Conjugate Gradient Methods for Toeplitz Systems 论文

摘要

In this expository paper, we survey some of the latest developments on using preconditioned conjugate gradient methods for solving Toeplitz systems. One of the main results is that the complexity of solving a large class of n-by-n Toeplitz systems is reduced to O(n log n) operations as compared to O(n log 2 n) operations required by fast direct Toeplitz solvers. Different preconditioners proposed for Toeplitz systems are reviewed. Applications to Toeplitz-related systems arising from partial differential equations, queueing networks, signal and image processing, integral equations, and time series analysis are given. Key words. Toeplitz matrices, preconditioners, preconditioned conjugate gradient methods, differential equations, signal and image processing, time series, queueing problems, integral equations. AMS Subject classifications. 45E10, 62M10, 65F10, 65N22, 65P05, 68U10, 93E11 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong. Research suppo...