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A novel algorithm for accelerating the iterative solution of integral equations governing scattering from surfaces is presented. The proposed fast steepest descent path algorithm (FASDPA) complements previously developed fast solvers, notably the fast multipole method (FMM) and the matrix decomposition algorithm (MDA). Whereas the computational complexity per iteration of a two-level FMM and MDA is O(N3/2), the FASDPA permits a matrix-vector multiplication in O(N4/3) operations.