What happens if we choose our Initial Guess Foolishly?
- Do all initial guesses go to a root?
- Do some go off to infinity?
- Do some bounce around forever?
- What root does each initial guess lead to?
Here's the graph of a simple function, a fourth-order polynomial:
f(x)=(x-3)(x-1)(x+1)(x+3)
There are four roots, which are at x values of -3, -1, 1, and 3.
Let's color the four roots red, purple, blue and green, respectively.
We can make a map, for this function, of where each initial guess leads.
For instance, the initial guess x=0.4 leads to the nearby blue root (x=1), so
we'll color the point 0.4 in as blue:
However, the initial guess x=0.13 hops over to the rightmost green root (x=3), so
we'll color the point 0.13 in as green:
If we continue this process of finding the root for each possible initial x-value,
we will get a map that looks like this:
Map of Initial Guesses for f(x)=(x-3)(x-1)(x+1)(x+3)
Wow! It turns out that you can have pockets of initial guesses that lead to faraway
roots. These "bad" guesses occur where the derivative gets small, since a shallow
slope will shoot the iterated value to a distant point on the axis.
This structure -- the fact that the different regions are mixed together near their boundary --
becomes even more surprising if we extend Newton's Method to the Complex Plane.