FIXED-POINT ITERATION feed the output back in until it stops moving
Write an equation as x = g(x) and just iterate: pick a start, apply g, feed the result back in, again and again. If g pulls points together (a contraction), the sequence homes in on the one value that g maps to itself — the fixed point. It is the bare skeleton every iterative method wears: Newton, Halley, even the humble average are all x = g(x) in disguise.
THE TECHNIQUE the value that is its own image
Iterate x = 1 + 1/x from x=1. The sequence 1, 2, 1.5, 1.667, … converges to the golden ratio φ — the number that satisfies φ = 1 + 1/φ, i.e. is its own image under the map. Watch it settle: live demo
HISTORY & CREDIT Banach 1922; the iteration is ancient
“Any rearrangement x = g(x) will converge.” — no. It converges only if g is a contraction near the root (|g’| < 1); rearrange the same equation the wrong way and the identical iteration diverges. The map matters, not just the fixed point. cited
ancient · iterating a map to a fixed value is old as arithmetic — the Babylonian / Heron square-root step x = (x + a/x)/2 is a fixed-point iteration. 1890 / 1922 · Emile Picard (successive approximations) and Stefan Banach — the contraction mapping theorem (1922): a contraction on a complete space has a unique fixed point, reached from any start. everywhere · Newton, Halley, Gauss-Seidel, PageRank, and the corpus’s own fold are all x = g(x) driven to a fixed point.
A fixed point is a self-map: g(x*) = x*, the value unchanged by the transform — the same shape as a palindrome unchanged by reversal, or the corpus’s ROOT unchanged by another fold. Banach 1922
RECOMMEND FOR I-13 converges to phi, computed
The iteration x = 1 + 1/x converges to φ on the canonical compiler:
$ i13 run fixed.i13 # x = 1 + 1/x, from x=1, 40 iterations
phi = 1.618033988749895 -- the golden ratio: the value that is its own image, phi = 1 + 1/phi
Recommend: fixed-point iteration is LIT and the skeleton of the whole numeric library — verified x = 1 + 1/x converges to φ = 1.6180339887 in native recursion, the value that maps to itself. It is not one method but the form of every iterative method already in I-13: the Babylonian sqrt (dart 026), Newton, Halley (171), the Taylor-sum refinements — each is a contraction driven to its fixed point. The corpus’s own signature is here too: a fixed point is the value unchanged by the transform, the same self-identity a palindrome has under reversal and ROOT_0 has under the fold.