Newton’s method uses a function’s first derivative to approximate a zero of that function; geometrically it can be realized as a sequence of lines tangent to the function’s graph intersecting the axis closer and closer to the zero. Halley’s method is a generalization of Newton’s method that uses a function’s first and second derivative to approximate a zero of that function; geometrically it can be realized as a sequence of hyperbolas tangent to the function’s graph intersecting the axis closer and closer to the zero.
To be precise, suppose \(f\) is a differentiable function with zero \(r.\) The hyperbola \[ y = \frac{\Bigl(2\bigl(f'(c)\bigr)^{2}-f''(c)f(c)\Bigl)(x-c)+2f(c)f'(c)}{-f''(c)(x-c)+2f'(c)} \] will have the same first and second derivative as \(f\) at \(x=c\) — it will be oscillate to the graph of \(f\) at \(x=c\) — and its \(x\)-intercept will typically be closer to \(r\) than \(c.\) Choose some \(c_0\) close to the zero \(r\) to serve as the seed to initiate Halley’s method and compute the \(x\)-intercept of that oscillating hyperbola at \(c_0.\) Call that number \(c_1.\) Then compute the \(x\)-intercept of the oscillating hyperbola at \(c_1\) and call that \(c_2,\) and then again with \(c_2\) in place of \(c_1,\) and again with \(c_3\) in place of \(c_2,\) and so on: for each number \(c_n\) iteratively compute the next number as \[ c_{n+1} = c_n - \frac{2f(c_n)f'(c_n)}{2\bigl(f'(c_n)\bigr)^2-f(c_n)f''(c_n)}\,. \] This sequence of values \(\{c_0, c_1, c_2, c_3, \dotsc\}\) will (typically) converge to the exact value of the zero \(r.\)