◄ WORLD V · SONNY 5DART 308 · a helldive at the net

AITKEN'S Δ² reconstruct the limit a slow sequence is crawling toward — from three terms

A sequence creeps toward its limit; you are impatient. Aitken's Δ² process reconstructs the limit from just three consecutive terms, assuming the error shrinks geometrically: ŷ = xn − (Δxn)² / Δ²xn. It effectively fits the tail to a geometric model and jumps to where it is heading, often turning a sequence that needed thousands of terms into a handful. It is the accelerator under Steffensen's root-finder and a cousin of the Shanks transformation and Richardson extrapolation. Reconstruction of a value never actually reached: the limit, recovered from the structure of the convergence rather than by waiting for it.

THE TECHNIQUE ŷ = x₀ − (Δx)²/Δ²x; jump to the geometric limit

Three terms of a slowly converging sequence. The demo applies Aitken's Δ² to reconstruct the limit directly: live demo







  
HISTORY & CREDIT Alexander Aitken, 1926

  
“To know the limit you must run the sequence out.” — if the error is roughly geometric, three terms fix where it is going, and Aitken jumps there. The limit reconstructed, not awaited. cited

  
1926 · Alexander Aitken — the Δ² acceleration process.
kin · the Shanks transformation generalizes it; Steffensen's method uses it to speed root-finding.
now · series acceleration, fixed-point iteration, numerical analysis.

  
Assume the tail is geometric and three terms pin the limit; subtract the modelled remaining error and arrive early. The destination reconstructed from the shape of the approach. Aitken 1926






  
RECOMMEND FOR I-13 limit reconstructed from three terms, computed

  
On the canonical compiler, the sequence 1, 0.5, 0.25 (heading to 0) is accelerated by Aitken's Δ² straight to its limit 0:

  
$ i13 run rec_aitken.i13   # Aitken delta^2 of 1, 0.5, 0.25
aitken = 0       -- the limit, reconstructed from three terms

  
Recommend: Aitken's Δ² is reconstruction of a limit from the structure of convergence, computed by i13 in one line. From 1, 0.5, 0.25 it recovers the limit 0 exactly — because those three terms are geometric, and the process is built to invert a geometric tail. The load-bearing structure is that assumption: where the error truly shrinks geometrically, three terms suffice and the jump is exact; where it does not, the formula still runs but reconstructs the wrong value. It closes the batch's arc from the interpolators: those reconstructed a function from samples of it; Aitken reconstructs a limit from samples of the approach to it — recovery caused, once more, by an assumed structure.