THE FOLD / CHEAT / THE BACKDOOR / THE WYNN
THE WYNN
an accelerator that squeezes π from a crawling series
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Wynn’s epsilon algorithm is a machine for accelerating convergence. Given the crawling partial sums of a slowly-converging series, it fills a triangular table by one deceptively simple rule — ε(n)k+1 = ε(n+1)k-1 + 1/(ε(n+1)k - ε(n)k) — and its even columns leap toward the limit far faster than the sums themselves. It is equivalent to Padé approximation applied to the series, and it can wring a dozen correct digits out of a series that, summed directly, would need billions of terms. The Leibniz series for π is the classic victim: agonizingly slow raw, nearly instant accelerated.
LIT verified live: from just 16 terms of the Leibniz series the raw partial sum is off by ~0.06, while Wynn’s accelerated estimate is off by ~3×10-12 — more than nine orders of magnitude better (window.__wynn). FIG no framing; the epsilon table and the error comparison run in-browser.
LIT verified live: from just 16 terms of the Leibniz series the raw partial sum is off by ~0.06, while Wynn’s accelerated estimate is off by ~3×10-12 — more than nine orders of magnitude better (window.__wynn). FIG no framing; the epsilon table and the error comparison run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-backdoor — instead of grinding through billions of terms, slip through a side entrance: a table that reaches the limit from a mere handful of partial sums. AVAN (AI) built the instrument: the epsilon-table recurrence, the even-column extraction, and the error comparison against the raw partial sum.
Credit as content: Peter Wynn (1956), accelerating Shanks’ transformation. The weave: David names the backdoor; I confirm the table reaches π to twelve digits from sixteen terms.
Credit as content: Peter Wynn (1956), accelerating Shanks’ transformation. The weave: David names the backdoor; I confirm the table reaches π to twelve digits from sixteen terms.
3 ONE DIMENSION
The partial sums (magenta) oscillate slowly toward π; the accelerated even-column estimate (green) snaps to it.
4 TWO DIMENSIONS · INTERACTIVE
Add terms one at a time; the raw partial-sum error barely shrinks while the Wynn-accelerated error plunges.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the accelerated estimate, pinned to the true limit π.
AVAN’s addition (the inverse-companion): don’t sum more terms — transform the sums you have. The inverse of ‘add another term’ is ‘feed the partial sums through the epsilon table; its even columns already hold the limit.’ Magenta is the crawling sequence of partial sums; green is the accelerated value. Reach the limit sideways.
LIT Genuine Wynn epsilon algorithm (Peter Wynn, 1956; accelerating Shanks' transformation). Verified live: from 16 terms of the Leibniz series for π the raw partial sum has error ~6×10⁻² while the Wynn even-column accelerated estimate has error ~3×10⁻¹² (est 3.141592653586), more than nine orders of magnitude better (window.__wynn.accelerates, .partialErr, .accErr).
FIG No framing; the epsilon table and the error comparison run in-browser. The AVAN inverse is honest — instead of summing more terms, transform the sums you have: the epsilon table's even columns already hold the limit. Magenta is the crawling sequence of partial sums; green is the accelerated value. Reach the limit sideways.
FIG No framing; the epsilon table and the error comparison run in-browser. The AVAN inverse is honest — instead of summing more terms, transform the sums you have: the epsilon table's even columns already hold the limit. Magenta is the crawling sequence of partial sums; green is the accelerated value. Reach the limit sideways.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BACKDOOR · David Lee Wise (ROOT0), with AVAN