  We study the formation of dark matter halos in the concordance LCDM model
over a wide range of redshifts, from z=20 to the present. Our primary focus is
the halo mass function, a key probe of cosmology. By performing a large suite
of nested-box N-body simulations with careful convergence and error controls
(60 simulations with box sizes from 4 to 256 Mpc/h, we determine the mass
function and its evolution with excellent statistical and systematic errors,
reaching a few percent over most of the considered redshift and mass range.
Across the studied redshifts, the halo mass is probed over 6 orders of
magnitude (10^7 - 10^13.5 M_sun/h). Historically, there has been considerable
variation in the high redshift mass function as obtained by different groups.
We have made a concerted effort to identify and correct possible systematic
errors in computing the mass function at high redshift and to explain the
discrepancies between some of the previous results. We discuss convergence
criteria for the required force resolution, simulation box size, halo mass
range, initial and final redshift, and time stepping. Because of conservative
cuts on the mass range probed by individual boxes, our results are relatively
insensitive to simulation volume, the remaining sensitivity being consistent
with extended Press-Schechter theory. Previously obtained mass function fits
near z=0, when scaled by linear theory, are in good agreement with our results
at all redshifts, although a mild redshift dependence consistent with that
found by Reed and collaborators exists at low redshifts.
