Finite-budget implicit-midpoint collocation for a nonlinear BVP

Fix a uniform mesh and the open nodal domain $D_h$ and normalized residual $G_h$ from Normalized implicit-midpoint collocation residual for a two-point BVP. If $f$ and $B$ are $C^1$ on the required domains (total differentiability with continuous derivative entries), then $G_h$ is a $C^1$ square map from the open subset $D_h\subseteq {\mathbb{R}}^{d(N+1)}$ into ${\mathbb{R}}^{d(N+1)}$. Given an initial node vector $Y_0\in D_h$, tolerance $\tau \ge 0$, and correction budget $M\in {\mathbb{N}}_0$, apply Finite-budget Newton for a square vector residual to $G_h$ using its exact Jacobian. Return the last node vector and its root, singular-jacobian, invalid-trial, or iteration-budget-exhausted status as defined by that procedure. A root status means only that $\mathrm{\parallel }{G}_h(Y)\mathrm{\parallel }\le \tau$; it is not a branch, convergence, or error certificate.