Nearly every numerical analysis algorithm has computational complexity that scales exponentially in

the underlying physical dimension. The separated representation, introduced previously, allows many operations to be

performed with scaling that is formally linear in the dimension. In this paper we further develop this representation by:

(i) discussing the variety of mechanisms that allow it to be surprisingly ecient (ii) addressing the issue of conditioning

(iii) presenting algorithms for solving linear systems within this framework and (iv) demonstrating methods for dealing

with antisymmetric functions, as arise in the multiparticle Schrodinger equation in quantum mechanics. Numerical

examples are given.-Nearly every numerical analysis algorithm has computational complexity that scales exponentially in

the underlying physical dimension. The separated representation, introduced previously, allows many operations to be

performed with scaling that is formally linear in the dimension. In this paper we further develop this representation by:

(i) discussing the variety of mechanisms that allow it to be surprisingly ecient　(ii) addressing the issue of conditioning

(iii) presenting algorithms for solving linear systems within this framework　and (iv) demonstrating methods for dealing

with antisymmetric functions, as arise in the multiparticle Schrodinger equation in quantum mechanics. Numerical

examples are given.