  The FIND algorithm is a fast algorithm designed to calculate certain entries
of the inverse of a sparse matrix. Such calculation is critical in many
applications, e.g., quantum transport in nano-devices. We extended the
algorithm to other matrix inverse related calculations. Those are required for
example to calculate the less-than Green's function and the current density
through the device. For a 2D device discretized as an N_x x N_y mesh, the best
known algorithms have a running time of O(N_x^3 N_y), whereas FIND only
requires O(N_x^2 N_y). Even though this complexity has been reduced by an order
of magnitude, the matrix inverse calculation is still the most time consuming
part in the simulation of transport problems. We could not reduce the order of
complexity, but we were able to significantly reduce the constant factor
involved in the computation cost. By exploiting the sparsity and symmetry, the
size of the problem beyond which FIND is faster than other methods typically
decreases from a 130x130 2D mesh down to a 40x40 mesh. These improvements make
the optimized FIND algorithm even more competitive for real-life applications.
