  We present a comparative analysis of several methods, known as local
Lagrangian approximations, which are aimed to the description of the nonlinear
evolution of large-scale structure. We have investigated various aspects of
these approximations, such as the evolution of a homogeneous ellipsoid,
collapse time as a function of initial conditions, and asymptotic behavior. As
one of the common features of the local approximations, we found that the
calculated collapse time decreases asymptotically with the inverse of the
initial shear. Using these approximations, we have computed the cosmological
mass function, finding reasonable agreement with N-body simulations and the
Press-Schechter formula.
