Preconditioned iterative methods for solving Toeplitz-plus-Hankel systems

The use of preconditioned iterative methods to solve a system of equations with a Toeplitz-plus-Hankel coefficient matrix is studied. A preconditioner for Toeplitz-plus-Hankel matrices is proposed, and the spectral properties of preconditioned rational Toeplitz-plus-Hankel matrices are examined. It is shown that the eigenvalues of the preconditioned matrix are clustered around unity except a finite number of outliers depending on the orders of the rational generating functions, and the clustering radius is proportional to the magnitude of the last elements in Toeplitz and Hankel matrices. With the spectral regularities, an N*N rational Toeplitz-plus-Hankel system can be solved by preconditioned iterative methods with O(N log N) operations.<>