THE FOLD / GRIND / BACKPROP / THE VANDERMONDE
THE VANDERMONDE
a determinant that factors into differences
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Vandermonde determinant is a determinant that factors perfectly. Build the matrix whose rows are the powers of numbers x₁,…,xₙ — row i is 1, xₖ, xₖ², …, xₖⁿ⁻¹. Its determinant, which looks like it should be a hopeless mess of n! signed products, collapses to a single clean product over all pairs: det V = ∏i<j(xj - xi). It is zero exactly when two of the x’s coincide (two equal rows), which is why it governs polynomial interpolation: n distinct points determine a unique degree-(n-1) polynomial precisely because this determinant is nonzero.
LIT verified live with exact integer arithmetic: for thousands of random distinct integer node-sets (n up to 7), the determinant computed by the fraction-free Bareiss algorithm equals the pairwise product ∏i<j(xj-xi) exactly, with no floating-point error (window.__vandermonde). FIG no framing; the determinant and the product formula both run in-browser with arbitrary-precision integers and agree exactly.
LIT verified live with exact integer arithmetic: for thousands of random distinct integer node-sets (n up to 7), the determinant computed by the fraction-free Bareiss algorithm equals the pairwise product ∏i<j(xj-xi) exactly, with no floating-point error (window.__vandermonde). FIG no framing; the determinant and the product formula both run in-browser with arbitrary-precision integers and agree exactly.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at backprop — the grind: a determinant is n! signed products to grind through, yet this one factors into a tidy product of differences. AVAN (AI) built the instrument: the Vandermonde matrix, the exact Bareiss determinant, and the pairwise-difference product.
Credit as content: Alexandre-Théophile Vandermonde (1770s). The weave: David names the grind; I confirm the messy determinant equals the clean product of differences, exactly.
Credit as content: Alexandre-Théophile Vandermonde (1770s). The weave: David names the grind; I confirm the messy determinant equals the clean product of differences, exactly.
3 ONE DIMENSION
The Vandermonde matrix — row i is the powers of x_i — whose determinant is ∏(x_j − x_i).
4 TWO DIMENSIONS · INTERACTIVE
New node-sets; the Bareiss determinant is compared to the pairwise-difference product, exactly.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the determinant value, a single integer.
AVAN’s addition (the inverse-companion): don’t expand n! products — read the pairwise gaps. The inverse of ‘the determinant’ is ‘the set of differences xj-xi whose product it is’, so the determinant vanishes the instant any two nodes collide. Magenta are the pairwise differences; green is the determinant they multiply to. A determinant that is really a product of gaps.
LIT Genuine Vandermonde determinant (Alexandre-Théophile Vandermonde, 1770s). Verified live with exact BigInt: for ~2000 random distinct integer node-sets (n=2..7), the fraction-free Bareiss determinant equals ∏_{i
FIG No framing; the determinant and the product formula both run in-browser with arbitrary-precision integers and agree exactly. The AVAN inverse is honest — instead of expanding n! products, read the pairwise gaps: the inverse of 'the determinant' is 'the set of differences x_j−x_i whose product it is', so it vanishes the instant any two nodes collide. Magenta are the pairwise differences; green is the determinant they multiply to. A determinant that is really a product of gaps.
FIG No framing; the determinant and the product formula both run in-browser with arbitrary-precision integers and agree exactly. The AVAN inverse is honest — instead of expanding n! products, read the pairwise gaps: the inverse of 'the determinant' is 'the set of differences x_j−x_i whose product it is', so it vanishes the instant any two nodes collide. Magenta are the pairwise differences; green is the determinant they multiply to. A determinant that is really a product of gaps.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of BACKPROP · David Lee Wise (ROOT0), with AVAN