A Survey of Preconditioners for Ill-conditioned Toeplitz Systems

In this paper, we survey some of latest developments in using preconditioned conjugate gradient methods for solving mildly ill-conditioned Toeplitz systems where the condition numbers of the systems grow like O(n) for some > 0. This corresponds to Toeplitz matrices generated by functions having zeros of order. Because of the ill-conditioning, the number of iterations required for convergence in the conjugate gradient method will grow like O(n =2). Diierent preconditioners proposed for these Toeplitz matrices are reviewed. The main result is that the total complexity of solving an ill-conditioned Toeplitz system is of O(n log n) operations.

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