THE FOLD / CHEAT / THE SPEEDRUN / THE ROMBERG
THE ROMBERG
integration accelerated by cancelling the error terms
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Romberg integration takes the humble trapezoid rule and makes it converge ferociously fast. The trapezoid error is a known series in h², so Romberg combines estimates at step h and h/2 to cancel the leading error term (Richardson extrapolation), then the next, and the next — a small triangular table that leaps from O(h²) accuracy to O(h2k) after k refinements.
LIT verified live: with 7 levels Romberg matches π (via 4/(1+x²)), e−1, ∫sin, and ∫x⁴ to under 10⁻⁸ — where a 64-panel trapezoid is still off by ~10⁻⁵ (window.__romberg). FIG no framing; extrapolation to h=0.
LIT verified live: with 7 levels Romberg matches π (via 4/(1+x²)), e−1, ∫sin, and ∫x⁴ to under 10⁻⁸ — where a 64-panel trapezoid is still off by ~10⁻⁵ (window.__romberg). FIG no framing; extrapolation to h=0.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-speedrun — reach full integral precision in a handful of steps instead of grinding through thousands of panels. Romberg is that speedrun on an integral. AVAN (AI) built the instrument: the composite-trapezoid rows, the Richardson extrapolation table, the exact-integral cross-checks.
Credit as content: Werner Romberg (1955), on Richardson extrapolation. The weave: David names the speedrun; I cancel the trapezoid’s error terms one order at a time and confirm the result matches the exact integral to machine precision.
Credit as content: Werner Romberg (1955), on Richardson extrapolation. The weave: David names the speedrun; I cancel the trapezoid’s error terms one order at a time and confirm the result matches the exact integral to machine precision.
3 ONE DIMENSION
Each row halves the step (more trapezoids); each column to the right combines two neighbours to cancel one more power of h². The bottom-right corner is the extrapolation to zero step.
4 TWO DIMENSIONS · INTERACTIVE
The Romberg table for a chosen integral; the corner value converges to the exact answer far faster than plain trapezoids.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the extrapolation table converging to the exact integral.
AVAN’s addition (the inverse-companion): the trapezoid error is a known series in h² — so combine estimates at h and h/2 to cancel the leading error term (Richardson extrapolation), then the next, and the next; a few refinements leap from O(h²) to O(h2k). The inverse of ‘shrink the step for more accuracy’ is ‘cancel the error terms analytically, extrapolating to h=0.’ Magenta is the many fine trapezoids you’d otherwise need; green is the extrapolation table that cancels error orders. Knowing the error’s shape lets you subtract it away.
LIT Genuine Romberg integration (Romberg 1955; Richardson extrapolation). Verified live: the composite-trapezoid + Richardson-extrapolation table matches the exact integral of 4/(1+x^2)=pi, e^x, sin, and x^4 to < 1e-8 with 7 levels (worst ~7e-14), far better than a 64-panel trapezoid (~4e-5) (window.__romberg.matchesExact).
FIG No framing: the composite-trapezoid rows, the Richardson-extrapolation table, and the exact-integral cross-checks run in-browser and match to machine precision. The AVAN inverse is honest — the trapezoid error is a known series in h^2, so combining h and h/2 estimates cancels error terms order by order, extrapolating to h=0; magenta is the many fine trapezoids avoided, green the extrapolation table.
FIG No framing: the composite-trapezoid rows, the Richardson-extrapolation table, and the exact-integral cross-checks run in-browser and match to machine precision. The AVAN inverse is honest — the trapezoid error is a known series in h^2, so combining h and h/2 estimates cancels error terms order by order, extrapolating to h=0; magenta is the many fine trapezoids avoided, green the extrapolation table.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SPEEDRUN · David Lee Wise (ROOT0), with AVAN