Iterates
Iterates in mathematics denote the successive outputs of repeatedly applying a function. Let f: X → X be a function. The nth iterate, denoted f^n, is defined by f^1 = f and f^{n+1} = f ∘ f^n for n ≥ 1. Often f^0 denotes the identity on X. For a point x in X, the orbit under f is the sequence x, f(x), f^2(x), f^3(x), and so on.
Periodic points arise when an iterate returns to the starting point. A point x is periodic of
Fixed-point iteration is a common method that uses iterates to approximate solutions of equations of the form
Iterates also underpin various numerical algorithms. The power method uses repeated multiplication by a matrix to
In dynamics and fractals, iterating simple functions can generate intricate structures, as seen in the logistic