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A probing method for computing the diagonal of a matrix inverse
Author(s) -
Tang Jok M.,
Saad Yousef
Publication year - 2012
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.779
Subject(s) - inverse , diagonal , mathematics , diagonal matrix , matrix (chemical analysis) , computation , simple (philosophy) , block matrix , algorithm , sparse matrix , eigenvalues and eigenvectors , geometry , computational chemistry , philosophy , materials science , physics , epistemology , quantum mechanics , chemistry , composite material , gaussian
SUMMARY The computation of some entries of a matrix inverse arises in several important applications in practice. This paper presents a probing method for determining the diagonal of the inverse of a sparse matrix in the common situation when its inverse exhibits a decay property, i.e. when many of the entries of the inverse are small. A few simple properties of the inverse suggest a way to determine effective probing vectors based on standard graph theory results. An iterative method is then applied to solve the resulting sequence of linear systems, from which the diagonal of the matrix inverse is extracted. The results of numerical experiments are provided to demonstrate the effectiveness of the probing method. Copyright © 2011 John Wiley & Sons, Ltd.