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We discuss the numerical solution of block systems x b by preconditioned conjugate gradient methods where are block Toeplitz matrices with ! ! Toeplitz blocks (BTTB). These systems occur in a variety of applications, such as two-dimensional image processing and the discretization of two-dimensional partial differential equations. In this paper, we propose the BTTB matrix " $ as a preconditioner for the above system. We prove that the resulting preconditioned matrix " $ will have clustered spectrum. Numerical results show that our preconditioners are more efficient than circulant preconditioners.

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