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.

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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.

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