In recent years, fast multiple based methods, fast low-rank compression methods, and FFT-based methods have been developed that dramatically reduce the memory requirement of the iterative integral-equation (IE) solvers to O(N), and the CPU time to O(NlogN) for electrodynamic problems, with N being the number of unknowns. Compared to iterative solvers, direct solvers have advantages when the number of iterations is large or the number of right hand sides is large. In this work, an H{sup}2 matrix [1] was constructed to represent the dense system matrix resulting from integral-equation-based analysis of electrodynamic problems. The H- and H{sup}2-matrix framework is a general mathematical framework that enables a highly compact representation and efficient numerical computation of dense matrices. It was shown that the storage requirements and matrix-vector products of H{sup}2 matrices are of complexity O(N) [1-3]. However, the complexity of H{sup}2-matrix-based inverse has not been clearly established in the literature. In this work, based on the H{sup}2-matrix representation, a direct inverse of O(N) operation count and O(N) memory complexity was developed to solve the dense system matrix arising from electromagnetics-based analysis. Both complexity and accuracy were demonstrated. The method is kernel independent, and hence is suitable for any integral-equation-based formulation. In addition, it is applicable to arbitrary structures.
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