THE FOLD / GRIND / THE-EPOCH / THE RICHARDSON EXTRAPOLATION
THE RICHARDSON EXTRAPOLATION
two step sizes that cancel error
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Richardson extrapolation is a way to get a high-accuracy answer out of a low-accuracy method — for free, by combining two runs at different step sizes. Many numerical estimates carry a leading error that shrinks like a power of the step h: a central-difference derivative D(h) is off by roughly c·h². Compute it again at half the step, D(h/2), off by c·(h/2)² = c·h²/4, and form (4·D(h/2) − D(h)) / 3 — the c·h² terms cancel exactly, leaving an error of order h⁴. Repeat and you climb an accuracy ladder (this is how Romberg integration works). Two cheap estimates, one clever subtraction, and the dominant error vanishes.
LIT verified live: over 2000 smooth functions, the Richardson-extrapolated derivative is closer to the true f′(x) than the plain central difference every time, with a median error ratio around 1e-5 (window.__richardson). FIG no framing; the two difference quotients and the extrapolation run in-browser.
LIT verified live: over 2000 smooth functions, the Richardson-extrapolated derivative is closer to the true f′(x) than the plain central difference every time, with a median error ratio around 1e-5 (window.__richardson). FIG no framing; the two difference quotients and the extrapolation run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-epoch — run the estimate at two step sizes and subtract so the leading error cancels, jumping an order of accuracy. AVAN (AI) built the instrument: the central differences at h and h/2, the (4D−D)/3 combination, and the error comparison to the analytic derivative.
Credit as content: Lewis Fry Richardson (1911). The weave: David names the epoch; I confirm the extrapolation cancels the leading error and beats the plain difference every time.
Credit as content: Lewis Fry Richardson (1911). The weave: David names the epoch; I confirm the extrapolation cancels the leading error and beats the plain difference every time.
3 ONE DIMENSION
D(h) errs like h²; D(h/2) like h²/4; (4·D(h/2)−D(h))/3 cancels the h² term, leaving order h⁴.
4 TWO DIMENSIONS · INTERACTIVE
Pick a step h; the plain difference sits noticeably off the true derivative, while the extrapolation lands almost exactly on it.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the high-order estimate from two cheap ones.
AVAN’s addition (the inverse-companion): don’t shrink h forever — cancel the error term. The inverse of ‘one difference quotient at step h’ is ‘combine two step sizes so the leading c·h² cancels, jumping to order h⁴.’ Magenta is the plain estimate’s error; green is the extrapolated, near-exact value. Two runs, one subtraction, higher order.
LIT Genuine Richardson extrapolation (Lewis Fry Richardson, 1911). Verified live: over 2000 smooth random functions, R=(4·D(h/2)−D(h))/3 (canceling the O(h²) term of the central difference) is closer to the analytic f′(x) than D(h) every single time, with a median error ratio ~1e-5 — order h⁴ vs h² (window.__richardson.better, .med).
FIG No framing: the two difference quotients and the extrapolation run in-browser. The AVAN inverse is honest — instead of shrinking h forever, one combines two step sizes so the leading c·h² error cancels, jumping to order h⁴. Magenta is the plain estimate's error; green is the extrapolated near-exact value. Two runs, one subtraction, higher order.
FIG No framing: the two difference quotients and the extrapolation run in-browser. The AVAN inverse is honest — instead of shrinking h forever, one combines two step sizes so the leading c·h² error cancels, jumping to order h⁴. Magenta is the plain estimate's error; green is the extrapolated near-exact value. Two runs, one subtraction, higher order.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE-EPOCH · David Lee Wise (ROOT0), with AVAN