Approximate Inverse Circulant-plus-Diagonal Preconditioners for Toeplitz-plus-Diagonal Matrices

We consider the solutions of Hermitian positive definite Toeplitz-plus-diagonal systems (T +D)x = b, where T is a Toeplitz matrix and D is diagonal and positive. However, unlike the case of Toeplitz systems, no fast direct solvers have been developed for solving them. In this paper, we employ the preconditioned conjugate gradient method with approximate inverse circulant-plus-diagonal preconditioners to solving such systems. The proposed preconditioner can be constructed and implemented efficiently using fast Fourier transforms. We show that if the entries of T decay away exponentially from the main diagonals, the preconditioned conjugate gradient method applied to the preconditioned system converges very quickly. Numerical examples including spatial regularization for image deconvolution application are given to illustrate the effectiveness of the proposed preconditioner.