NSA: gate iter-1 convergence on inner solver retcode (#3817)

[singhharsh1708]

Fixes #3817. Standalone on master.

Bug

nlsolve! has an iteration-1 shortcut:

if (iter == 1 && ndz < 1.0e-5) || ...

ndz is the displacement of the iterate. For a plain (modified-)Newton inner solver a tiny first-iterate displacement means the initial guess was already the solution, so the shortcut is sound. For a globalized inner solver it is not: a TrustRegion step that gets rejected or truncated leaves the iterate essentially unmoved, which produces the same tiny ndz while the solver has not actually decided anything. The outer loop reads that as convergence and accepts the stage with no correction applied.

Reproduced on current master, Robertson with an analytic jac, FBDF + NonlinearSolveAlg(TrustRegion()), fixed dt = 1.0:

retcode: Success
naccept: 1000
u[end]:  [1.0, 0.0, 0.0]     # identical to u0 — never moved
njacs:   2
nnonliniter: 1000            # exactly one inner iteration per step

Every step "converges" instantly, the state never changes, and the solve reports Success. NonlinearSolveAlg(NewtonRaphson()) on the same problem correctly reports ConvergenceFailure.

Fix

Only take the iter-1 shortcut when the inner cache's own retcode says it actually converged:

if (iter == 1 && ndz < 1.0e-5 && _nsa_inner_converged(nlsolver)) || ...

Iterations past the first derive η/θ from real step history and are not vulnerable to the false positive, so they are untouched — an earlier attempt that also gated the η * ndz < κ branch cost roughly one extra outer iteration per solve and pushed marginal solves over the default max_iter = 10, so the gate is deliberately confined to iter == 1.

The predicate excludes AbstractSimpleNonlinearSolveAlgorithm inner algs. Those caches are solved to completion rather than stepped and never update their own retcode, so without the exclusion they would lose the shortcut and run two full inner solves per stage.

Result

TrustRegion now reaches the same verdict as NewtonRaphson everywhere tested — no more silent Success on a frozen state:

dt	TrustRegion (before)	TrustRegion (after)	NewtonRaphson
0.01	Success, u frozen at u0	ConvergenceFailure	ConvergenceFailure
0.1	Success, u frozen at u0	ConvergenceFailure	ConvergenceFailure
1.0	Success, u frozen at u0	ConvergenceFailure	ConvergenceFailure
10.0	Success, u frozen at u0	ConvergenceFailure	ConvergenceFailure

Where the solves genuinely do converge (dt = 2^-15 … 2^-18), results are bit-identical to before the change, with no extra f-evals.

Tests

Two testsets added to nsa_jacobian_reuse_tests.jl: one asserting a frozen state is never reported as a successful solve and that TrustRegion agrees with NewtonRaphson on the verdict, one asserting TrustRegion still converges to the reference where it should. Both fail on master and pass with this change.

The OrdinaryDiffEqNonlinearSolve suite passes, as does OrdinaryDiffEqMultirate (88/88), which shares nlsolve! for its implicit MRI-GARK stages.

AI Disclosure

Claude assisted with this work.

[ChrisRackauckas]

don't stack this

[singhharsh1708]

don't stack this

done.