The Mandelbrot Set is built around a simple iterative equation.

 z(1)   = c
 z(n+1) = z(n)^2 + c

For any complex c, we can continue this iteration until either abs(z(n+1)) > 2 or n == lim, then return the iteration count n.

• If c = 0 and lim = 3, then z = [0 0 0] and n = 3.
• If c = 1 and lim = 5, then z = [1 2], and n = length(z) or 2.
• If c = 0.5 and lim = 5, then z = [0.5000 0.7500 1.0625 1.6289] and n = 4.

For a matrix of complex numbers C, return a corresponding matrix N such that each element of N is the iteration count n for each complex number c in the matrix C, subject to the iteration count limit of lim.

If C = [0 0.5; 1 4] and lim = 5, then N = [5 4; 2 1]

Cleve Moler has a whole chapter on the Mandelbrot set in his book Experiments with MATLAB: Chapter 10, Mandelbrot Set (PDF)