Sparse factorization of the TM_z impedance matrix in an overlapped localizing basis

It has been observed that localized solution modes provide sparse factored representations of the discrete integral equations encountered in the simulation of electromagnetic phenomena at low frequencies. This paper extends these results by incorporating overlapped localizing modes. For TM_z scattering from a rectangular array of perfectly conducting obstacles, it is observed that the complexity scaling of the resulting factorization is significantly reduced relative to previously reported results. The memory complexity of the resulting factored representation scales approximately as O(N) for electrically small arrays. Limitations and possible extensions of these results are discussed.