The extension of the concept of the generating function to a class of preconditioned Toeplitz matrices

Serra, Stefano

doi:10.1016/s0024-3795(97)80047-x

Linear Algebra and its Applications

1997

DOI: 10.1016/s0024-3795(97)80047-x

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The extension of the concept of the generating function to a class of preconditioned Toeplitz matrices

Stefano Serra¹

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Cited by 10 publications

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A Korovkin-type theory for finite Toeplitz operators via matrix algebras

Serra¹

1999

Numerische Mathematik

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Preconditioned conjugate gradients (PCG) are widely and successfully used methods for solving a Toeplitz linear system A n x = b [59,9,20,5,34,62,6,10,28,45,44,46,49]. Frobenius-optimal preconditioners are chosen in some proper matrix algebras and are defined by minimizing the Frobenius distance from A n . The convergence features of these PCG have been naturally studied by means of the Weierstrass-Jackson Theorem [17,36,49], owing to the profound relationship between the spectral features of the matrices A n , generated by the Fourier coefficients of a continuous function f , and the analytical properties of the symbol f itself. In this paper, we capsize this point of view by showing that the optimal preconditioners can be used to define both new and just known linear positive operators uniformly approximating the function f . On the other hand, by modifying the Korovkin Theorem to study the Frobenius-optimal preconditioning problem, we provide a new and unifying tool for analyzing all Frobenius-optimal preconditioners in any generic matrix algebra related to trigonometric transforms. Finally, the multilevel case is sketched and discussed by showing that a Korovkin-type Theory also holds in a multivariate sense. Subject Classification (1991): 65F10, 47B25, 41A36, 15A18 Mathematics

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A Korovkin-type theory for finite Toeplitz operators via matrix algebras

Serra¹

1999

Numerische Mathematik

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Korovkin theorems and linear positive Gram matrix algebra approximations of Toeplitz matrices

Serra‐Capizzano

1998

Linear Algebra and its Applications

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An ergodic theorem for classes of preconditioned matrices

Serra‐Capizzano

1998

Linear Algebra and its Applications

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The extension of the concept of the generating function to a class of preconditioned Toeplitz matrices

Cited by 10 publications

References 37 publications

A Korovkin-type theory for finite Toeplitz operators via matrix algebras

A Korovkin-type theory for finite Toeplitz operators via matrix algebras

Korovkin theorems and linear positive Gram matrix algebra approximations of Toeplitz matrices

An ergodic theorem for classes of preconditioned matrices

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