Fix fixed-step Newton tolerance for complex states

[ChrisRackauckas-Claude]

Please ignore this PR until it has been reviewed by @ChrisRackauckas.

What changed and why

Fixed-step in-place Newton derived its LinearSolve relative tolerance with eps(eltype(dz)). For complex states, dz is complex and eps(ComplexF64) is undefined. This derives the tolerance from the corresponding real scalar instead and adds fixed-step complex-state regressions for both ImplicitEM and ISSEM.

Fixes SciML/DifferentialEquations.jl#1161

Verification

Failing before

The regression body matching the checked-in in-place problem was run before the source change:

julia +1.12 --project=lib/StochasticDiffEq -e 'using ADTypes, StochasticDiffEq, Test; tspan=(0.0,1.0); u0=ones(2,4)+im*ones(2,4); function f(du,u,p,t); t isa Complex && error("time is complex"); du .= 1.01u; end; prob=SDEProblem(f,f,u0,tspan); @testset "issue 1161 regression" begin for alg in (ImplicitEM(autodiff=AutoFiniteDiff()),ISSEM(autodiff=AutoFiniteDiff())); sol=solve(prob,alg;adaptive=false,dt=0.01); @test eltype(sol.t)==Float64; @test eltype(sol.u[end])==ComplexF64; end; end'

issue 1161 regression: Error During Test
  MethodError: no method matching eps(::Type{ComplexF64})
  ... OrdinaryDiffEqNonlinearSolve/src/newton.jl:620
Test Summary:         | Error  Total
issue 1161 regression |     1      1

Both algorithms were also reproduced independently on clean master with the same eps(ComplexF64) error. The introducing commit was f231f6c.

Passing after

The identical regression command passed:

Test Summary:         | Pass  Total   Time
issue 1161 regression |    4      4  19.4s

The full relevant Stochastic group passed without using shared compiled-module caches:

JULIA_PKG_PRECOMPILE_AUTO=0 GROUP=Interface1 julia +1.12 --project=. -e 'using Pkg; Pkg.test(; julia_args=["--compiled-modules=no"])'

Complex Number Tests | 28 / 28 passed
Solver Reversal Tests | 152 / 152 passed
Testing StochasticDiffEq tests passed

Formatting and spelling checks all exited 0 with no output:

julia +1.12 -m Runic --check lib/OrdinaryDiffEqNonlinearSolve/src/newton.jl lib/StochasticDiffEq/test/complex_tests.jl
typos lib/OrdinaryDiffEqNonlinearSolve/src/newton.jl lib/StochasticDiffEq/test/complex_tests.jl
git diff --check

Pre-existing local check failures

• GROUP=Core in lib/OrdinaryDiffEqNonlinearSolve: 112 passed, 1 failed. The failure is a deterministic njacs assertion caused by resolving registered sibling releases. All-local master is already fixed by Fix clean-master solver package test regressions #4144.
• GROUP=QA in lib/OrdinaryDiffEqNonlinearSolve on Julia 1.12.6: JET reported five possibly-undefined locals (12 passed, 1 failed, 33 broken). This reproduces on clean master with JET 0.11.5/0.11.6; Julia 1.11/JET 0.9.20 and JET 0.10.15 pass. A separate isolated control-flow refactor makes JET 0.11.6 pass without suppressions.
• The declared GROUP=StochasticDiffEq_QA route exited 0, but the current Stochastic subpackage runner has no QA branch and executes zero assertions.

Not verified

The docs build was not run because no docs, docstrings, or public API changed. GROUP=Everything, GPU, downstream, and Julia pre jobs were not run.

Reviewer attention

The original issue describes complex-valued time, but clean reproductions keep eltype(sol.t) == Float64; the complex value is the state correction dz. The fix is therefore in OrdinaryDiffEqNonlinearSolve, and uses the same real(one(T)) pattern already used for generic roundoff handling in the sublibrary.

[ChrisRackauckas-Claude]

CI triage after matrix completion

The first CI run completed with 125 passed, 54 failed, and 3 skipped checks.

The issue-relevant StochasticDiffEq Interface1 group passed in every lane:

• Julia lts: https://github.com/SciML/OrdinaryDiffEq.jl/actions/runs/31489640193/job/93776535849
• Julia current: https://github.com/SciML/OrdinaryDiffEq.jl/actions/runs/31489640193/job/93776535395
• Julia pre: https://github.com/SciML/OrdinaryDiffEq.jl/actions/runs/31489640193/job/93776535078

OrdinaryDiffEqNonlinearSolve Core passed on Julia lts, and its ModelingToolkit group passed:

• https://github.com/SciML/OrdinaryDiffEq.jl/actions/runs/31489640193/job/93776535490
• https://github.com/SciML/OrdinaryDiffEq.jl/actions/runs/31489640193/job/93776535105

Its Julia current/pre Core failures are the previously bisected registered-sibling njacs == 0 mismatch (112 passed, 1 failed), already fixed in source by #4144.

All 54 red checks were reproduced or matched on clean master/current dependency resolution and are unrelated to this two-file diff:

• Master docs failure: existing fix Render missing SciMLBase API references #4211
• Root and sublibrary downgrade failures: Downgrade sublibrary jobs resolve incompatible SciMLBase 3.43 / SciMLPublic 1.0.0 #4185
• Root CI clusters: JLArrays 0.3.2 breaks InterfaceV implicit solve via host LAPACK Ptr conversion #4179, Integrators_II resize test calls length on a DiffCache #4188, SDIRK2 CSC/diag CuArray loses CPU/GPU agreement after guarded-Hegedüs boundary #4215, and Newton-Krylov instability diagnostics call matrix-only helper on JVPCache #4218
• Downstream/integration clusters: ProbNumDiffEq downstream job uses Julia 1.10 for a Julia 1.12 test environment #4186, Instability logging throws on in-place problems with an analytic Jacobian: get_fresh_jacobian calls out-of-place calc_J #4155, Core4 ensemble tests rely on undeclared EnsembleSerial reexport SciMLSensitivity.jl#1587, ModelingToolkitBase discrete-system tests use SciMLBase without importing it ModelingToolkit.jl#4937, Preserve tracked array wrappers in restructure JuliaArrays/ArrayInterface.jl#498, and Bump the all-julia-dependencies group across 3 directories with 1 update NumericalMathematics/PositiveIntegrators.jl#216
• Other sublibrary clusters: DelayDiffEq QA fails tstop ownership and SciMLBase API-doc checks #4182, Fix RSwM stack resizing after state changes DiffEqNoiseProcess.jl#308, log_numerical_instability scalar-indexes CUDA state in WeakAdaptiveGPU #4217, and the existing QA tracking at ExplicitImports: drive sublibrary QA checks to green (ei_broken markers) #3776

Clean-base workflow evidence:

• CI: https://github.com/SciML/OrdinaryDiffEq.jl/actions/runs/31478956160
• Documentation: https://github.com/SciML/OrdinaryDiffEq.jl/actions/runs/31478955896
• Downgrade: https://github.com/SciML/OrdinaryDiffEq.jl/actions/runs/31478955776
• Downgrade sublibraries: https://github.com/SciML/OrdinaryDiffEq.jl/actions/runs/31478955799
• IntegrationTest: https://github.com/SciML/OrdinaryDiffEq.jl/actions/runs/31478955487

No unrelated CI fix was added to this branch.