Halley Cubic-Root Convergence

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.

Worked Example

Consider solving f(x) = x³ - 2 = 0 using Halley's Method starting at x = 1. The function and its derivatives are f(x) = x³ - 2, f'(x) = 3x², and f''(x) = 6x.

At x = 1, f(1) = 1 - 2 = -1.
The first derivative is f'(1) = 3(1)² = 3.
The second derivative is f''(1) = 6(1) = 6.
The original denominator is 2(f')² - f f'' = 2(3)² - (-1)(6) = 18 + 6 = 24.
The update term is -2 f f' / denominator = -2(-1)(3) / 24 = 6 / 24 = 0.25.
The new iterate is 1 + 0.25 = 1.25.

After one update, the exact result is 1.25. This yields a faster local convergence compared to Newton's method which would yield 1.3333333333333333.

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.

Limitations

While Halley's method converges cubically near a simple root, it involves evaluating the second derivative and a more complex denominator. The method can fail (producing a singular step) if the denominator 2(f')² - f f'' evaluates exactly to zero. Numerical convergence is not exact rational equality or guaranteed global convergence. Additionally, near-root cancellation can affect the denominator's evaluation in floating-point arithmetic.

References

For more information on root-finding and Halley's iteration, see the DLMF chapter on Numerical Methods:
https://dlmf.nist.gov/3.8.

Halley Cubic-Root Convergence