Finite budget RATTLE solve with scaled constraint residuals

The input state is consistent, $(q,p)\in \mathcal{M}$. For $h=0$, return $(q,p)$ with status exact-identity; this branch returns no multipliers or residual norms. For nonzero $h$, fix a compact phase subset $C\subset \mathcal{M}$ over a compact regular configuration chart with $(q,p)\in C$ and $0<\mathrm{\mid }h\mathrm{\mid }<h_0(C)$, using branch balls $U_q,U_p$ supplied by Scaled RATTLE equations have unique local multiplier branches. The admissible inputs also include initial guesses ${\lambda }_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}\in U_q$, ${\mu }_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}\in U_p$, tolerances ${\epsilon }_q,{\epsilon }_p\ge 0$, finite correction budgets $N_q,N_p\in {\mathbb{N}}_0$, and terminating exact factorization/solve backends ${\mathcal{B}}_q,{\mathcal{B}}_p$ for the two chord calls. Each backend accepts a square $m\times m$ Jacobian, returns a singularity report exactly when it is singular, and otherwise returns a retained factor object. For every right-hand side, solving on this object terminates with the exact unique solution for that matrix; factorization and retained-factor solves are separate operations, as in Finite-budget Newton and chord solves with scheduled factorization reuse. All evaluations of $c,J,\mathrm{\nabla }V$ and the displayed matrix operations in the updates, residual and exact residual-Jacobian evaluations, arithmetic, Euclidean norm evaluation/comparison, and domain-membership tests are returning primitives; no bound on their internal cost is specified. The multiplier domains $U_q,U_p\subseteq {\mathbb{R}}^m$ are open balls within the regular branch neighborhoods. Choose the balls so that $F_q$ is $C^1$ on $U_q$ and, for every $\lambda \in U_q$, the corresponding $F_p$ is $C^1$ on $U_p$ and all evaluations in the updates are defined and the corresponding positions remain in the ambient full-row-rank configuration region of the branch lemma. Inconsistent input, missing branch balls, or maps not defined on these domains are outside the algorithm's admissible inputs. For a nonzero-step success, the routine returns the approximate state, both multipliers, and both scaled residual norms; on failure it returns the residual solver's status, last valid multiplier iterate, and actual scaled residual norm, together with any position-stage values already computed.

1. Form $F_q(\lambda )=h^{-2}c\text{\hspace{0.17em}⁣}(q+h{M}^{-1}(p-\frac{h}{2}\mathrm{\nabla }V(q)-hJ(q{)}^T\lambda ))\mathrm{.}$ Run Finite-budget Newton and chord solves with scheduled factorization reuse on $F_q$ in chord mode with refresh period $1$, domain $U_q$, initial value ${\lambda }_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}$, tolerance ${\epsilon }_q$, budget $N_q$, and backend ${\mathcal{B}}_q$. If it returns anything other than root, return its exact status (iteration-budget-exhausted, singular-jacobian, or invalid-trial), last valid multiplier iterate, and actual scaled residual norm; do not continue.

2. For its returned $\tilde{\lambda }$, compute ${\tilde{p}}_{1\mathrm{/}2}=p-\frac{h}{2}\mathrm{\nabla }V(q)-hJ(q{)}^T\tilde{\lambda }$ and ${\tilde{q}}_1=q+h{M}^{-1}{\tilde{p}}_{1\mathrm{/}2}$. Form $F_p(\mu )=h^{-1}J({\tilde{q}}_1)M^{-1}({\tilde{p}}_{1\mathrm{/}2}-\frac{h}{2}\mathrm{\nabla }V({\tilde{q}}_1)-hJ({\tilde{q}}_1{)}^T\mu )\mathrm{.}$ Run the same supplier procedure on $F_p$ in chord mode with refresh period $1$, domain $U_p$, initial value ${\mu }_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}$, tolerance ${\epsilon }_p$, budget $N_p$, and backend ${\mathcal{B}}_p$. If it returns anything other than root, return its exact status (iteration-budget-exhausted, singular-jacobian, or invalid-trial), last valid multiplier iterate, actual scaled tangency residual norm, and the successful position-stage values; do not continue.

3. If both calls meet their residual tolerances, set ${\tilde{p}}_1={\tilde{p}}_{1\mathrm{/}2}-\frac{h}{2}\mathrm{\nabla }V({\tilde{q}}_1)-hJ({\tilde{q}}_1{)}^T\tilde{\mu }$ and return $({\tilde{q}}_1,{\tilde{p}}_1,\tilde{\lambda },\tilde{\mu })$, the norms $\mathrm{\parallel }{F}_q(\tilde{\lambda }){\mathrm{\parallel }}_2$, $\mathrm{\parallel }{F}_p(\tilde{\mu }){\mathrm{\parallel }}_2$, and status “residual tolerances met”.

For nonzero $h$, leaving a branch ball returns invalid-trial, a singular factorization returns singular-jacobian, and exhausting either correction budget returns iteration-budget-exhausted; the supplier supplies the last valid iterate and its evaluated residual norm in each case. No undefined-evaluation status is needed because the admissible domains require all evaluations to be defined. A successful output with nonzero residuals need not lie in $\mathcal{M}$; applying the routine again requires a consistent input, for example one restored by Holonomic phase residuals control distance to the consistent phase manifold. The supplier status root means only that its stated residual tolerance was met; it does not certify an exact root or a forward-error bound. No finite budget is claimed to succeed on every admissible input.