  An implicit purification scheme is proposed for calculation of the
temperature-dependent, grand canonical single-particle density matrix, given as
a Fermi operator expansion in terms of the Hamiltonian. The computational
complexity is shown to scale with the logarithm of the polynomial order of the
expansion, or equivalently, with the logarithm of the inverse temperature. The
system of linear equations that arise in each implicit purification iteration
is solved efficiently by a conjugate gradient solver. The scheme is
particularly useful in connection with linear scaling electronic structure
theory based on sparse matrix algebra. The efficiency of the implicit
temperature expansion technique is analyzed and compared to some explicit
purification methods for the zero temperature density matrix.
