There is one complication here: Any local apparent horizon finding algorithm may fail if the initial guess is not good
enough, even if the desired surface is actually present. The solution is to use the constant-expansion surface for a slightly
larger expansion
as an initial guess, gradually “walking down” the value of
to find the minimum value
. Thornburg [156
, Appendix C] describes such a “continuation-algorithm binary search” algorithm in
detail.