Newton Fractal Basins

Worked Example: Single Real Step from z = 1/2 vs. Derivative Pole at z = 0

Trace the exact rational arithmetic of a single Newton step from z0 = 1/2 and compare it with the singular origin z0 = 0:

1. One step from z0 = 1/2 (budget 1):
   P(1/2) = (1/2)3 - 1 = -7/8, and P'(1/2) = 3(1/2)2 = 3/4.
   The first iterate jumps to z1 = 1/2 - (-7/8)/(3/4) = 1/2 + 7/6 = 5/3 ≈ 1.66666666666667 with residual magnitude |P(5/3)| = |(5/3)3 - 1| = 98/27 ≈ 3.62962962962963 > 10-8. After 1 iteration, the status is Iteration budget reached and root classification remains Unavailable.
2. Convergence from z0 = i (budget 12):
   Starting at z0 = i (where |P(i)| = |-i - 1| = √2 ≈ 1.4142135623731), 6 iterations reduce |z3 - 1| ≤ 10-8, converging to the Upper root -0.5 + 0.86602540378444i.
3. Derivative singularity at z0 = 0:
   At z0 = 0, |P(0)| = 1 > 10-8 and P'(0) = 0, so the iteration halts immediately at 0 iterations with status Derivative singular and Unavailable root classification.

Interactive Experiment

Explore complex basin geometry and stopping conditions in the Newton Fractal Basins interactive workspace:

1. 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).
2. Switch Probe seed = One minus i (1 - i) with Iteration budget = 12. Even though 1 - i lies in the lower half-plane, its trajectory crosses basin filaments and reaches the Real root 1 in 8 iterations.
3. Set Probe seed = Half and Iteration budget = 1 to observe the single-step overshoot z1 = 5/3 ≈ 1.66666666666667 with Status = Iteration budget reached and Root classification = Unavailable.
4. Compare Probe seed = Zero at Iteration budget = 4 (Status = Derivative singular) against Probe seed = One at Iteration budget = 0 (which satisfies |P(1)| = 0 ≤ 10-8 before the budget guard and immediately classifies as Real root).

Limitations and Assumptions

In the complex plane, the basin boundary (Julia set) of Newton's method for z3 - 1 has the property that every boundary point is a common boundary point of all three basins of attraction. The 161 × 161 pixel grid over [-2, 2]2 and finite iteration budget provide a discrete resolution of these basins, not an infinite-precision fractal boundary or a proof of global convergence. A point is classified as a root only when its residual satisfies |z3 - 1| ≤ 10-8 within the selected budget.

References

The global complex behavior of Newton's method on z3 - 1 and the fractal structure of its three root basins are analyzed in the following reference:

• Cleve Moler, Fractal Global Behavior of Newton's Method (Cleve's Corner, MathWorks)