Matrix analysis
A proposal for toeplitz matrix calculations
Studies in Applied Mathematics
Toeplitz system associated with the product of a formal Laurent series and a Laurent polynomial
SIAM Journal on Matrix Analysis and Applications
Superfast solution of real positive definite toeplitz systems
SIAM Journal on Matrix Analysis and Applications
Toeplitz equations by conjugate gradients with circulant preconditioner
SIAM Journal on Scientific and Statistical Computing
The spectrum of a family of circulant preconditioned Toeplitz systems
SIAM Journal on Numerical Analysis
Circulant preconditioners for Hermitian Toeplitz systems
SIAM Journal on Matrix Analysis and Applications
Displacement structure: theory and applications
SIAM Review
Conjugate Gradient Methods for Toeplitz Systems
SIAM Review
A Fast Stable Solver for Nonsymmetric Toeplitz and Quasi-Toeplitz Systems of Linear Equations
SIAM Journal on Matrix Analysis and Applications
Iterative Methods for Toeplitz Systems (Numerical Mathematics and Scientific Computation)
Iterative Methods for Toeplitz Systems (Numerical Mathematics and Scientific Computation)
A look-ahead Levinson algorithm for general Toeplitz systems
IEEE Transactions on Signal Processing
Design and analysis of Toeplitz preconditioners
IEEE Transactions on Signal Processing
Noniterative and fast iterative methods for interpolation andextrapolation
IEEE Transactions on Signal Processing
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This paper explores a seemingly counter-intuitive idea: the possibility of accelerating the solution of certain linear equations by adding even more equations to the problem. The basic insight is to trade-off problem size by problem structure. We test this idea on Toeplitz equations, in which case the expense of a larger set of equations easily leads to circulant structure. The idea leads to a very simple iterative algorithm, which works for a certain class of Toeplitz matrices, each iteration requiring only two circular convolutions. In the symmetric definite case, numerical experiments show that the method can compete with the preconditioned conjugate gradient method (PCG), which achieves O(nlogn) performance. Because the iteration does not converge for all Toeplitz matrices, we give necessary and sufficient conditions to ensure convergence (for not necessarily symmetric matrices), and suggest an efficient convergence test. In the positive definite case we determine the value of the free parameter of the circulant that leads to the fastest convergence, as well as the corresponding value for the spectral radius of the iteration matrix. Although the usefulness of the proposed iteration is limited in the case of ill-conditioned matrices, we believe that the results show that the problem size/problem structure trade-off deserves further study.