberg quasi-Toeplitz matrix can be written as the sum of a Toeplitz matrix and a rank-one matrix, such that the fast solver involves O<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_6791_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">\((n\log n)\)</EquationSource> </InlineEquation> operators for solving the Toeplitz linear system can be adopted. Finally, numerical results prove the superiority and accuracy of our algorithms by comparing the values of residual and CPU time with existing algorithms.</p>

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Two fast algorithms for finding the solution of the lower Hessenberg quasi-Toeplitz linear system from Markov chain

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月1日

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  • Yaru Fu,
  • Xiaoyu Jiang,
  • Yanpeng Zheng,
  • Zhaolin Jiang

摘要

We present two fast algorithms for finding the solution of the nonsingular lower Hessenberg quasi-Toeplitz linear system stem from Markov chain. And we confirm the complexity of these two algorithms is both O \((n\log n)\) based on the fact that a lower Hessenberg quasi-Toeplitz matrix can be written as the sum of a Toeplitz matrix and a rank-one matrix, such that the fast solver involves O \((n\log n)\) operators for solving the Toeplitz linear system can be adopted. Finally, numerical results prove the superiority and accuracy of our algorithms by comparing the values of residual and CPU time with existing algorithms.