  A purification algorithm for expanding the single-particle density matrix in
terms of the Hamiltonian operator is proposed. The scheme works with a
predefined occupation and requires less than half the number of matrix-matrix
multiplications compared to existing methods at low (<10%) and high (>90%)
occupancy. The expansion can be used with a fixed chemical potential in which
case it is an asymmetric generalization of and a substantial improvement over
grand canonical McWeeny purification. It is shown that the computational
complexity, measured as number of matrix multiplications, essentially is
independent of system size even for metallic materials with a vanishing band
gap.
