Fix optimizer bugs and preserve original figures

[wyli]

Scope

Fix correctness and reproducibility bugs while preserving the repository's original figure layout and Matplotlib style.

The regenerated assets retain:

• labeled black contour lines for quadratic.png
• the original Newton / modified-Newton / rank-one overlay for quasi_newton.png
• the single filled YlGnBu Rosenbrock contour with every method overlaid for cg.png
• the original legend labels, start/optimum annotations, and output sizes

Bugs fixed

• return the backtracking step that actually satisfied Armijo instead of contracting it once more
• replace the unsafe interpolating line search with safeguarded quadratic interpolation
• record the accepted SR1 iterate instead of duplicating the previous point
• stop BFGS and SR1 using the actual gradient norm rather than an unchanged variable
• skip numerically unsafe SR1 updates instead of dividing by an arbitrary epsilon
• restart quasi-Newton/CG directions if numerical error makes them point uphill
• solve Newton systems directly instead of explicitly forming matrix inverses
• reject nonsymmetric quadratic matrices whose declared gradient would be mathematically incorrect
• support current NumPy arrays without deprecated numpy.matrix assumptions

Figure regeneration

MPLBACKEND=Agg python main.py regenerates the same three original-style figures. Repeated runs in the pinned environment produce byte-identical PNGs.

Validation

• MPLBACKEND=Agg pytest -q — 13 passed
• finite-difference Rosenbrock derivative check
• Newton one-step quadratic check
• BFGS/SR1 convergence checks
• accepted-step checks for both line searches
• SR1 trajectory regression for the duplicated-point bug
• Rosenbrock convergence checks from the figure start
• figure smoke test locking the original output dimensions
• GitHub Actions on Python 3.9