On the Computation of Square Roots and Inverse Square Roots of Gram Matrices for Surface Integral Equations in Electromagnetics

Surface integral equations (SIEs) are widely used for modeling electromagnetic scattering problems. However, after their discretization via the boundary element method (BEM), the spectra and eigenvectors of resulting matrices are not usually representative of those of underlying surface integral operators, which can be problematic for methods that rely heavily on spectral properties. To address this issue, the spectrum of the integral operators can be recovered, while preserving symmetry, using square roots and inverse square roots of Gram matrices. In this work, several algorithms are delineated for the computation of square roots and inverse square roots of several relevant Gram matrices. The algorithms we detail rely on properly chosen polynomial expansions of the scalar square root and inverse square root functions and the theory of matrix functions. Tables containing different sets of expansion coefficients are provided along with comparative numerical experiments that evidence the advantages and disadvantages of the different approaches. In addition, the use of the proposed techniques as a tool for the recovery of the spectrum of integral operators is illustrated in the case of a spherical geometry for which the analytic spectrum is known.