[NAThompson]

Near a singular J, Newton and Halley crept to max_iter; they now reach the local minimum of |F| and report it.
Closes #1628

Once the backtracked Newton step t·a is shorter than the Cauchy step −(‖g‖²/‖Jg‖²)g, g = JᵀF, the line search restarts along the Cauchy step (the dogleg rule, with t‖a‖ as the radius), and the Armijo test and quadratic interpolation use the actual slope gᵀp. Acceptance is now strict, so a step too short to change ‖F‖ is not accepted.

A test on the angle between the Newton step and −g, as suggested in the issue, breaks Powell's badly scaled problem, where that angle is ~1e-8 throughout but Newton is right.

Calls of f on Freudenstein–Roth. Bold: ran to max_iter = 1000. Every other run ends at the non-root minimum, ‖F‖ = 6.9989, with evaluation_error.

start	Newton, develop	Newton, PR	Halley, develop	Halley, PR
(15, −2)	96	84	6098	127
(8.424166, −0.933377)	6249	121	6249	121
(9.15, −0.8964)	7001	123	7001	123
(0.5, −2)	96	84	5700	208

On the existing test problems (circle_line, rosenbrock, broyden10, powell, squares) the iterations and calls are unchanged, except Newton on squares from (1/1024, 0.5): 11 → 17 calls.

Freudenstein–Roth from 400×300 starts, x1 ∈ [−5, 20], x2 ∈ [−4, 6], max_iter = 1000:

	root	evaluation_error	max_iter	total calls
develop Newton	45200	71278	3522	43.3M
develop Halley	45200	67544	7256	65.4M
PR Newton	45684	74316	0	8.32M
PR Halley	45200	74800	0	8.60M

🤖 Generated with Claude Code