Iterative Methods for Toeplitz-like Matrices

In this paper we will give a survey on iterative methods for solving linear equations with Toeplitz matrices. We introduce a new class of Toeplitz matrices for which clustering of eigenvalues and singular values can be proved. We consider optimal (ω)circulant preconditioners as a generalization of the circulant preconditioner. For positive definite Toeplitz matrices, especially in the real case, there is a hard competition between the fast, superfast, and iterative solvers. Therefore, it is necessary to get optimal implementations of the iterative solver. We will show different ways to get improved preconditioned conjugate gradient algorithms, and compare the number of flops for the three concurrent methods. Furthermore, we show different ways to deal with nearsingular Toeplitz matrices.