Press cos on a calculator, over and over, from any start — the number stops moving at 0.739. That’s a FIXED POINT: a value that maps to itself, x = g(x). Iterating g settles onto it whenever g doesn’t stretch distances (|slope| < 1) — a ‘contraction.’ It’s the simplest solver there is, and the idea behind everything from PageRank to equilibrium. Slide to iterate and watch it spiral onto the fixed point.
Fixed-point iteration solves x=g(x) by simply iterating xₙ₊₁=g(xₙ). It converges to a fixed point x* whenever g is a CONTRACTION near it — |g′(x*)|<1 — by the Banach fixed-point theorem, at a LINEAR rate set by |g′| (smaller = faster); if |g′|>1 it diverges. Iterating cos settles on the Dottie number 0.739085…. Most solvers are fixed-point schemes in disguise: [[the-newton-method|Newton]], [[the-gauss-seidel|Gauss–Seidel]], value iteration, and PageRank all iterate a map to its fixed point. A fail-loud self-check throws unless iterating cos reaches 0.739085 to 1e−8. ◆ real numerical methods, node-verified.
Convergence needs |g′|<1 near the root and only LINEAR speed; the same equation can be rearranged into a diverging g. The contraction condition and the cobweb dynamics are exact.