The quadratic convergence that is established by Newton’s method is based upon hypotheses that are difficult to verify a priori. One of them is that f’(r) is not equal to 0. If f’(r) is equal to 0 then r would be a zero of f and f’. Such a zero is termed a multiple zero of f. As mentioned in our text book (pg 96), Newton’s method for a multiple zero converges only linearly.

If one knew the multiplicity in advance Newton’s method can be accelerated by modifying the equation to read, Xn+1 = Xn- m*f(Xn)/f’(Xn) , where m = multiplicity of the zero

In general to accelerate Newton’s method, numerical analysis texts frequently suggest using higher order approximations to f at Xn. But instead we could take a different approach and ask the following questions:
1) For what class of functions does Newton’s method perform particularly well?
2) How can we modify a given function in such a way that the order of convergence is increased? (Once the class of functions has been identified)
(We assume that f is sufficiently differentiable, the initial approximation X0 is sufficiently close to r and f has a simple root at x = r)

The first question can be answered by using the theorem below.
In addition to the hypotheses above assume that f”(r) = f”’(r) = … = f^(m-1)(r) = 0, and f^(m)(r) not equal to 0. Then the Newton sequence yields,
|Xn+1 - r| <= C * |Xn - a|^m

This roughly says that the more f looks like a linear function, the faster the Newton iterations will converge.

Secondly, we want to mold a given function into a new one in such a way that the roots remain unchanged, but it looks nearly linear in a neighborhood of the root so that the convergence of Newton’s method will be accelerated.
We could use a theorem which is as follows,
Let f(r) = 0, f’(r) > 0, f”(r) = …. = f^(m-1)(r) = 0 and let f^(m)(r) not equal to 0.
Then the function, F(x) = f(x)/ (f’(x))^1/m satisfies F(r) = 0, F’(r) > 0, F”(r)= ….= F^(m)(r) = 0.The determination of m is not always easy. However it is reassuring to know that the performance of Newton’s method for F(X) will never be worse than that of f(x) itself regardless of the choice of m.

Further details and mathematical proofs can be found from the sources below:

Main source:
“Accelerated Convergence in Newton’s Method” by Jurgen Gerlach, SIAM review Vol 36, June 1994
(Can be obtained from the Cornell library database)
Other sources:
Numerical Mathematical and Computing, Ward Cheney & David Kincaid
http://math.fullerton.edu/mathews/n2003/NewtonImprovedMod.html