Theorem 4 (Convergence of Newton-Raphson Method)

Theorem 4: Let be a function which is twice continuous on the interval with and for a in the interval . Then there exists a such that the Newton-Raphson Method generates a converging sequence with for and for any initial approximation in the interval .

Proof:

Let us write the Newton-Raphson sequence as with

> g:=x->x-f(x)/D(f)(x);

> g(x);

We now need to find a value in the interval and an interval such that the conditions for a unique fixed point are satisfied. First, since there must be an subinterval of about where the first derivative is not zero: . In this interval, is defined and continuous and

> dg:=D(g);

This means that , and since is a continuous function, there must be a value such that on the interval with .

We then need to show that lies in the interval for all values of in that interval.

Since and, from the Mean Value Theorem,

> g:='g':expr1:=abs(g(x)-p);

> expr1=abs(g(x)-g(p));

> expr1=abs(D(g)(zeta))*abs(x-p);

with in the interval . Then,

> expr1<=K*abs(x-p);

> expr1<abs(x-p);

Since is in the interval , and therefore, so that lies in the interval for all in that interval.

Now, satisfies all the condition of Theorem 3, so that the Newton-Raphson sequence converges to for any initial value in the specified interval.