Newton Fractal Basins
Color complex starting points by the cubic root they reach and distinguish a converged orbit from a finite iteration limit.
Free theory articleColor complex starting points by the cubic root they reach and distinguish a converged orbit from a finite iteration limit.
Concept & interpretation
Complex Newton iteration applies the root-finding recurrence zk+1 = zk - P(zk) / P'(zk) in the complex plane. For the cubic polynomial P(z) = z3 - 1 with derivative P'(z) = 3z2, the iteration step is:
Interactive Experiment
Explore complex basin geometry and stopping conditions in the Newton Fractal Basins interactive workspace:
- Select Probe seed = i and Iteration budget = 12. Verify that the orbit converges in Iterations used = 6 to Upper root (
Last real = -0.5,Last imaginary ≈ 0.86602540378444). - Switch Probe seed = One minus i (
1 - i) with Iteration budget = 12. Even though1 - ilies in the lower half-plane, its trajectory crosses basin filaments and reaches the Real root1in8iterations. - Set Probe seed = Half and Iteration budget = 1 to observe the single-step overshoot
z1 = 5/3 ≈ 1.66666666666667with Status = Iteration budget reached and Root classification = Unavailable. - Compare Probe seed = Zero at Iteration budget = 4 (Status = Derivative singular) against Probe seed = One at Iteration budget = 0 (which satisfies
|P(1)| = 0 ≤ 10-8before the budget guard and immediately classifies as Real root).