solutions, we use a variant of the overdetermined shooting method developed by the author and collaborators in previous work to study several related problems: time-periodic solutions of the Benjamin-Ono equation [@benj1; @benj2] and the vortex sheet with surface tension [@vtxs1; @vtxs2]; Hopf bifurcation and stability transitions in mode-locked lasers [@lasers]; cyclic steady-states in rolling treaded tires [@tires1]; self-similarity (or lack thereof) at the crests of large-amplitude standing water waves [@water1]; harmonic resonance and spontaneous nucleation of new branches of standing water waves at critical depths [@water2]; and three-dimensional standing water waves [@water3d]. The three approaches developed in these papers are the adjoint continuation method [@benj1; @lasers], a Newton-Krylov shooting method [@tires1], and a trust region shooting method [@water2] based on the Levenberg-Marquardt algorithm [@nocedal]. We adopt the latter method here to exploit an opportunity to consolidate the work in computing the Dirichlet-Neumann operator for many columns of the Jacobian simultaneously, in parallel. One computational novelty of this work is that we search directly for large-amplitude solutions of a nonlinear two-point boundary value problem, without using numerical continuation to get there. This is generally difficult. However, in the present case, numerical continuation is also difficult due to non-smooth bifurcation “curves” riddled 