Newton’s Method

Newton’s method is a technique to find the roots of a function commonly taught in beginning calculus courses. Given a function \(f(x)\) and an initial guess \(x_0\) at a root — that is, a value of \(x\) for which \(f(x) = 0\) — we can make a better guess: \(x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}\). If we take \(x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}\), and repeat this recursively to create an \(x_n\) for every positive integer \(n\), then the sequence \((x_n)\) converges (assuming the function and initial guess are somewhat well-behaved) to a root of \(f\).

It turns out that Newton’s method generalizes to polynomials defined over the complex numbers, \(\mathbb{C}\). This lends itself naturally to images, since \(\mathbb{C}\) is easily represented as a two-dimensional plane. For a given function \(f\), we color every point in the plane based on which root of \(f\) it eventually converges to when chosen as the initial guess. We also give it a brightness based on how long it takes to converge at all: bright for a short time and dark for a long one.

This applet applies Newton’s method to polynomials only. The white dots represent the roots, so a configuration of them uniquely determines the (monic) polynomial with those roots. The polynomials \(f(z) = z^n - 1\) have their roots spread evenly around the point \(0\), and have plots that look particularly nice, so there’s a button to align the roots this way. Finally, to produce a higher-resolution image that doesn’t have the white dots (and to see the polynomial at work), use the download button below.

Higher Resolution