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Recursive self preconditioning method based on Schur complement for Toeplitz matrices

机译：基于Schur补的Toeplitz矩阵递归自预处理方法

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摘要

In this paper, we propose to solve the Toeplitz linear systems T n x = b by a recursive-based method. The method is based on repeatedly dividing the original problem into two subproblems that involve the solution of systems containing the Schur complement of the leading principal submatrix of the previous level. The idea is to solve the linear systems S m y = d, where S m is the Schur complement of T 2m (the principal submatrix of T n ), by using a self preconditioned iterative methods. The preconditioners, which are the approximate inverses of S m , are constructed based on famous Gohberg–Semencul formula. All occurring matrices are represented by proper generating vectors of their displacement rank characterization. We show that, for well conditioned problems, the proposed method is efficient and robust. For ill-conditioned problems, by using some iterative refinement method, the new method would be efficient and robust. Numerical experiments are presented to show the effectiveness of our new method.
机译：在本文中，我们建议通过基于递归的方法求解Toeplitz线性系统T n x = b。该方法基于将原始问题反复分为两个子问题，这些子问题涉及包含上一级前导主要子矩阵的Schur补码的系统的解决方案。这个想法是通过使用自我预处理的迭代方法来求解线性系统S m y = d，其中S m是T 2m的Schur补码（T n的主要子矩阵）。预条件是S m的近似逆，是根据著名的Gohberg-Semencul公式构造的。所有出现的矩阵均由其位移秩表征的适当生成矢量表示。我们表明，对于条件良好的问题，所提出的方法是有效且鲁棒的。对于病态问题，通过使用一些迭代细化方法，新方法将是有效且健壮的。数值实验表明了该方法的有效性。

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