Halley's Method -- from Wolfram MathWorld

Halley's Method

Halley's method is a root-finding algorithm also known as the tangent hyperbolas method or Halley's rational formula. As in Halley's irrational formula, take the second-order Taylor series

(1)

A root of satisfies , so

(2)

Now write

(3)

giving

(4)

Using the result from Newton's method,

(5)

gives

(6)

so the iteration function is

(7)

This satisfies where is a root, so it is third order for simple zeros. Curiously, the third derivative

$H_f^(''')(alpha)=-{(f^(''')(alpha))/(f^'(alpha))-3/2[(f^('')(alpha))/(f^'(alpha))]^2}$

(8)

is the Schwarzian derivative. Halley's method may also be derived by applying Newton's method to . It may also be derived by using an osculating curve of the form

(9)

Taking derivatives,

			(10)
			(11)
			(12)

which has solutions

			(13)
			(14)
			(15)

so at a root, and

(16)

which is Halley's method.

Basti and Cremaschi (2026) proved that the quartic function

(17)

has four distinct roots, but its Halley map has an extraneous attracting two-cycle, with one point in and the other in . The absolute value of its multiplier is less than . Consequently, a nonempty open basin of attraction fails to converge to any root. The same polynomial also has an extraneous superattracting two-cycle for Newton's method.

References

Basti, A. and Cremaschi, T. "Simultaneous Global Convergence Failure of Newton's and Halley's Methods in Degree Four." 2 Oct 2026. https://arxiv.org/abs/2610.03246.

Scavo, T. R. and Thoo, J. B. "On the Geometry of Halley's Method." Amer. Math. Monthly 102, 417-426, 1995.

Halley's Method

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