Iterative solutions for the gravitational lens equation in the strong
  deflection limit
  Two exact lens equations have been recently shown to be equivalent to each
other, being consistent with the gravitational deflection angle of light from a
source to an observer, both of which can be within a finite distance from a
lens object [Phys. Rev. D 102, 064060 (2020)]. We examine methods for iterative
solutions of the gravitational lens equations in the strong deflection limit.
It has been so far unclear whether a convergent series expansion can be
provided by the gravitational lens approach based on the geometrical optics for
obtaining approximate solutions in the strong deflection limit in terms of a
small offset angle. By using the ratio of the lens mass to the lens distance,
we discuss a slightly different method for iterative solutions and the behavior
of the convergence. Finite distance effects begin at the third order in the
iterative method. The iterative solutions in the strong deflection limit are
estimated for Sgr $A^{*}$ and M87. These results suggest that only the linear
order solution can be relevant with current observations, while the finite
distance effects at the third order may be negligible in the Schwarzschild lens
model for these astronomical objects.
