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Zbl 0863.65013
Chan, Raymond H.; Ng, Michael K.
Conjugate gradient methods for Toeplitz systems. (English)
[J] SIAM Rev. 38, No.3, 427-482 (1996). ISSN 0036-1445; ISSN 1095-7200/e

The use of preconditioned conjugate gradient methods to solve linear systems of equations with Toeplitz matrices is discussed. Using this iterative method, the complexity is reduced from $O(n\log^2n)$ operations for fast direct Toeplitz solvers to $O(n \log n)$. The authors review the use of preconditioners based on circulant matrices, embedding, minimization of norms, optimal transform (like the fast Fourier or sine transform) and band Toeplitz matrices. The various techniques are applied for solving partial differential equations, queuing problems, signal and image restoration, integral equations and time series analysis (filtering).
[W.Gander (Zürich)]

MSC 2000:: *65F10 Iterative methods for linear systems
68U10 Image processing
65F35 Matrix norms, etc. (numerical linear algebra)
65R20 Integral equations (numerical methods)
65N30 Finite numerical methods (BVP of PDE)
65C99 Numerical simulation

Keywords: fast Fourier transform; fast sine transform; preconditioned conjugate gradient methods; Toeplitz matrices; complexity; circulant matrices; queuing problems; signal and image restoration; integral equations; time series analysis; filtering

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