Calculator
Compute which attractor a starting point converges to using Newton’s method, and optionally estimate basin proportions over a sampled region.
Example data table
Sample outputs for the preset z³ − 1 with a 60×60 scan centered at (0,0).
| Cluster | Attractor (approx.) | Count | Percent | Note |
|---|---|---|---|---|
| C1 | 1.00000000 + 0.00000000i | 1200 | 33.33% | avg iters 7.8 |
| C2 | -0.50000000 + 0.86602540i | 1185 | 32.92% | avg iters 7.9 |
| C3 | -0.50000000 - 0.86602540i | 1215 | 33.75% | avg iters 7.7 |
Formula used
A basin of attraction is the set of starting points that converge to the same attractor under an iterative rule.
Here, the iteration is Newton’s method applied to a polynomial f(z) on the complex plane:
Convergence is detected when the step size Δ = |zₙ₊₁ − zₙ| or the residual |f(zₙ)| falls below ε.
How to use this calculator
- Choose a preset polynomial, or select a custom one.
- Enter the real and imaginary parts of your starting point.
- Set tolerance, maximum iterations, and damping α if needed.
- Enable region scan to estimate basin proportions across a grid.
- Press Compute basin to see results above the form.
- Use the download buttons to export CSV or PDF.
FAQs
1) What is a basin of attraction?
A basin of attraction is the collection of initial values that eventually converge to the same long‑term outcome, called an attractor, under repeated iteration.
2) Why does Newton’s method create multiple basins?
Polynomials often have multiple roots. Newton’s method converges to different roots depending on the starting point, so each root has its own basin.
3) What does the damping factor α do?
α scales the Newton update. Values below 1 reduce step size and can help convergence near problematic points, but may require more iterations.
4) What does “did not converge” mean in the summary?
Those sample points failed to meet the tolerance within the iteration limit or encountered a near‑zero derivative. They are reported separately from convergent basins.
5) How is the basin summary computed?
Each sampled starting point is iterated until convergence or failure. Converged endpoints are grouped into clusters representing attractors, then counts and percentages are computed.
6) Why might two roots merge into one cluster?
If tolerance is very loose, clustering may group close attractors together. Reduce ε and increase iterations to separate nearby roots more reliably.
7) Is the region scan a full basin plot?
No. It estimates basin proportions by sampling a grid. A visual fractal plot needs image rendering, while this tool focuses on numeric classification and export.
8) What settings work well for most cases?
Try ε = 1e‑6, max iterations 50–80, and α = 1. If you see many failures, lower α to 0.7 and raise max iterations modestly.