Finite-budget Newton for a square vector residual

Let $U\subseteq {\mathbb{R}}^n$ be open, $n\ge 1$, and let $F:U\to {\mathbb{R}}^n$ be $C^1$ (Continuously differentiable maps, local inverses, and local diffeomorphisms) with Jacobian $J_F$ as in The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case. The matrix of $DF(x)$ in the standard basis is $J_F(x)$ by A total derivative computes every directional derivative, and its matrix is the Jacobian. Inputs: $x_0\in U$, residual tolerance $\epsilon \ge 0$, and update budget $N\in {\mathbb{N}}_0$. Output: the last valid iterate and one of root, singular-jacobian, invalid-trial, or iteration-budget-exhausted.

For $k=0,1,\dots ,N$, in this order:

1. Compute $F(x_k)$. If $\mathrm{\parallel }F(x_k){\mathrm{\parallel }}_2\le \epsilon$, return $(x_k,\mathtt{\text{root}})$. Here root means the residual tolerance was met; it does not certify an exact zero when $\epsilon >0$.
2. If $k=N$, return $(x_k,\mathtt{\text{iteration-budget-exhausted}})$.
3. Call Newton correction for a square vector residual with $A=J_F(x_k)$ and $r=F(x_k)$. If it returns singular-jacobian, return $(x_k,\mathtt{\text{singular-jacobian}})$.
4. Let $s_k$ be its correction. If $x_k+s_k\mathrm{\notin }U$, return $(x_k,\mathtt{\text{invalid-trial}})$. Otherwise set $x_{k+1}=x_k+s_k$.

It makes no arbitrary-start, global-convergence, or discretization-error claim.