Halley Cubic-Root Convergence
Compare Newton and Halley root updates while exposing singular steps and local convergence.
Free theory articleCompare Newton and Halley root updates while exposing singular steps and local convergence.
Concept & interpretation
Newton's method replaces a curved graph near the current estimate with its tangent line, then uses the line's x-intercept as the next estimate. Halley's method approximates the function with a hyperbola, producing a third-order rational root iteration.
Experiment
Explore the convergence and singular behavior using the interactive module:
- Select Cubic target as 2 and Initial x as 0.
- Observe that the Method set to Halley results in a Singular update status, as the denominator vanishes.
- Change the Initial x to -1. The denominator again vanishes and results in a Singular update.
- Set Method to Newton and observe that at x = 0 it also produces a Singular update, but it does not at x = -1.