How to prove that a preconditioner cannot be superlinear

Authors:
S. Serra Capizzano;E. Tyrtyshnikov
Affiliations:
Dipartimento di Chimica, Fisica e Matematica, Università Dell'Insubria - Sede di Como, Via Valleggio 11, 22100 Como, Italy;Institute of Numerical Mathematics, Russian Academy of Sciences, Gubkina 8, Moscow 117333, Russia
Venue:
Mathematics of Computation
Year:
2003

Citing 10
Cited 2

On the rate of convergence of the preconditioned conjugate gradient method

Numerische Mathematik
The rate of convergence of conjugate gradients

Numerische Mathematik
On a matrix algebra related to the discrete Hartley transform

SIAM Journal on Matrix Analysis and Applications
Displacement structure: theory and applications

SIAM Review
Multigrid Methods for Symmetric Positive Definite Block Toeplitz Matrices with Nonnegative Generating Functions

SIAM Journal on Scientific Computing
Conjugate Gradient Methods for Toeplitz Systems

SIAM Review
Multigrid Method for Ill-Conditioned Symmetric Toeplitz Systems

SIAM Journal on Scientific Computing
Toeplitz Preconditioners Constructed from Linear Approximation Processes

SIAM Journal on Matrix Analysis and Applications
Any Circulant-Like Preconditioner for Multilevel Matrices Is Not Superlinear

SIAM Journal on Matrix Analysis and Applications
Optimal Kronecker Product Approximation of Block Toeplitz Matrices

SIAM Journal on Matrix Analysis and Applications

Matrix algebra preconditioners for multilevel Toeplitz systems do not insure optimal convergence rate

Theoretical Computer Science - Algebraic and numerical algorithm
Spectral equivalence and matrix algebra preconditioners for multilevel Toeplitz systems: a negative result

Contemporary mathematics

Quantified Score

Hi-index	0.00

Visualization

Abstract

In the general case of multilevel Toeplitz matrices, we recently proved that any multilevel circulant preconditioner is not superlinear (a cluster it may provide cannot be proper). The proof was based on the concept of quasi-equimodular matrices, although this concept does not apply, for example, to the sine-transform matrices. In this paper, with a new concept of partially equimodular matrices, we cover all trigonometric matrix algebras widely used in the literature. We propose a technique for proving the nonsuperlinearity of certain frequently used preconditioners for some representative sample multilevel matrices. At the same time, we show that these preconditioners are, in a certain sense, the best among the sublinear preconditioners (with only a general cluster) for multilevel Toeplitz matrices.