Newtons's Method

Question

If when you are using Newton's method and your results are just going back and forth between two values, say $0$ and $1$. It is $f(x)=x^3 -2x+2$ starting with $x=1$. What is the reasoning behind this?

Answer 1

In terms of the theorem about convergence of Newton's method, this is simply one of the issues that can occur if your initial guess for the root is not good enough. We are only guaranteed convergence for initial guesses which are already sufficiently close to the root.

In terms of what it means about the function, it doesn't mean very much analytically. It can be interpreted geometrically, if you want. The tangent line to $f$ at $0$ is $y=-2x+2$ while the tangent line to $f$ at $1$ is $y=(x-1)+1=x$. These have their roots at $1$ and $0$ respectively. Newton's method finds the root of the tangent line, so starting at $0$ sends you to $1$, then back to $0$, etc. This geometric observation correctly suggests that we may see cycles if we start out relatively close to an extremum of $f$ (since the derivative changes sign between the two values that are being hit).

Answer 2

For illustration, see https://en.wikipedia.org/wiki/Newton%27s_method#Starting_point_enters_a_cycle

with the picture

A Newton fractal with discussion of this example can also be found in https://www.math.sunysb.edu/~scott/Newton.ps.gz