BANACH a contraction has exactly one fixed point — and iteration always finds it
If a map pulls every pair of points closer — |f(x)−f(y)| ≤ c|x−y| with c<1 — then it has exactly one fixed point, and iterating from any starting point converges to it, with the error shrinking by the factor c every step. This is the Banach contraction principle (1922), and it is the theorem that makes fixed-point iteration trustworthy: existence, uniqueness, and a convergence rate, all from one inequality. It underwrites Newton, Picard's proof that differential equations have solutions, and every “iterate until it stops moving” algorithm.
THE TECHNIQUE |f(x)−f(y)| ≤ c|x−y|, c<1 ⇒ unique fixed point, error ×c/step
The demo iterates the contraction f(x)=x/2+1 (c=½) — the error halves each step toward the unique fixed point 2: live demo
HISTORY & CREDIT Stefan Banach, 1922
“Iteration might converge, to one of several answers, at some rate.” — a contraction guarantees all three: one fixed point, from anywhere, at rate c. Certainty from a single inequality. cited
1922 · Stefan Banach — the contraction mapping principle (in his thesis, founding functional analysis). gives · existence + uniqueness + a geometric error bound, from c<1. underwrites · Picard–Lindelöf (ODEs have solutions), Newton's local convergence, numerical iteration.
Pull everything closer and there is one still point, reached from anywhere, at a known rate. The theorem that makes settling safe. Banach 1922
RECOMMEND FOR I-13 the contraction's unique point, on the compiler
On the canonical compiler, x→x/2+1 from 0 converges to the unique fixed point 2, the error halving each step:
$ i13 run f_banach.i13 # x -> x/2 + 1 (c = 1/2)
RUN OK · 654 step(s) · peak stack 4 · call depth 41
fp = 1.999999999998181 -- the unique fixed point, error x 1/2 each step
Recommend: Banach is the guarantee under every fixed-point dart — contraction gives one point, from anywhere, at rate c — and i13 converges to 2. Not a keeper (existence-and-uniqueness of the fixed point is a theorem true of every contraction — witnessed, and the unique point is a normal form → confluence). The dart that certifies the rest of the batch actually settles.