Visualization and post-root-finding diagnostics for polynomials in the complex plane.

[ratwolfzero]

δ-Normalized Root Influence Field + Newton Flow

This project provides powerful visual and numerical insights into the local strength and global geometry of polynomial roots — going well beyond simply computing their locations.

---

The Core Idea: The δ Parameter

After locating the roots, we compute a natural characteristic scale δ for each root of multiplicity m:

$$ \delta = \left| \frac{m!}{f^{(m)}(a)} \right|^{1/m} $$

For simple roots (m = 1), this reduces to:

$$ \delta \approx \frac{1}{|f'(a)|} $$

• Small δ → Strong, well-conditioned root (steep local slope, fast Newton convergence)
• Large δ → Weak, sensitive root (flat local behavior, larger influence zone, slower convergence)

Effectively acting as the classical condition number for simple roots, this parameter quantifies the "personality" of each root and enables fair structural comparisons across roots of varying strengths:

---

Features

• Automatic computation of asymptotic triplets (a, m, δ)
• δ-normalized root influence field
• Newton flow streamlines (mapping the precise fractal boundaries of each root's basin of attraction)- δ-circles showing each root’s natural radius of influence
• Root strength classification (Strong / Normal / Weak)
• Post-root-finding diagnostics and refinement guidance
• High-quality matplotlib visualizations (two complementary figures)
• Support for real and complex coefficients

---

Why This Project Exists

Most polynomial solvers stop after returning a list of complex numbers.
This tool answers the next important questions:

• How strong or weak is each root numerically?
• How large is its reliable basin of attraction?
• Which roots require high-precision care during refinement?
• How do the roots compete for spatial dominance in the complex plane?

Visualizing the δ-normalized field reveals topological asymmetries and interactions that standard scatter plots completely miss—even when all roots share a multiplicity of 1.

---

Example

Polynomial:
f(x) = 1.17x⁴ − 0.00034x³ − x² + x + 1

Roots & Diagnostics:

Root (approx)	m	δ	Strength	Notes
-0.8326 ± 0.0975i	1	1.328	Weak	Large influence zone
0.8328 ± 0.7229i	1	0.179	Strong	Tight, well-conditioned

The two conjugate pairs exhibit dramatically different local behavior despite having identical multiplicities.

And the Classic Wilkinson Polynomial (distinct roots 1–20)

---

See: https://github.com/ratwolfzero/Param_Poly_Root