Newton-Raphson Method

Let us consider the solution of . Assume is a solution of this equation with . Create a fixed-point iteration sequence, with and . is a function we can choose at will. Clearly, when is bounded near , then . So for the iterative process to converge quadratically, we require that . This means

> g:=x->x-phi(x)*f(x);

> D(g);

so that

> D(g)(p);

> eq:=subs(f(p)=0,D(g)(p)=0);

> solve(eq,phi(p));

This lead to the iterative scheme

> p[n+1]=p[n]-f(p[n])/(D(f)(p[n]));

which we should recognise as the Newton-Raphson Method. This is yet an alternative way to derive the Newton-Raphson formula and shows that it is a quadratic method by construction.