  As it stands, density matrix purification is a powerful tool for linear
scaling electronic structure calculations. The convergence is rapid and depends
only weakly on the band gap. However, as will be shown in this paper, there is
room for improvements. The key is to allow for non-monotonicity in the
recursive polynomial expansion. Based on this idea, new purification schemes
are proposed that require only half the number of matrix-matrix multiplications
compared to previous schemes. The speedup is essentially independent of the
location of the chemical potential and increases with decreasing band gap.
