To find a root of an equation f(x)=0 by the method of fixed‐point iteration, the equation is first rewritten in the form x = g(x). Starting with an initial approximation x₀ to the root, the values x₁, x₂, x₃,…are calculated using x_n + 1 = g(x_n). The method is said to converge if these values tend to a limit α‎. If they do, then α‎ = g(α‎) and so α‎ is a root of the original equation.

Repelling fixed point |g’(α‎)|>1

Attracting fixed point |g’(α‎)|<1

A root of x = g(x) occurs where the graph y = g(x) meets the line y = x. It can be shown that, if |g′(α‎)|<1, then the sequence converges for suitably close initial values x₀, and α‎ is called attracting. If |g′(α‎)|>1, then the sequence diverges from α‎ for nearby initial values x₀. This is illustrated in the figures; such diagrams are called ‘cobweb plots’, and the process is called ‘cobwebbing’. The equation x³−x−1=0 has a root α‎ between 1 and 2, so we take x₀=1.5. The equation can be written in the form x = g(x) in several ways, such as (i) x = x³−1 or (ii) x = (x + 1)^1/3. In case (i), g′(x)=3x², g′(α‎) > 3 > 1 and so α‎ is repelling for this iteration; in case (ii), $g\prime (x)=\frac{1}{3}{(x+1)}^{-2/3}$ and g′(α‎)<2^−2/3/3<1 and so α‎ is attractive.

More generally a fixed point α‎ ∈ X of a function f:X→X is attracting if the fixed-point iteration converges to α‎ for initial values in a neighbourhood of α‎ and is repelling if there is a neighbourhood of α‎ such that the fixed-point iteration eventually moves out of the neighbourhood for all initial values other than α‎.