kangwonlee / kangwonlee/nmisp

Pedagogy: add complex-plane initial-guess picker to 10_root_finding/45_newton_raphson_complex

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Description

Add an interactive widget to `10_root_finding/45_newton_raphson_complex.ipynb` that lets learners click (or slide) an initial guess on the complex plane and see which root the Newton-Raphson iteration converges to, with the iterate trajectory drawn on top.

## Scope

- Interactive control: ipywidgets sliders for `Re(z₀)` and `Im(z₀)`, **or** matplotlib `mpl_connect` click handler if the latter renders well in Colab.
- Background: faint Newton fractal (basins of attraction colored by which root) so the click position is meaningful.
- Foreground: iterate trajectory `z₀ → z₁ → z₂ → …` overlaid as connected dots until convergence or max iterations.
- Annotation of which root was reached, and how many iterations it took.

## Why

Newton fractals are the canonical "wow" visualization of complex root finding — basin boundaries are fractal, neighboring initial guesses can converge to different roots, and the iterate trajectory zigzagging across basin boundaries before settling is genuinely surprising. A static plot doesn't carry that. The interactive picker lets learners discover sensitivity for themselves.

In-curriculum (`45_newton_raphson_complex.ipynb` already exists) and validated slider pattern (#353/#354) — no new theory needed beyond what the notebook already covers.

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