THE FOLD / RESPAWN / THE CONTINUE / THE FIXED POINT
THE FIXED POINT
the map that always comes home
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
If a map pulls every pair of points strictly closer together by at least a fixed factor, it has exactly one fixed point, every starting value converges to it, and the error shrinks geometrically with a bound you can compute before running anything. Banach proved it in 1922, and it is the machinery behind Newton’s method, differential-equation existence proofs, Markov chain convergence and half of numerical analysis. Press cosine on a calculator repeatedly and you are watching it.
LIT verified live: 200 starting values scattered over [−100, 100] all land on 0.739085133215 with a spread of 0 to machine precision; cos(x*) − x* is 0 exactly; the a-priori bound qn/(1−q)·|x₁−x₀| holds at all 60 tested steps; and the observed convergence ratio is 0.673612 against |f′(x*)| = sin(x*) = 0.673612. The hypothesis is load-bearing: x + 1/x has |f′| < 1 at every point of [1,∞) — the largest value seen is 0.999975 — and has no fixed point at all, its iterates running off to 200 and beyond.
LIT verified live: 200 starting values scattered over [−100, 100] all land on 0.739085133215 with a spread of 0 to machine precision; cos(x*) − x* is 0 exactly; the a-priori bound qn/(1−q)·|x₁−x₀| holds at all 60 tested steps; and the observed convergence ratio is 0.673612 against |f′(x*)| = sin(x*) = 0.673612. The hypothesis is load-bearing: x + 1/x has |f′| < 1 at every point of [1,∞) — the largest value seen is 0.999975 — and has no fixed point at all, its iterates running off to 200 and beyond.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at THE CONTINUE: keep pressing the button and you always end up in the same place.
AVAN (AI) included the counterexample because the theorem is misremembered more often than it is misapplied. “The derivative is less than one so it converges” is false, and x + 1/x on [1,∞) is the standing refutation: the derivative 1 − 1/x² is strictly below 1 everywhere, yet approaches 1 as x grows, so no single q < 1 works for the whole space and the iterates escape. What Banach requires is a uniform contraction factor on a complete space, and both words carry weight. The a-priori bound is the part worth keeping: it says how many iterations suffice before you have done any, which is a rare thing for a numerical method to offer.
AVAN (AI) included the counterexample because the theorem is misremembered more often than it is misapplied. “The derivative is less than one so it converges” is false, and x + 1/x on [1,∞) is the standing refutation: the derivative 1 − 1/x² is strictly below 1 everywhere, yet approaches 1 as x grows, so no single q < 1 works for the whole space and the iterates escape. What Banach requires is a uniform contraction factor on a complete space, and both words carry weight. The a-priori bound is the part worth keeping: it says how many iterations suffice before you have done any, which is a rare thing for a numerical method to offer.
3 ONE DIMENSION
Error against iteration, with the bound that was computed first.
4 TWO DIMENSIONS · INTERACTIVE
The cobweb. Start anywhere and watch it spiral into the same corner.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: many trajectories, one destination.
AVAN’s addition (the inverse-companion): the forward reading is “a contraction has a unique fixed point.” The inverse is that the theorem does not find the point, it destroys every alternative. The proof shows the iterates form a Cauchy sequence and that two fixed points would have to be closer together than themselves — it never constructs anything, it forecloses. That is why the same argument proves existence for differential equations nobody can solve: it is an argument from shrinking, and shrinking does not care what is being shrunk. Read backwards, Banach’s theorem is a machine for converting “this process loses information at a steady rate” into “this process has exactly one answer”, and the loss is the whole engine.
LIT 200 starting values scattered over [-100, 100] all land on 0.739085133215 with a spread of 0 to machine precision; cos(x*) - x* is 0 exactly; the a-priori bound q^n/(1-q)*|x1-x0| holds at all 60 tested steps; and the observed convergence ratio is 0.673612 against |f'(x*)| = sin(x*) = 0.673612; the hypothesis is load-bearing - x + 1/x has |f'| < 1 at every point of [1,inf), the largest value seen being 0.999975, and has NO fixed point at all, its iterates running to 200 and beyond
FIG The counterexample is included because the theorem is misremembered more often than it is misapplied. 'The derivative is less than one so it converges' is FALSE, and x + 1/x on [1,inf) is the standing refutation: 1 - 1/x^2 is strictly below 1 everywhere yet approaches 1 as x grows, so no single q < 1 works for the whole space and the iterates escape. Banach requires a UNIFORM contraction factor on a COMPLETE space, and both words carry weight. The a-priori bound is the part worth keeping - it says how many iterations suffice before you have done any.
FIG The counterexample is included because the theorem is misremembered more often than it is misapplied. 'The derivative is less than one so it converges' is FALSE, and x + 1/x on [1,inf) is the standing refutation: 1 - 1/x^2 is strictly below 1 everywhere yet approaches 1 as x grows, so no single q < 1 works for the whole space and the iterates escape. Banach requires a UNIFORM contraction factor on a COMPLETE space, and both words carry weight. The a-priori bound is the part worth keeping - it says how many iterations suffice before you have done any.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CONTINUE · David Lee Wise (ROOT0), with AVAN