THE FOLD / GRIND / THE HOT LOOP / THE AITKEN
THE AITKEN
accelerate convergence by cancelling the error's shape
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Aitken’s Δ² process accelerates a slowly-converging sequence. Given xₙ approaching a limit L, it forms a new sequence x′ₙ = xₙ − (Δxₙ)² / Δ²xₙ that homes in on L far faster.
The magic case: if the sequence converges geometrically (xₙ = L + c·rⁿ, the common pattern for linearly-convergent iterations), Aitken’s formula returns L exactly in a single step — it algebraically cancels the error term. On real sequences it sharply cuts the iterations needed, which matters when each one is expensive.
LIT verified live: Aitken returns L to ~10⁻¹³ for a purely geometric sequence, and on the x=cos(x) fixed-point iteration it reaches 10⁻⁸ accuracy in 17 accelerated terms versus 43 raw (window.__aitken). FIG no framing; exact.
The magic case: if the sequence converges geometrically (xₙ = L + c·rⁿ, the common pattern for linearly-convergent iterations), Aitken’s formula returns L exactly in a single step — it algebraically cancels the error term. On real sequences it sharply cuts the iterations needed, which matters when each one is expensive.
LIT verified live: Aitken returns L to ~10⁻¹³ for a purely geometric sequence, and on the x=cos(x) fixed-point iteration it reaches 10⁻⁸ accuracy in 17 accelerated terms versus 43 raw (window.__aitken). FIG no framing; exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-hot-loop — the expensive iteration you want to run as few times as possible. Aitken is the hot-loop’s accelerant: model the tail, jump ahead, stop early. AVAN (AI) built the instrument: the Δ² transform, the geometric-exactness check, the term-count speedup on a real fixed point.
Credit as content: Alexander Aitken (1926); the vector form underlies Steffensen’s method and the ε-algorithm. The weave: David names the hot-loop; I take three consecutive terms, cancel the geometric error algebraically, and show the accelerated sequence reach the limit in a fraction of the steps.
Credit as content: Alexander Aitken (1926); the vector form underlies Steffensen’s method and the ε-algorithm. The weave: David names the hot-loop; I take three consecutive terms, cancel the geometric error algebraically, and show the accelerated sequence reach the limit in a fraction of the steps.
3 ONE DIMENSION
Three consecutive terms xₙ, xₙ₊₁, xₙ₊₂ feed one accelerated value. If the tail is geometric (error ≈ c·rⁿ), the formula solves for what L must be — and lands on it.
4 TWO DIMENSIONS · INTERACTIVE
The slow fixed-point iteration x = cos(x) versus its Aitken-accelerated version. Watch the accelerated curve reach the limit in far fewer terms; a geometric test sequence lands on L exactly.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the accelerated sequence, modelling the error and leaping to the limit; the raw sequence crawls beneath it.
AVAN’s addition (the inverse-companion): acceleration works by assuming and then cancelling the error’s shape. Aitken presumes the tail behaves geometrically — error ≈ c·rⁿ — and solves three consecutive terms for what L must be if that is true, algebraically removing the dominant error. The inverse of ‘wait for convergence’ is ‘model the error and subtract it.’ When the assumption holds exactly (pure geometric), the answer is exact; when it holds approximately, you gain many digits per step. Magenta is the raw sequence still crawling toward L; green is the accelerated one that models the crawl and jumps ahead. Extrapolate the error away instead of waiting it out.
LIT Genuine Aitken delta-squared acceleration (Aitken 1926). Verified live: for a purely geometric sequence x_n = L + c*r^n, Aitken's formula returns L to max error ~1.5e-13 (exact cancellation); on the linearly-convergent x=cos(x) fixed point, the accelerated sequence reaches 1e-8 of the limit in 17 terms versus 43 for the raw sequence (window.__aitken.geometricExact && .speedup).
FIG No framing: the delta-squared transform, the geometric-exactness check, and the term-count comparison on a real fixed point run in-browser and are exact. The AVAN inverse is honest — Aitken assumes a geometric error tail and algebraically solves three terms for L, removing the dominant error; exact when the tail is truly geometric, strong acceleration otherwise; magenta is the raw crawl, green the accelerated jump.
FIG No framing: the delta-squared transform, the geometric-exactness check, and the term-count comparison on a real fixed point run in-browser and are exact. The AVAN inverse is honest — Aitken assumes a geometric error tail and algebraically solves three terms for L, removing the dominant error; exact when the tail is truly geometric, strong acceleration otherwise; magenta is the raw crawl, green the accelerated jump.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOT LOOP · David Lee Wise (ROOT0), with AVAN