From early on, young mathematicians have been tasked with finding the roots of various functions. Probably the earliest interesting example was back in high school when they needed to use the quadratic equation to find the roots of degree 2 polynomials. However for many this is the end of the road for for solving polynomial equations outside of some specific cases where everything turns out nice. This is for good reason! It’s provable that you can’t solve any polynomials of degree 5 or higher directly. But where does that leave us to be able to find the roots of such functions? Luckily Sir Isaac Newton found a solution decades ago. While we can’t find the exact solutions, we can approximate them to an arbitrary amount of accuracy.
To begin, consider an arbitrary polynomial p : C → C. Since we are interested in the roots of p it is best to work in C so that we stand to hopefully find all of them. We will also need a function, called the Newton Map of p, defined as
Then Newton says we can construct a sequence of points which, if we are lucky, will get closer and closer to a root of p. To start the construction, pick a point z0 so that
Then find a new point
and we repeat! Get
and
Depending on what z0 is picked determines the whole sequence of points and also will change which root of p we approach. We call this process iterating Np. What we can do now is color the complex plane based of which root of p our point z0 iterates towards.
Here the polynomial is
Depending on if you choose z0 in a red, blue, or green region will change which root of p you ultimately approximate.
There is a flaw with Newton’s method. We do not always approach a root if z0 isn’t chosen at least a little carefully.
For example if the polynomial is
As before the colored regions are when z0 iterates towards a root. However now if you pick z0 = −0.00696−0.00021i (center of the black blob in the middle one will get
In other words the sequence just goes between 3 points and doesn’t settle down to a point close to a root of p.
While this is interesting, Newton’s method for polynomials is well understood by mathematicians. One example property is it is known that there can only be at most one of these troublesome cycles of points. For reader who have seen some topology; properties such as the connectedness of the boundaries of these colorful regions or that these colorful regions solve variations of the Lake of Wada problem.
Where my research has lead me is to ask questions about what if instead of a polynomial, what if I start with a rational function and applying Newtons method to it to find it roots. In other words, given two polynomials p(z) and q(z), let
Then can the roots of f(z) be found by iterating the Newton map
Since this is uncharted territory, it’s best to start with functions which have some level of predictability. Take
Obviously f(0) = 0 but does Newton’s method always find this root? Sadly no.
Here the green region is the points that will iterate towards 0, while the red region
is all the points which go to ∞. Arguably Newton’s method is still doing what it
was designed to do since
This doesn’t even start to consider if other cycles of points which don’t find a root. Let alone determining how many distinct cycles can exist at once. While Newton’s method has clear application to other fields, mathematicians care about it because it helps shine light on deeper structures that lurking in the background. In addition to being able to study some truly beautiful mathematics.
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