A probing method for computing the diagonal of a matrix inverse |
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Authors: | Jok M Tang Yousef Saad |
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Institution: | Department of Computer Science and Engineering, University of Minnesota, 200 Union Street S.E, Minneapolis, MN 55455, U.S.A. |
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Abstract: | 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. |
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Keywords: | matrix diagonal extraction probing sparse approximate inverses graph theory Krylov‐subspace methods Green's functions covariance matrices |
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