Halley's Method

Halley's method is an algorithm for finding roots, also known as the tangent hyperbolas method or Halley's rational formula. As in Halley's irrational formula, take the second-order Taylor series of an analytic function

A root of satisfies , so

Setting the quadratic approximation to zero gives

giving

Using the estimate from Newton's method,

gives

so the iteration function is

At a simple root , , giving at least cubic convergence for starting values sufficiently close to . Curiously, the third derivative

is the negative of the Schwarzian derivative at . 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

Taking derivatives,

which has solutions

so at a root, and

which is Halley's method.

Basti and Cremaschi (2026) proved that the quartic function

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.