### 8.1 Zero-finding in spherical symmetry

In a spherically symmetric slice, any apparent horizon must also be spherically symmetric, so the
apparent horizon equation (16) becomes a 1-dimensional nonlinear algebraic equation for the
horizon radius . For example, adopting the usual (symmetry-adapted) polar-spherical spatial coordinates
, we have [154, Equation (B7)]
Given the geometry variables , , , and , this equation may be
easily and accurately solved using one of the zero-finding algorithms discussed in
Appendix A.
Zero-finding has been used by many researchers, including [141, 142, 143, 144, 119, 47, 139, 9, 154,
155].
For example, the apparent horizons shown in Figure 2 were obtained using this algorithm. As another
example, Figure 9 shows and at various times in a (different) spherically symmetric collapse
simulation.