Preconditioners for Symmetrized Toeplitz and Multilevel Toeplitz Matrices
Preconditioners for Symmetrized Toeplitz and Multilevel Toeplitz Matrices
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DOI：
10.1137/18m1205406
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发表时间：
2018-12
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影响因子：
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通讯作者：
J. Pestana
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文献类型：
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作者：
J. Pestana
在用Krylov子空间方法求解具有非对称Toeplitz或多层Toeplitz矩阵的线性系统时，系数矩阵可以是对称的。然后将预条件MINRES方法应用于该对称系统，从而可以获得MINRES迭代次数的严格上界。然而，对于对称（多层）Toeplitz矩阵，缺乏有效的预条件。在此，我们提出了新的理想预条件，并研究了预条件矩阵的谱。我们展示了这些预条件是如何被近似的，并通过数值实验证明了它们的有效性。
When solving linear systems with nonsymmetric Toeplitz or multilevel Toeplitz matrices using Krylov subspace methods, the coefficient matrix may be symmetrized. The preconditioned MINRES method can then be applied to this symmetrized system, which allows rigorous upper bounds on the number of MINRES iterations to be obtained. However, effective preconditioners for symmetrized (multilevel) Toeplitz matrices are lacking. Here, we propose novel ideal preconditioners, and investigate the spectra of the preconditioned matrices. We show how these preconditioners can be approximated and demonstrate their effectiveness via numerical experiments.