A survey and some extensions of T. Chan’s preconditioner

For any given n-by-n matrix An, T. Chan’s circulant preconditioner cF(An) proposed by T. Chan [T. Chan, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Statist. Comput. 9 (1988) 766–771] is defined to be the solution ofminCnAn-Cn‖Fover all n-by-n circulant matrices Cn. The cF(An), called the optimal circulant preconditioner by T. Chan, has been proved to be a good preconditioner for a large class of structured systems. A generalization of T. Chan’s circulant preconditioner was given by Huckle [T. Huckle, Circulant and skew circulant matrices for solving Toeplitz matrix problems, SIAM J. Matrix Anal. Appl. 13 (1992) 767–777] which can be used for solving some general linear systems. In this paper, we review some old and develop some new properties of T. Chan’s circulant preconditioner.