  In this paper we introduce a new algorithm for solving perturbed nonlinear
functional equations which admit a right-invertible linearization, but with an
inverse that loses derivatives and may blow up when the perturbation parameter
$\epsilon$ goes to zero. These equations are of the form $F_\epsilon(u)=v$ with
$F_\epsilon(0)=0$, $v$ small and given, $u$ small and unknown. The main
difference with the by now classical Nash-Moser algorithm is that, instead of
using a regularized Newton scheme, we solve a sequence of Galerkin problems
thanks to a topological argument. As a consequence, in our estimates there are
no quadratic terms. For problems without perturbation parameter, our results
require weaker regularity assumptions on $F$ and $v$ than earlier ones, such as
those of Hormander. For singularly perturbed functionals, we allow $v$ to be
larger than in previous works. To illustrate this, we apply our method to a
nonlinear Schrodinger Cauchy problem with concentrated initial data studied by
Texier-Zumbrun, and we show that our result improves significantly on theirs.
