Layer 4 V4 — Newton globalization (line search)

Plain Newton (Layer 4 V3) is fragile when the warm-start x is in the "wrong region" of a stiff nonlinearity — the full step can overshoot and make the residual worse. Layer 4 V4 adds backtracking line search to globalize Newton.

Algorithm

1. Compute Newton direction:  J · dx = -f
2. Try α = 1.
3. If ||f(x + α · dx)||_∞ ≥ ||f(x)||_∞:
       α *= 0.5
       repeat (up to 8 backtracks)
4. If no α reduced the residual, fall back to α = 1
   (plain Newton's behaviour at that step)
5. Update: x := x + α · dx

This is "Armijo-lite": only checks that the residual norm decreases, no Wolfe conditions, no trust region. Simple, robust, no extra parameters to tune.

API

struct SimulationOptions {
    // ... existing fields ...
    bool enable_newton_line_search = false;   // NEW
};

Vector solve_with_newton_b_extra(
    const PwlSegment& seg,
    const NonlinearRefreshFn& refresh,
    const Graph& graph,
    const DevicePool& pool,
    const Vector& x_init,
    const Vector& b_extra,
    Size max_iters = 50,
    Real tol_dx  = 1e-9,
    Real tol_res = 1e-9,
    bool enable_line_search = false);   // NEW

When the flag is false (default), V4 behaviour matches V3 bit-identically.

Cost

Each backtrack iteration costs one matrix-vector multiply + one residual norm — no factor/solve (those are once per Newton iter, not per backtrack). For well-behaved problems, the first α = 1 trial succeeds and there's zero overhead.

Verified

• Line-search OFF reproduces V3: same DC diode load-line result with both paths.
• Line-search ON on well-behaved DC problem: converges to the same operating point as plain Newton.
• All regression tests stay green.

V0 limitations

The sinusoidal half-wave rectifier with κ=20 sigmoid (a stiff problem at zero-crossings) still fails even with line search — Newton oscillates around the steep transition. Two realistic future paths:

1. Trust-region methods (Dogleg, Levenberg-Marquardt). More robust globalization but more complex.
2. Continuation methods (homotopy from κ=2 to κ=20 over sub-steps). The natural way to handle stiff nonlinearities.

Both are research-grade follow-up OpenSpecs.

Status

Layer	Cases	Assertions
0	19	80
1	36	126
2	36	93
3	16	61
4 V0	24	58
5 V0	21	2069
4 V1	32	76
5 V1	17	59
5 V2.1	18	42
4 V2	9	520
4 V3	5	13
5 V4	4	60 ← +2 / +5 globalization tests
Total	237	3257