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Conjugate gradient methods for Toeplitz systems. (English) Zbl 0863.65013
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(nlog 2 n) operations for fast direct Toeplitz solvers to O(nlogn). 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).
MSC:
65F10Iterative methods for linear systems
68U10Image processing (computing aspects)
65F35Matrix norms, conditioning, scaling (numerical linear algebra)
65R20Integral equations (numerical methods)
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65C99Probabilistic methods, simulation and stochastic differential equations (numerical analysis)