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This paper presents a single-level matrix compression algorithm, termed IE-QR, based on a low-rank approximation to speed up the electric field integral equation (EFIE) formulation. It is shown, with the number of groups chosen to be proportional to N12/, where N is the number of unknowns, the memory and CPU time for the resulting algorithm are both O(N1.5). The unique features of the algorithm are: a. The IE-QR algorithm is based on the near-rank-deficiency property for well-separated groups. This near-rank-deficiency assumption holds true for many integral equation methods such as Laplacian, radiation, and scattering problems in electromagnetics (EM). The same algorithm can be adapted to other applications outside EM with few or no modifications; and, b. The rank estimation is achieved by a dual-rank process, which ranks the transmitting and receiving groups, respectively. Thus, the IE-QR algorithm can achieve matrix compression without assembling the entire system matrix. Also, a "geometric-neighboring" preconditioner is presented in this paper. This "geometric-neighboring" preconditioner when used in conjunction with GMRES is proven to be both efficient and effective for solving the compressed matrix equations.