NonlinearSolveAlg silently fails with quasi-Newton inner solvers (Broyden, Klement)

[singhharsh1708]

NonlinearSolveAlg with Broyden never moves off u0:

using OrdinaryDiffEqSDIRK, NonlinearSolve
using OrdinaryDiffEqNonlinearSolve: NonlinearSolveAlg

prob = ODEProblem((du, u, p, t) -> (du .= -u), [1.0], (0.0, 1.0))
solve(prob, TRBDF2(nlsolve = NonlinearSolveAlg(Broyden())))

Unstable, u(1) = 1.0 after one accepted step, exact is 0.36787944. Klement() does the
same thing. NewtonRaphson() and TrustRegion() are fine on it.

Not a stiffness thing, it fails the same way on u' = diag(-L, -1) u at L = 1, 1e2, 1e4, 1e6.

It's also why FastShortcutNonlinearPolyalg dies (Unstable, relerr 1.41 on Robertson) where
RobustMultiNewton gets 5e-7. Both build the same 6 branch polyalg cache and both sit on
branch 1 the whole time, the only difference is FastShortcut's branch 1 is Broyden. Broyden
solves the stage system on its own fine, so it isn't the algorithm.

Is NonlinearSolveAlg supposed to work with Jacobian free inner solvers at all?

[singhharsh1708]

The existing guards can't see it because they're all displacement based. Printing either side
of the inner step! on the repro above:

Broyden  iter=1  ndisp=0.5        fprev=2.83e-5  fnow=120711.2
Broyden  iter=2  ndisp=0.5000001  fprev=120711.2 fnow=0.01544
Newton   iter=1  ndisp=7.07e-6    fprev=7.07e-6  fnow=6.828
Newton   iter=2  ndisp=7.07e-6    fprev=6.828    fnow=7.99e-10

Broyden's first inner step takes the residual from 2.8e-5 to 1.2e5. The divergence check is
theta = ndz/ndzprev > 2 on the displacement though, and 0.5 to 0.5000001 gives theta about 1,
so it never fires. stage_unsolved is true throughout and inner_solve_failed is false, so
nothing rejects the stage and dt just collapses.

A residual based check catches this, but Newton's residual also jumps on its first iteration
here (7e-6 to 6.8) before dropping to 8e-10, so "residual got worse" on its own would throw out
good Newton steps. What would you want the criterion to be?