Newton fractal

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The Newton fractal is a boundary set in the complex plane which is characterized by Newton's method applied to a fixed polynomial p(z) ∈ ℂ[z] or transcendental function. It is the Julia set of the meromorphic function z ↦ z − ⁠p(z)/p′(z)⁠ which is given by Newton's method. When there are no attractive cycles (of order greater than 1), it divides the complex plane into regions Gk, each of which is associated with a root ζk of the polynomial, k = 1, …, deg(p). In this way the Newton fractal is similar to the Mandelbrot set, and like other fractals it exhibits an intricate appearance arising from a simple description. It is relevant to numerical analysis because it shows that (outside the region of quadratic convergence) the Newton method can be very sensitive to its choice of start point. Almost all points of the complex plane are associated with one of the deg(p) roots of a given polynomial in the following way: the point is used as starting value z0 for Newton's iteration zn + 1 := zn...

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