  We present a detailed analysis of a new, iterative density reconstruction
algorithm. This algorithm uses a decreasing smoothing scale to better
reconstruct the density field in Lagrangian space. We implement this algorithm
to run on the Quijote simulations, and extend it to (a) include a smoothing
kernel that smoothly goes from anisotropic to isotropic, and (b) a variant that
does not correct for redshift space distortions. We compare the performance of
this algorithm with the standard reconstruction method. Our examinations of the
methods include cross-correlation of the reconstructed density field with the
linear density field, reconstructed two-point functions, and BAO parameter
fitting. We also examine the impact of various parameters, such as smoothing
scale, anisotropic smoothing, tracer type/bias, and the inclusion of second
order perturbation theory. We find that the two reconstruction algorithms are
comparable in most of the areas we examine. In particular, both algorithms give
consistent fittings of BAO parameters. The fits are robust over a range of
smoothing scales. We find the iterative algorithm is significantly better at
removing redshift space distortions. The new algorithm will be a promising
method to be employed in the ongoing and future large-scale structure surveys.
