Fixed Point Iteration

Numerical Analysis · Axiom Academy

Transform root-finding into a simple iteration: xₙ₊₁ = g(xₙ) A fixed point is where a function returns its input unchanged. Graphically, it's where y = g(x) intersects the line y = x. Example: For g(x) = cos(x), the fixed point is where cos(p) = p, which is approximately p ≈ 0.739085. 2. Converting Roots to Fixed Points Any equation f(x) = 0 can be rewritten as x = g(x) for some function g. Then the root of f is a fixed point of g! The iteration xₙ₊₁ = g(xₙ) creates a beautiful "cobweb" pattern when visualized: If g is continuous on [a, b], g([a, b]) ⊆ [a, b], and |g'(x)| < 1 for all x in (a, b), then fixed-point iteration converges to the unique fixed point for any starting point in [a, b]. Different rearrangements of the same equation can converge or diverge: Good Choice: g(x) = (x + 2/x)/2

This is the written version of the interactive lesson above. See the full Numerical Analysis course.