THE FOLD / SPAWN / THE TOOLCHAIN / THE NEVILLE
THE NEVILLE
polynomial interpolation by a triangle of blends
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Neville’s algorithm evaluates the unique polynomial through a set of data points at a query x — by a triangle of linear blends, never forming the polynomial explicitly. Start with the y-values; each step combines two neighboring lower-degree interpolants, weighted by distance to x, into one of higher degree, until a single value remains. It is numerically friendly and, like de Casteljau, replaces coefficients with repeated interpolation.
LIT verified live: over 300 random datasets, Neville’s value passes exactly through every data point and matches the Lagrange interpolation at random x (to ~10−13) — window.__neville. FIG no framing; exact.
LIT verified live: over 300 random datasets, Neville’s value passes exactly through every data point and matches the Lagrange interpolation at random x (to ~10−13) — window.__neville. FIG no framing; exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-toolchain — the numerical toolkit that fits a curve through points without solving a linear system. Neville’s algorithm is that fit. AVAN (AI) built the instrument: the interpolation triangle, the distance-weighted blends, the passes-through check, and the Lagrange cross-check.
Credit as content: Eric Harold Neville (1934). The weave: David names the toolchain; I combine neighboring interpolants into higher-degree ones by distance-weighted blends and confirm the result passes through every point and equals the Lagrange value.
Credit as content: Eric Harold Neville (1934). The weave: David names the toolchain; I combine neighboring interpolants into higher-degree ones by distance-weighted blends and confirm the result passes through every point and equals the Lagrange value.
3 ONE DIMENSION
Bottom row: the y-values (degree-0 interpolants). Each level up blends two adjacent entries by their distance to x, raising the degree. The apex is the interpolated value P(x).
4 TWO DIMENSIONS · INTERACTIVE
Data points and the interpolating curve from Neville’s algorithm; it is checked to pass through every point and match Lagrange.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: an interpolant built by a triangle of blends.
AVAN’s addition (the inverse-companion): evaluate the interpolating polynomial without ever computing its coefficients — blend neighboring lower-degree interpolants by their distance to x, climbing a triangle to the answer. The inverse of ‘solve for polynomial coefficients, then evaluate’ is ‘blend interpolants pairwise up a triangle.’ Magenta is the linear system you never solve; green is the blending triangle. Interpolation without coefficients.
LIT Genuine Neville's algorithm (Neville 1934). Verified live: over 300 random datasets, the distance-weighted interpolation triangle passes exactly through every data point and matches Lagrange interpolation at random query points (worst deviation ~1e-13) (window.__neville.passesThrough && .matchesLagrange).
FIG No framing: the interpolation triangle, the distance-weighted blends, the passes-through check, and the Lagrange cross-check run in-browser and agree to machine precision. The AVAN inverse is honest — blending neighboring lower-degree interpolants up a triangle evaluates the interpolant with no coefficients; magenta is the linear system you never solve, green the blending triangle. Interpolation without coefficients.
FIG No framing: the interpolation triangle, the distance-weighted blends, the passes-through check, and the Lagrange cross-check run in-browser and agree to machine precision. The AVAN inverse is honest — blending neighboring lower-degree interpolants up a triangle evaluates the interpolant with no coefficients; magenta is the linear system you never solve, green the blending triangle. Interpolation without coefficients.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE TOOLCHAIN · David Lee Wise (ROOT0), with AVAN