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“Preconditioned Iterative Methods for Solving Toeplitz-Plus-Hankel Systems”
by Ta-Kang Ku and C.-C. Jay Kuo
June 1991
The use of preconditioned iterative methods to solve a system of equations with a Toeplitz-plus-Hankel coefficient matrix is studied. We propose a new preconditioner suitable for Toeplitz-plus-Hankel matrices, and examine the spectral properties of preconditioned rational Toeplitz-plus-Hankel matrices. We show 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 x N rational Toeplitz-plus-Hankel system can be solved by preconditioned iterative methods with O(N log N) operations. Numerical experiments are given to demonstrate the efficiency of the proposed preconditioner.