Finite-budget pseudo-arclength continuation with a certified local corrector

Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^{n+1}$ be open, and let $F:U\to {\mathbb{R}}^n$ be $C^2$ in the componentwise sense of $C^k$ Euclidean maps and diffeomorphisms on a declared tube; its total derivative $DF$ has the linear first-order meaning of The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder. Require $C^3$ on any subtube where a fold check is requested. A valid seed consists of a center ${\bar{z}}_0$, a radius $r_0$, and a certified ball containing exactly one zero $z_0^{\ast }$ of an initial augmented section $G_{-1}(z):=(F(z),{\tau }_{-1}^T(z-z_{p,-1})),$ where ${\tau }_{-1}$ is unit and the section is transverse to the regular local solution branch. Thus the seed certifies one solution point of $F=0$ by intersecting the branch with a hyperplane; it does not claim that $F=0$ has an isolated zero in ${\mathbb{R}}^{n+1}$. The seed also includes a neighborhood in which every zero of $F$ lies on the certified local curve chart and a verified rank-$n$ condition on that chart.

Supply a unit previous orientation, step length $h>0$, requested step count $N_{\mathrm{r}\mathrm{e}\mathrm{q}}\in {\mathbb{N}}_0$, continuation budget $N_s\in {\mathbb{N}}_0$, corrector budget $N_c\in {\mathbb{N}}_0$, residual tolerance $ϵ\ge 0$, and angle margin $c\in (0,1]$. At each current certified section ball $B_k={\bar{B}}_2({\bar{z}}_k,r_k)$, its unique branch point is denoted $z_k^{\ast }$. A tangent routine must provide a unit candidate ${\widehat{t}}_k$ and a certified error bound to a unit vector $t_k\in \ker DF(z_k^{\ast })$, plus a verified inverse bound for $DF(z_k^{\ast }){\mathrm{\mid }}_{t_k^{\perp }}$. Missing rank, tangent, or error bounds are failures; they are never treated as zero error.

At a queried point $z$, an evaluator returns either exact $(F(z),DF(z))$ or certified data $(\widehat{F}(z),{\eta }_F,\widehat{J}(z),{\eta }_J)$ with $\mathrm{\parallel }\widehat{F}(z)-F(z){\mathrm{\parallel }}_2\le {\eta }_F,\mathrm{\parallel }\widehat{J}(z)-DF(z){\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le {\eta }_J\mathrm{.}$ All supplied tangent, evaluator, chart, certificate and requested fold callbacks must return a result or their declared failure in finite time for a termination claim. Their mathematical bounds must be sound; finite outer budgets cannot force a nonreturning callback to finish. For every root ball proposed by a continuation attempt, the certificate routine must also supply an inverse bound for the approximate augmented Jacobian, a Jacobian-Lipschitz constant on the whole ball, and an isolating branch neighborhood/angle enclosure. If any bound is missing, return root-certificate-inconclusive for that candidate. Initialize the fold field as not-requested when no fold check was asked, and as inconclusive while a requested check remains unresolved.

Output: a finite list of certified local section-root balls; a continuation status from finite-trace, rank-or-tangent-unverified, tangent-angle-failure, tube-exit, singular-corrector, corrector-budget-exhausted, root-certificate-inconclusive, or continuation-budget-exhausted; and a fold field with value not-requested, certified, or inconclusive. The initial seed record retains its supplied unique-section-root certificate, center, radius, section and transverse local chart/rank evidence; its center error is at most $r_0$. Any stronger error or contraction bounds proved by the supplied seed certificate are retained, but no contraction bounds are required or inferred for the seed. Every newly appended ball records its center, radius, section, strict contraction bounds and center-error bound from A bounded residual and derivative error give a local root certificate, the local chart and angle enclosure identifying it with the unique next point on the traced branch, and the evaluation-error bounds used.

Procedure:

1. On entry only, initialize the list with the valid seed ball, accepted count $a=0$ and attempted continuation count $s=0$, and use the supplied unit previous orientation at the seed. On each visit to this step, denote the current previous orientation by ${\tau }_k$; retain the list, counts and orientation when returning from step6. If $a=N_{\mathrm{r}\mathrm{e}\mathrm{q}}$, including $N_{\mathrm{r}\mathrm{e}\mathrm{q}}=0$, finalize with status finite-trace. Otherwise, if $s=N_s$, finalize with continuation-budget-exhausted. Increment $s$ before making any callback for the next attempt. Thus $N_s$ caps attempts, not just successful steps.
2. At the unique current root $z_k^{\ast }$, query the tangent routine. Verify rank $DF(z_k^{\ast })=n$, a unit candidate ${\widehat{t}}_k$, a unit true kernel vector $t_k$, its certified error $\mathrm{\parallel }{\widehat{t}}_k-t_k{\mathrm{\parallel }}_2\le e_k$, and the complement inverse bound. Missing evidence returns rank-or-tangent-unverified. Orient ${\widehat{t}}_k$ and its corresponding $t_k$ together, keeping ${\tau }_k$ fixed. Certify ${\tau }_k^T{t}_k\ge c$ and $a_k:={\tau }_k^T{\widehat{t}}_k>0$; an unavailable or failed angular test returns tangent-angle-failure.
3. Set $z_p={\bar{z}}_k+h{\widehat{t}}_k$ and retain the previous orientation as section normal: $G_k(z)=(F(z),{\tau }_k^T(z-z_p))\mathrm{.}$ The chart certificate supplies a unit-speed chart $c_k$ with $c_k(0)=z_k^{\ast }$, $c_k^{\mathrm{\prime }}(0)=t_k$, an interval $[-r_{\mathrm{c}\mathrm{h}}^{-},r_{\mathrm{c}\mathrm{h}}^{+}]$, a number $\kappa >0$, and a neighborhood $V_k$ in which every zero of $F$ belongs to that chart interval. Verify ${\tau }_k^T{c}_k^{\mathrm{\prime }}(u)\ge \kappa$ throughout the interval and $h{a}_k>r_k,h{a}_k+r_k\le \kappa r_{\mathrm{c}\mathrm{h}}^{+}\mathrm{.}$ These checks bound $d={\tau }_k^T(z_p-z_k^{\ast })$ between $h{a}_k-r_k$ and $h{a}_k+r_k$, so the selected section has one forward intersection on the supplied chart by A transverse predictor hyperplane selects one nearby point of a regular branch. A failed chart angle gives tangent-angle-failure; failed offset/length evidence gives root-certificate-inconclusive. If the predictor, proposed ball or required chart leaves the declared tube, return tube-exit.
4. With exact evaluations, call Finite-budget Newton for a square vector residual on $G_k$ from $z_p$, tolerance $ϵ$ and update budget $N_c$. Map singular-jacobian to singular-corrector and invalid-trial to tube-exit. For root or iteration-budget-exhausted, retain the last valid iterate as a candidate. If the predictor is certified inside the invariant ball of A transverse pseudo-arclength section has a locally quadratic Newton corrector, every attempted exact update has its quadratic error bound; otherwise this is only a candidate generator. Apply the root certificate of A bounded residual and derivative error give a local root certificate to the last candidate. Accept only if every inequality passes and the whole ball lies in $V_k$. Unavailable inputs give root-certificate-inconclusive; failed tests after a root residual stop give that same status; failed tests after exhausted updates give corrector-budget-exhausted. Neither residual tolerance nor budget alone is a root proof. A passing ball proceeds to step6, without an earlier append.
5. With bounded evaluation errors, set $y_0=z_p$. For $j=0,\dots ,N_c$, query the supplied certified residual/Jacobian and root-ball data at $y_j$. Unavailable certificate inputs give root-certificate-inconclusive. Apply the same strict contraction tests and whole-ball inclusion in $V_k$ as in step4. A passing ball proceeds to step6, without an earlier append. If the tests fail at $j=N_c$, return corrector-budget-exhausted. Otherwise call Newton correction for a square vector residual on the supplied approximate augmented residual and matrix. Map singular-jacobian to singular-corrector; otherwise let $s_j$ be its returned correction. A trial $y_j+s_j$ outside the tube gives tube-exit; otherwise set $y_{j+1}=y_j+s_j$. This bounded-error branch has no convergence promise.
6. Append the passing ball once, with its section, error bounds and chart/angle record; increase $a$ by one. For the next attempt use the already certified unit candidate ${\widehat{t}}_k$ as the new previous orientation. It is not silently declared the exact tangent at the new root: step2 must certify its angle against the next root's tangent. Return to step1.
7. At every finalization, including every early return, retain the continuation status already assigned. For a requested fold check require a returned certified record satisfying all hypotheses of A simple fold has a unique local Lyapunov–Schmidt chart and nonzero parameter curvature at its specified point: $F=0$, the required $C^3$ neighborhood, $\mathrm{rank}{F}_x=n-1$, normalized right and nonzero left nullvectors, and strict enclosures excluding zero for $w^T{F}_{\lambda }$ and $w^T{F}_{xx}[v,v]$, together with the stated complement/parameter normalization. Missing or failed evidence leaves the fold field inconclusive; otherwise set it to certified and record only the local fold chart and curvature formula. Without a fold request keep not-requested. This finalization makes finitely many callback requests and performs no unbounded search.

The local predictor-defect estimate in A regular solution curve has a unique local arclength chart and a quadratic predictor defect applies to an exact root and exact unit tangent. Tangent and seed-center errors in an actual step are handled by the supplied residual/error enclosures and the root certificate, not silently set to zero. The exact Newton branch preserves the supplier's root convention: residual tolerance was met, not necessarily an exact zero. No finite trace certifies that a connected component is exhausted or that another component is absent.