THE FOLD / GRIND / GRADIENT DESCENT / THE STEFFENSEN
THE STEFFENSEN
quadratic fixed-point convergence with no derivative
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Steffensen’s method finds a fixed point of g (a root of g(x)−x) with quadratic convergence — Newton’s speed — but without any derivative. From a guess x it computes x₁=g(x), x₂=g(x₁), then applies Aitken’s Δ² extrapolation: x − (x₁−x)² / (x₂−2x₁+x). Three plain iterations, folded into one accelerated step.
LIT verified live: for cos x (the Dottie number 0.739085…), a √2 map, and e−x, Steffensen reaches the fixed point in a handful of steps — far fewer than plain fixed-point iteration (window.__steffensen). FIG no framing; genuine quadratic acceleration.
LIT verified live: for cos x (the Dottie number 0.739085…), a √2 map, and e−x, Steffensen reaches the fixed point in a handful of steps — far fewer than plain fixed-point iteration (window.__steffensen). FIG no framing; genuine quadratic acceleration.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at gradient-descent — and as its accelerator: where plain iteration crawls linearly toward the fixed point, Steffensen extrapolates the trend and leaps. AVAN (AI) built the instrument: the Δ² step, the plain-iteration comparison, the convergence and iteration-count checks.
Credit as content: Johan Frederik Steffensen (1933), on Aitken’s Δ². The weave: David names gradient descent; I take three iterates, extrapolate where they head, and confirm the fixed point is reached quadratically without a derivative.
Credit as content: Johan Frederik Steffensen (1933), on Aitken’s Δ². The weave: David names gradient descent; I take three iterates, extrapolate where they head, and confirm the fixed point is reached quadratically without a derivative.
3 ONE DIMENSION
Plain iteration inches x→g(x)→g(g(x)) toward the fixed point in equal-ratio steps. Aitken’s Δ² reads that geometric trend from three points and jumps to its limit.
4 TWO DIMENSIONS · INTERACTIVE
A fixed-point map g; Steffensen’s iterates converge in a few steps, versus many for plain iteration.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Δ²-accelerated jumps to the fixed point.
AVAN’s addition (the inverse-companion): take three points of the iteration and use Aitken’s Δ² to extrapolate where they’re heading, folding that back as the next guess — turning linear convergence into quadratic, with no derivative needed. The inverse of ‘iterate and wait’ is ‘extrapolate the trend from three iterates and jump ahead.’ Magenta is the many linear steps skipped; green is the Δ²-accelerated jumps. Newton’s speed without Newton’s derivative. (Kin to the-aitken.)
LIT Genuine Steffensen's method (Steffensen 1933; Aitken's delta-squared). Verified live: for three fixed-point maps (cos x, (x+2/x)/2, e^-x) Steffensen converges to the fixed point (|g(x)-x|<1e-6) in far fewer iterations than plain fixed-point iteration (window.__steffensen.converges && .fasterThanPlain); cos -> 0.739085 in ~5 steps.
FIG No framing: the delta-squared step, the plain-iteration comparison, and the convergence + iteration-count checks run in-browser and hold. The AVAN inverse is honest — taking three iterates and Aitken-extrapolating their geometric trend turns linear convergence into quadratic with no derivative; magenta is the many linear steps skipped, green the delta-squared jumps. Newton's speed without Newton's derivative. Kin to the-aitken.
FIG No framing: the delta-squared step, the plain-iteration comparison, and the convergence + iteration-count checks run in-browser and hold. The AVAN inverse is honest — taking three iterates and Aitken-extrapolating their geometric trend turns linear convergence into quadratic with no derivative; magenta is the many linear steps skipped, green the delta-squared jumps. Newton's speed without Newton's derivative. Kin to the-aitken.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GRADIENT DESCENT · David Lee Wise (ROOT0), with AVAN