Conditioning of generalized source integral equation formulations

Recent years have seen an increasing interest in the development of fast direct integral equation solvers. These do not rely on the convergence of iterative procedures for obtaining the solution. Instead, they compute a compressed factorized form of the impedance matrix resulting from the discretization of an underlying integral equation. The compressed form can then be applied to multiple right-hand sides, at a relatively low additional cost. The most common class of direct integral equation solvers exploits the rank-deficiency of off-diagonal blocks of the impedance matrix, in order to express them in a compressed manner. Such rank deficiency is inherent to problems of small size compared to the wavelength, as well as to problems of reduced dimensionality, e.g., elongated and quasi-planar problems.