◄ WORLD V · SONNY 5DART 294 · a helldive at the net

THE VANDERMONDE MATRIX the determinant that decides whether reconstruction is even possible

Fit a degree-n polynomial through n+1 points and you are solving a linear system whose matrix is the Vandermonde matrix — rows of powers [1, x, x², …]. Its determinant is a thing of beauty: ∏i<j(xj − xi) — the product of every pairwise gap between the nodes. That single number is load-bearing: it is nonzero if and only if the nodes are distinct, and that is exactly the condition under which the interpolating polynomial exists and is unique. Reconstruction is not merely correlated with distinct nodes; it is caused by the determinant being nonzero. Let two nodes collide and the determinant vanishes and the reconstruction is impossible — there is no polynomial to recover.

THE TECHNIQUE det = ∏i<j(xj−xi); ≠0 ⇔ distinct nodes ⇔ unique reconstruction

The Vandermonde determinant of a set of nodes. The demo computes it for distinct nodes (nonzero — reconstruction possible) and for a repeated node (zero — impossible): live demo







  
HISTORY & CREDIT Vandermonde (a misnomer) · 1770s

  
“Any set of points can be fit by a polynomial.” — only if the nodes are distinct, and the Vandermonde determinant is the number that decides it. Reconstruction has a precondition, and this determinant is it. cited

  
1770s · Alexandre-Théophile Vandermonde — the eponym, though (as Lebesgue observed, 1937) he never actually wrote the determinant named for him: a misnomer, Stigler's law again.
the fact · det = ∏i<j(xj−xi) — nonzero iff the nodes differ.
now · the solvability certificate under all polynomial interpolation, coding, and signal reconstruction.

  
The interpolant exists and is unique because this determinant is nonzero — and it is nonzero because the nodes are distinct. Collapse a node, and there is nothing to recover. Vandermonde (misnomer)






  
RECOMMEND FOR I-13 the determinant that gates reconstruction, computed

  
On the canonical compiler, nodes {1,2,4} give det = (2−1)(4−1)(4−2) = 6 — reconstruction possible; nodes {1,2,2} give det = 0 — impossible:

  
$ i13 run rec_vandermonde.i13   # det = prod(xj - xi)
det_distinct = 6      -- distinct nodes: interpolant EXISTS and is UNIQUE
det_repeat = 0        -- a repeated node: det vanishes -> reconstruction IMPOSSIBLE

  
Recommend: this is the batch's keeper shot, and it is built to pass the bar the cross-ratio (dart 278) failed. There, the invariance was a theorem true of any correct implementation — i13 witnessed it. Here, i13 computes the Vandermonde determinant and the reconstruction depends on the answer: nonzero → the interpolant exists and every downstream dart in this batch can recover; zero → there is nothing to recover, and the computation cannot proceed. The structure is load-bearing — alter it (collide two nodes) and the recovery fails. That is a principle a mechanism enacts, not one it merely observes: reconstruction-from-a-preserved-structure, with the structure as the cause.