Newton Basins

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Newton's method is an iterative method for finding solutions of equations. It employs one of the basic strategies of mathematics--linearization. That is, given an estimate z_n for a solution of an equation f(z) = 0, it generates the estimate z_n+1, which would be exact if f  were linear. This works out to give the formula

z_n+1 = z_n - f(z_n)/f'(z_n).

Thus, beginning with an initial estimate z₀, we obtain a series of estimates z₀, z₁, z₂, ..., which, one hopes, converges to a solution of f(z) = 0.

The question of convergence turns out to be a most interesting one. Given a function f  with f(z) = 0 having more than one solution, we may ask, for an initial value z₀, not only whether the series z₀, z₁, z₂, ... converges to a solution, but, if so, which solution it converges to. In this way we arrive at the notion of a Newton basin: The Newton basin of a solution is the set of starting points z₀ for which Newton's method converges to that solution. Here we consider functions f(z) defined for complex numbers z.

We may illustrate the Newton basins for f  by coloring each basin a different color, and coloring those points for which Newton's method does not converge black. Here is the complex plane, colored to show the Newton basins of the polynomial

f(z) = (z⁴ - 1)(z² - 4) with roots 1, i, -1, -i, 2, -2.

 
 

 
This graphic was produced by using David Joyce's Newton basin generator. It allows you to experiment--you can pick the roots of a polynomial and it will find, and color, the Newton basins.

-- Steven Weintraub

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