Suppose I have a root finding method on some function on the complex plane , okay, that gets us partitioned into Wada basins, and more to the point, those basins are associated with particular roots of . Which means that if a basin is associated with the root , then .

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 ? (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**.

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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.

oops ch01.pdf https://docs.google.com/viewer?url=http://www.math.miami.edu/~ec/book/ch01.pdf

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?