Trigonometric Polynomial Approximations for BTTB Systems

We discuss the numerical solution of block systems T_m,n x = b by preconditioned conjugate gradient methods where T_m,n are m × m block Toeplitz matrices with n × n 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 T^-1 as a preconditioner for the above system. We prove that the resulting preconditioned matrix T^-1 will have clustered spectrum. Numerical results show that our preconditioners are more efficient than circulant preconditioners.