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Analysis of a circulant based preconditioner for a class of lower rank extracted systems
Author(s) -
Salapaka S.,
Peirce A.,
Dahleh M.
Publication year - 2005
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.390
Subject(s) - preconditioner , toeplitz matrix , circulant matrix , mathematics , conjugate gradient method , positive definite matrix , matrix (chemical analysis) , rank (graph theory) , convolution (computer science) , iterative method , combinatorics , algorithm , pure mathematics , eigenvalues and eigenvectors , computer science , physics , materials science , quantum mechanics , machine learning , artificial neural network , composite material
This paper proposes and studies the performance of a preconditioner suitable for solving a class of symmetric positive definite systems, A p x = b , which we call lower rank extracted systems ( LRES ), by the preconditioned conjugate gradient method. These systems correspond to integral equations with convolution kernels defined on a union of many line segments in contrast to only one line segment in the case of Toeplitz systems. The p × p matrix, A p , is shown to be a principal submatrix of a larger N × N Toeplitz matrix, A N . The preconditioner is provided in terms of the inverse of a 2 N × 2 N circulant matrix constructed from the elements of A N . The preconditioner is shown to yield clustering in the spectrum of the preconditioned matrix similar to the clustering results for iterative algorithms used to solve Toeplitz systems. The analysis also demonstrates that the computational expense to solve LRE systems is reduced to O ( N log N ). Copyright © 2004 John Wiley & Sons, Ltd.