The main factors affecting the overall efficiency of any numerical procedure for the solution of large antenna or scattering problems, that is, the problem size, the memory occupation, and the computational cost, are introduced and are briefly discussed. It is shown how the size can be rigorously defined and estimated and the corresponding minimum, ideal computational cost is determined. Then the problem of developing algorithms approaching the ideal limit is examined, and possible ways to achieve the goal are enumerated. In particular, it is shown that in the case of large metallic scatterers in free space, the method of auxiliary sources, coupled to some kind of multilevel fast multipole algorithm, can allow development of numerical procedures whose effectiveness approaches the ideal limit.