On the rate of convergence of the preconditioned conjugate gradient method
Numerische Mathematik
A proposal for toeplitz matrix calculations
Studies in Applied Mathematics
Circulant preconditioners for Hermitian Toeplitz systems
SIAM Journal on Matrix Analysis and Applications
A new preconditioner for the parallel solution of positive definite Toeplitz systems
SPAA '90 Proceedings of the second annual ACM symposium on Parallel algorithms and architectures
Circulant preconditioners constructed from kernels
SIAM Journal on Numerical Analysis
On a matrix algebra related to the discrete Hartley transform
SIAM Journal on Matrix Analysis and Applications
Fast band-Toeplitz preconditioners for Hermitian Toeplitz systems
SIAM Journal on Scientific Computing
Analysis of preconditioning techniques for ill-conditioned Toeplitz matrices
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Preconditioning of Block Toeplitz Matrices by Sine Transforms
SIAM Journal on Scientific Computing
Superlinear PCG methods for symmetric Toeplitz systems
Mathematics of Computation
Effective Methods for Solving Banded Toeplitz Systems
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
Theoretical Computer Science - Algebraic and numerical algorithm
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The paper studies fast and efficient solution algorithms for nxn symmetric ill conditioned Toeplitz systems T"n(f)x=b where the generating function f is known a priori, real valued, nonnegative, and has isolated roots of even order. The preconditioner that we propose is a product of a band Toeplitz matrix and matrices that belong to a certain trigonometric algebra. The basic idea behind the proposed scheme is to combine the advantages of all components of the product that are well known when every component is used as a stand-alone preconditioner. As a result we obtain a flexible preconditioner which can be applied to the system T"n(f)x=b infusing superlinear convergence to the PCG method. The important feature of the proposed technique is that it can be extended to cover the 2D case, i.e. ill-conditioned block Toeplitz matrices with Toeplitz blocks. We perform many numerical experiments, whose results confirm the theoretical analysis and effectiveness of the proposed strategy.