Aitken's method to fixed-point iteration is known as Steffensen's method, and it can be shown that Steffensen's method yields a rate of convergence that is at least quadratic. Chaos game The term chaos game refers to a method of generating the fixed point of any iterated function system (IFS). Starting with any point x0, successive iterations are formed as xk+1 = fr(xk), where fr is a member of the given IFS randomly selected for each iteration. Hence the chaos game is a randomized fixed-point iteration. The chaos game allows plotting the general shape of a fractal such as the Sierpinski triangle by repeating the iterative process a large number of times. More mathematically, the iterations converge to the fixed point of the IFS. Whenever x0 belongs to the attractor of the IFS, all iterations xk stay inside the attractor and, with probability 1, form a dense set in the latter. See also Fixed-point combinator Cobweb plot Markov chain Infinite compositions of analytic functions Rate of convergence References Further reading External links Fixed-point algorithms online Fixed-point iteration online calculator (Mathematical Assistant on Web) Root-finding algorithms Iterative methods