Facing the Newton-Raphson method

Newton's method is the main numerical method for the approximation of roots of nonlinear equations. The function is linearly approximated at each iteration to obtain a better estimate of the zero point.

Using the Newton-Raphson algorithm for root-finding

Given a nonlinear function f and an initial approximation x0, Newton's method generates a sequence of approximations {xk} k > 0 by constructing, for each k, a linear model of the function f in a neighborhood of xk and approximating the function with the model itself. This model can be constructed starting from Taylor's development of the function f at a point x belonging to a neighborhood of the iterated current point xk, as follows:

Truncating Taylor's first-order development gives us the following linear model:

The previous equation remains valid in a sufficiently small neighborhood of xk.

Given x0 as the initial data, the first iteration consists of...