  Surface integral equations (SIEs)-based boundary element methods are widely
used for analyzing electromagnetic scattering scenarii. However, after
discretization of SIEs, the spectrum and eigenvectors of the boundary element
matrices are not usually representative of the spectrum and eigenfunctions of
the underlying surface integral operators, which can be problematic for methods
that rely heavily on spectral properties. To address this issue, we delineate
some efficient algorithms that allow for the computation of matrix square roots
and inverse square roots of the Gram matrices corresponding to the
discretization scheme, which can be used for revealing the spectrum of standard
electromagnetic integral operators. The algorithms, which are based on properly
chosen expansions of the square root and inverse square root functions, are
quite effective when applied to several of the most relevant Gram matrices used
for boundary element discretizations in electromagnetics. Tables containing
different sets of expansion coefficients are provided along with comparative
numerical experiments that evidence advantages and disadvantages of the
different approaches. In addition, to demonstrate the spectrum-revealing
properties of the proposed techniques, they are applied to the discretization
of the problem of scattering by a sphere for which the analytic spectrum is
known.
