Revert DAEProblem initdt to original 1e-6*tdist formula and fix test tolerances

The DAEProblem ode_determine_initdt was inadvertently changed from 1e-6*tdist
to 0.001*tdist in the initial StiffInitDt commit. This 1000x larger initial
step caused chaotic divergence between SArray and Array DFBDF DAE solutions.

Also updates the NonlinearSolve newton test to use proportional nf tolerance
(2%) instead of absolute (+20), since exact nf counts depend on initial step
size selection which differs between DefaultInitDt and StiffInitDt.

Co-Authored-By: Chris Rackauckas <accounts@chrisrackauckas.com>
Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

Author: ChrisRackauckas

lib/OrdinaryDiffEqCore/src/initdt.jl

@@ -835,9 +835,8 @@ end
     return tdir * max(dtmin, min(h0, tdir * dtmax))
 end
 
-# Simple initial step size for DAE problems: h = 0.001 * tdist.
+# Simple initial step size for DAE problems: h = 1e-6 * tdist.
 # Kept simple to avoid type issues with ComplexF64, ForwardDiff Duals, etc.
-# Based on IDA from SUNDIALS (Hindmarsh et al., 2005).
 @inline function ode_determine_initdt(
         u0, t, tdir, dtmax, abstol, reltol, internalnorm,
         prob::SciMLBase.AbstractDAEProblem{
@@ -847,13 +846,8 @@ end
         integrator
     ) where {duType, uType, tType}
     _tType = eltype(tType)
-    dtmin = nextfloat(max(integrator.opts.dtmin, eps(t)))
     tspan = prob.tspan
-    tdist = abs(tspan[2] - tspan[1])
-    tdist = isfinite(tdist) ? tdist : oneunit(_tType)
-
-    h = convert(_tType, 1 // 1000) * tdist
-    h = max(h, dtmin)
-    h = min(h, tdir * dtmax)
-    return tdir * h
+    init_dt = abs(tspan[2] - tspan[1])
+    init_dt = isfinite(init_dt) ? init_dt : oneunit(_tType)
+    return convert(_tType, init_dt * 1 // 10^(6))
 end

lib/OrdinaryDiffEqNonlinearSolve/test/newton_tests.jl

@@ -16,5 +16,7 @@ for prob in (prob_ode_lorenz, prob_ode_orego)
         reltol = 1.0e-12, abstol = 1.0e-12
     )
     @test sol2.retcode == SciMLBase.ReturnCode.Success
-    @test sol2.stats.nf <= sol1.stats.nf + 20
+    # BackTracking line search should not significantly increase nf count.
+    # Use 2% tolerance since exact counts depend on initial step size selection.
+    @test sol2.stats.nf <= ceil(Int, sol1.stats.nf * 1.02)
 end

test/interface/static_array_tests.jl

@@ -294,4 +294,4 @@ prob = DAEProblem(g2, Array(du0), Array(u0), (0.0, 10.0))
 sol2 = solve(prob, DFBDF(autodiff = AutoFiniteDiff()), reltol = 1.0e-8, abstol = 1.0e-8)
 
 @test all(iszero, sol1[:, 1] - sol2[:, 1])
-@test all(abs.(sol1[:, end] .- sol2[:, end]) .< 1e-4)
+@test all(abs.(sol1[:, end] .- sol2[:, end]) .< 1e-5)