A proposal for toeplitz matrix calculations
Studies in Applied Mathematics
Circulant preconditioners for Hermitian Toeplitz systems
SIAM Journal on Matrix Analysis and Applications
On the spectrum of a family of preconditioned block Toeplitz matrices
SIAM Journal on Scientific and Statistical Computing
Circulant preconditioners constructed from kernels
SIAM Journal on Numerical Analysis
Optimal and superoptimal circulant preconditioners
SIAM Journal on Matrix Analysis and Applications
Circulant and skewcirculant matrices for solving Toeplitz matrix problems
SIAM Journal on Matrix Analysis and Applications
Iterative solution methods and preconditioners for block-tridiagonal systems of equations
SIAM Journal on Matrix Analysis and Applications
A family of block preconditioners for block systems
SIAM Journal on Scientific and Statistical Computing
FFT-based preconditioners for Toeplitz-block least squares problems
SIAM Journal on Numerical Analysis
Conjugate Gradient Methods for Toeplitz Systems
SIAM Review
A note on construction of circulant preconditioners from kernels
Applied Mathematics and Computation
Any Circulant-Like Preconditioner for Multilevel Matrices Is Not Superlinear
SIAM Journal on Matrix Analysis and Applications
New Band Toeplitz Preconditioners for Ill-Conditioned Symmetric Positive Definite Toeplitz Systems
SIAM Journal on Matrix Analysis and Applications
The Best Circulant Preconditioners for Hermitian Toeplitz Systems
SIAM Journal on Numerical Analysis
Two-level Toeplitz preconditioning: approximation results for matrices and functions
SIAM Journal on Scientific Computing
A V-cycle Multigrid for multilevel matrix algebras: proof of optimality
Numerische Mathematik
An Introduction to Iterative Toeplitz Solvers (Fundamentals of Algorithms)
An Introduction to Iterative Toeplitz Solvers (Fundamentals of Algorithms)
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In this paper, we consider applying the preconditioned conjugate gradient (PCG) method to solve system of linear equations $$T x = \mathbf b $$ where $$T$$ is a block Toeplitz matrix with Toeplitz blocks (BTTB). We first consider Level-2 circulant preconditioners based on generalized Jackson kernels. Then, BTTB preconditioners based on a splitting of BTTB matrices are proposed. We show that the BTTB preconditioners based on splitting are special cases of embedding-based BTTB preconditioners, which are also good BTTB preconditioners. As an application, we apply the proposed preconditioners to solve BTTB least squares problems. Our preconditioners work for BTTB systems with nonnegative generating functions. The implementations of the construction of the preconditioners and the relevant matrix-vector multiplications are also presented. Finally, Numerical examples, including image restoration problems, are presented to demonstrate the efficiency of our proposed preconditioners.