Recursive self preconditioning method based on Schur complement for Toeplitz matrices

In this paper, we propose to solve the Toeplitz linear systems Tnx = 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 Smy = d, where Sm is the Schur complement of T2m (the principal submatrix of Tn), by using a self preconditioned iterative methods. The preconditioners, which are the approximate inverses of Sm, 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.

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