Fast iterative solvers for symmetric Toeplitz systems — A survey and an extension

Fast iterative Toeplitz solvers based on the preconditioned conjugate gradient (PCG) methods with circulant preconditioners were proposed in 1985. Since then, Sine-transform preconditioner and Hartley-transform preconditioner were proposed in 1990 and 1993, respectively. For solving a large family of Toeplitz systems Tnx = b, it requires only O(n log n) operations by using these preconditioners. In this paper, we give a brief survey and unify the analysis of all these preconditioners. An extension to the block Toeplitz systems is also given here.