Laurent expansion of the inverse of perturbed, singular matrices

Abstract

a b s t r a c tKeywords: Laurent series, Inverse, Radial basis functions, Interpolation.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.

Laurent series based RBF-FD method to avoid ill-conditioning

“…In the following, we use the algorithm described in [16] to compute the matrices H k in terms of the matrices A k in (5).…”

“…In fact, we have not been able to calculate the Laurent series of the perturbed matrix for stencils larger than 13 nodes in a reasonable time. These limitations disappear if the Laurent series is computed with the algorithm described in [16], which compu tational complexity grows algebraically with the number of nodes ( p n 3 ), but exponentially with the order of the singularity. In fact, the computational complexity of the proposed algorithm is Oð2n 3 BÞ, where…”

“…The algorithm is imple mented in Matlab using floating point arithmetic. The details of the implementation are described in [16]. Our method is semi analytical because the Laurent series is derived from analytical formulas, but the implementation described in [16] involves a quite significant amount of numerical computations.…”

“…The details of the implementation are described in [16]. Our method is semi analytical because the Laurent series is derived from analytical formulas, but the implementation described in [16] involves a quite significant amount of numerical computations. Once the Laurent series of the inverse of the RBF interpolation matrix has been obtained (as function of δ), it is straightforward to compute the RBF FD weights for any differential operator by just multiplying term by term the Laurent series by the Taylor series of the differential operator.…”

“…The recursive algorithm described in [16] is an efficient algo rithm compared to the computation of the inverse of a matrix using a symbolic language such as Mathematica. However, the approach proposed here to compute the RBF weights is not competitive with other existing algorithms (Contour Padé,RBF QR, RBF RA, RBF GA methods,…).…”