Factorized Banded Inverse Preconditioners for Matrices with Toeplitz Structure

In this paper, we study factorized banded inverse preconditioners for matrices with Toeplitz structure. We show that if a Toeplitz matrix T has certain off-diagonal decay property, then the factorized banded inverse preconditioner approximates T-1 accurately, and the spectra of these preconditioned matrices are clustered around 1. In nonlinear image restoration applications, Toeplitz-related systems of the form I + T*DT are required to solve, where D is a positive nonconstant diagonal matrix. We construct inverse preconditionersfor such matrices. Numerical results show that the performance of our proposed preconditioners are superior to that of circulant preconditioners. A two-dimensional nonlinear image restoration example is also presented to demonstrate the effectiveness of the proposed preconditioner.