Laurent expansion of the inverse of perturbed, singular matrices Articles uri icon

publication date

  • February 2015

start page

  • 307

end page

  • 319

volume

  • 299

international standard serial number (ISSN)

  • 0021-9991

electronic international standard serial number (EISSN)

  • 1090-2716

abstract

  • 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.