Preconditioners for symmetrized Toeplitz and multilevel Toeplitz matrices
This work provides a new preconditioning approach for solving linear systems with symmetrized Toeplitz matrices, benefiting researchers and practitioners in numerical linear algebra and related fields.
The paper addresses the lack of effective preconditioners for symmetrized (multilevel) Toeplitz matrices in Krylov subspace methods. It proposes novel ideal preconditioners, analyzes their spectral properties, and demonstrates their effectiveness through numerical experiments.
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.