A proposal for toeplitz matrix calculations
Studies in Applied Mathematics
Toeplitz equations by conjugate gradients with circulant preconditioner
SIAM Journal on Scientific and Statistical Computing
Circulant preconditioners for Hermitian Toeplitz systems
SIAM Journal on Matrix Analysis and Applications
Circulant preconditioners constructed from kernels
SIAM Journal on Numerical Analysis
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
Conjugate Gradient Methods for Toeplitz Systems
SIAM Review
New Band Toeplitz Preconditioners for Ill-Conditioned Symmetric Positive Definite Toeplitz Systems
SIAM Journal on Matrix Analysis and Applications
The Effectiveness of Band-Toeplitz Preconditioners: A Survey
WNAA '96 Proceedings of the First International Workshop on Numerical Analysis and Its Applications
Two-level Toeplitz preconditioning: approximation results for matrices and functions
SIAM Journal on Scientific Computing
Journal of Computational and Applied Mathematics
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In this paper, we consider solving the BTTB system ${\cal T}_{m,n}[f] {\bf{x}} = {\bf{b}}$ by the preconditioned conjugate gradient (PCG) method, where ${\cal T}_{m,n}[f]$ denotes the m 脳 m block Toeplitz matrix with n 脳 n Toeplitz blocks (BTTB) generated by a (2驴, 2驴)-periodic continuous function f(x, y). We propose using the BTTB matrix ${\cal T}_{m,n}[1/f]$ to precondition the BTTB system and prove that only O(m)驴+驴O(n) eigenvalues of the preconditioned matrix ${\cal T}_{m,n}[1/f] {\cal T}_{m,n}[f]$ are not around 1 under the condition that f(x, y)驴驴0. We then approximate 1/f(x, y) by a bivariate trigonometric polynomial, which can be obtained in O(m n log(m n)) operations by using the fast Fourier transform technique. Numerical results show that our BTTB preconditioner is more efficient than block circulant preconditioners.