Concrete mathematics: a foundation for computer science
Concrete mathematics: a foundation for computer science
Discrete cosine transform: algorithms, advantages, applications
Discrete cosine transform: algorithms, advantages, applications
Polynomial and matrix computations (vol. 1): fundamental algorithms
Polynomial and matrix computations (vol. 1): fundamental algorithms
Analysis of preconditioning techniques for ill-conditioned Toeplitz matrices
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Conjugate Gradient Methods for Toeplitz Systems
SIAM Review
Any Circulant-Like Preconditioner for Multilevel Matrices Is Not Superlinear
SIAM Journal on Matrix Analysis and Applications
Preconditioners for Ill-Conditioned Toeplitz Systems Constructed from Positive Kernels
SIAM Journal on Scientific Computing
New Band Toeplitz Preconditioners for Ill-Conditioned Symmetric Positive Definite Toeplitz Systems
SIAM Journal on Matrix Analysis and Applications
How to prove that a preconditioner cannot be superlinear
Mathematics of Computation
Stability of the notion of approximating class of sequences and applications
Journal of Computational and Applied Mathematics
Superlinear convergence for PCG using band plus algebra preconditioners for Toeplitz systems
Computers & Mathematics with Applications
A note on algebraic multigrid methods for the discrete weighted Laplacian
Computers & Mathematics with Applications
Antireflective boundary conditions for deblurring problems
Journal of Electrical and Computer Engineering - Special issue on iterative signal processing in communications
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In the last decades several matrix algebra optimal and superlinear preconditioners (those assuring a strong clustering at the unity) have been proposed for the solution of polynomially ill-conditioned Toeplitz linear systems. The corresponding generalizations for multilevel structures are neither optimal nor superlinear (see e.g. Contemp. Math. 281 (2001) 193). Concerning the notion of superlinearity, it has been recently shown that the proper clustering cannot be obtained in general (see Linear Algebra Appl. 343-344 (2002) 303; SIAM J. Matrix Anal. Appl. 22(1) (1999) 431; Math. Comput. 72 (2003) 1305). In this paper, by exploiting a proof technique previously proposed by the authors (see Contemp. Math. 323 (2003) 313), we prove that the spectral equivalence and the essential spectral equivalence (up to a constant number of diverging eigenvalues) are impossible too. In conclusion, optimal matrix algebra preconditioners in the multilevel setting simply do not exist in general and therefore the search for optimal iterative solvers should be oriented to different directions with special attention to multilevel/multigrid techniques.