# Constructing new functions from root finding methods

Suppose I have a root finding method on some function $w(z)$ on the complex plane $\mathbb{C}$, okay, that gets us  $\mathbb{C}$ partitioned into Wada basins, and more to the point, those basins are associated with particular roots of $w(z)$. Which means that if a basin is associated with the root $r$, then $w(r)=0$

Adding things that go to zero is a good way of making new functions. What if we were to take the sequence generated by the Newton’s  method iteration and then add together $w(z_{1}) +w(z_{2})+\ldots$? (or any other root-finding method.

for w(z) = 1-z^7, this gets us:

If a root finding method is a rhizogeny, then I’m going to call the class of functions generated this way metarhizogenous.

## 4 responses to “Constructing new functions from root finding methods”

1. 1. I don’t think this is just moire vibration: the top pic actually vibrates on my monitor’s output when i scroll up and down.

2. rhizogeny is a great word

3. math.miami.edu/~ec/book/ch1.pdf page 9

Are you just talking about inverting the function \$f\$ in \$f(x)=0\$? Except \$f\$ is approximated by \$f_1, f_2, f_3, \ldots\$ each being an iteration.

• So, any root finding method partitions the complex plane up into regions with an associated root. The root finding method has, for every point z in C, some convergents to a root z1, z2, … r. w(r)=0, so w(z1), w(z2)… should approach zero, so just add them up w(z1) + w(z2) + …. = u(z), and the u(z) is the metarhizogenous function.

• thanks. “regions with an associated root” means some shape which contains the root in question and no other roots?