Modified Newton–Raphson method
Consider function f of x with root x0 at f (x) = 0, and with an approximation, xapp, to x0. Expanding about the approximation
The standard Newton-Raphson method considers the first two terms of the expansion and inverts to make the unknown x0 – xapp explicit
More rapid convergence might be expected when the first three terms are taken into account. These represent a quadratic equation in the unknown x0 – xapp, with solutions
This must tend to the previous expression as ∂2f/∂x2 tends to zero, so that, by inspection on expanding the root by the binomial theorem, the positive option applies.