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Analysis of a novel preconditioner for a class of p ‐level lower rank extracted systems
Author(s) -
Salapaka S.,
Peirce A.
Publication year - 2006
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.468
Subject(s) - preconditioner , toeplitz matrix , mathematics , conjugate gradient method , circulant matrix , positive definite matrix , rank (graph theory) , matrix (chemical analysis) , convolution (computer science) , conjugate residual method , coefficient matrix , linear system , combinatorics , mathematical analysis , algorithm , pure mathematics , eigenvalues and eigenvectors , computer science , gradient descent , 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, Âx = b , which we call p ‐ level lower rank extracted systems ( p ‐ level LRES ), by the preconditioned conjugate gradient method. The study of these systems is motivated by the numerical approximation of integral equations with convolution kernels defined on arbitrary p ‐dimensional domains. This is in contrast to p ‐level Toeplitz systems which only apply to rectangular domains. The coefficient matrix, Â , is a principal submatrix of a p ‐level Toeplitz matrix, A , and the preconditioner for the preconditioned conjugate gradient algorithm is provided in terms of the inverse of a p ‐level circulant matrix constructed from the elements of A . The preconditioner is shown to yield clustering in the spectrum of the preconditioned matrix which leads to a substantial reduction in the computational cost of solving LRE systems. Copyright © 2006 John Wiley & Sons, Ltd.