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.

翻译：基于表面积分方程（SIE）的边界元方法广泛应用于电磁散射场景分析。然而，在SIE离散化后，边界元矩阵的谱与特征向量通常不能表征底层表面积分算子的谱与特征函数，这对高度依赖谱特性的方法可能造成问题。针对这一挑战，本文提出了若干高效算法，能够计算与离散化方案对应的Gram矩阵的矩阵平方根与逆平方根，从而揭示标准电磁积分算子的谱特性。这些算法基于平方根与逆平方根函数的合理展开式，在电磁学中几种最相关的边界元离散化Gram矩阵上展现出显著效果。文中提供了包含不同展开系数集的对照表，并通过数值实验对比展示了各方法的优劣。此外，为证明所提技术的谱揭示能力，将其应用于已知解析谱的球体散射问题离散化中。