Laurent expansion of the inverse of perturbed, singular matrices

In this paper we describe a numerical algorithm to compute the Laurent expansion of the inverse of singularly perturbed matrices. The algorithm is based on the resolvent formalism used in complex analysis to study the spectrum of matrices. The input of the algorithm are the matrix coefficients of th...

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Bibliographic Details
Published in:Journal of computational physics 2015-10, Vol.299, p.307-319
Main Authors: Gonzalez-Rodriguez, Pedro, Moscoso, Miguel, Kindelan, Manuel
Format: Article
Language:English
Subjects:
Algorithms
Computation
Formalism
Interpolation
Inverse
Laurent series
Mathematical analysis
Mathematical models
Radial basis functions
Singularities
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Summary:In this paper we describe a numerical algorithm to compute the Laurent expansion of the inverse of singularly perturbed matrices. The algorithm is based on the resolvent formalism used in complex analysis to study the spectrum of matrices. The input of the algorithm are the matrix coefficients of the power series expansion of the perturbed matrix. The matrix coefficients of the Laurent expansion of the inverse are computed using recursive analytical formulae. We show that the computational complexity of the proposed algorithm grows algebraically with the size of the matrix, but exponentially with the order of the singularity. We apply this algorithm to several matrices that arise in applications. We make special emphasis to interpolation problems with radial basis functions.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2015.07.006