A proposal for toeplitz matrix calculations
Studies in Applied Mathematics
Superfast solution of real positive definite toeplitz systems
SIAM Journal on Matrix Analysis and Applications
Fast iterative solvers for Toeplitz-plus-band systems
SIAM Journal on Scientific Computing
Conjugate Gradient Methods for Toeplitz Systems
SIAM Review
Matrix computations (3rd ed.)
How to Choose the Best Iterative Strategy for Symmetric Toeplitz Systems
SIAM Journal on Numerical Analysis
Factorized Banded Inverse Preconditioners for Matrices with Toeplitz Structure
SIAM Journal on Scientific Computing
Iterative Methods for Toeplitz Systems (Numerical Mathematics and Scientific Computation)
Iterative Methods for Toeplitz Systems (Numerical Mathematics and Scientific Computation)
Preconditioned Iterative Methods for Weighted Toeplitz Least Squares Problems
SIAM Journal on Matrix Analysis and Applications
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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.