THE FOLD / GRIND / BACKPROP / THE CHEBYSHEV
THE CHEBYSHEV
interpolate at clustered nodes to defeat Runge
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Chebyshev interpolation samples a function not on an even grid but at the Chebyshev nodes — points clustered toward the ends of the interval, the projected roots of the Chebyshev polynomials. This defeats Runge’s phenomenon: interpolating on an even grid can diverge wildly at the edges as you add points, but Chebyshev interpolation converges.
It is why spectral methods and Chebyshev approximation dominate high-accuracy numerics.
LIT verified live: on Runge’s function 1/(1+25x²), Chebyshev interpolation’s max error shrinks as nodes are added (to ~0.02 at n=21) while the equispaced error explodes (~60) — window.__chebyshev. FIG no framing; the error comparison is exact.
It is why spectral methods and Chebyshev approximation dominate high-accuracy numerics.
LIT verified live: on Runge’s function 1/(1+25x²), Chebyshev interpolation’s max error shrinks as nodes are added (to ~0.02 at n=21) while the equispaced error explodes (~60) — window.__chebyshev. FIG no framing; the error comparison is exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at backprop — beside Clenshaw, in the Chebyshev family: where you place the samples decides whether the fit converges, just as where you propagate gradients decides whether learning does. AVAN (AI) built the instrument: the barycentric interpolant, the Chebyshev vs equispaced node sets, the max-error comparison across n.
Credit as content: Pafnuty Chebyshev; Runge’s phenomenon (Carl Runge, 1901). The weave: David names backprop; I interpolate Runge’s function at Chebyshev nodes and confirm the error shrinks while the even-grid error blows up.
Credit as content: Pafnuty Chebyshev; Runge’s phenomenon (Carl Runge, 1901). The weave: David names backprop; I interpolate Runge’s function at Chebyshev nodes and confirm the error shrinks while the even-grid error blows up.
3 ONE DIMENSION
Chebyshev nodes are the projections of equally-spaced points on a semicircle onto the axis — dense near the ends, sparse in the middle. That clustering is exactly where an even grid’s error would otherwise blow up.
4 TWO DIMENSIONS · INTERACTIVE
Runge’s function with its Chebyshev interpolant (tracks it) and equispaced interpolant (oscillates at the edges); the max errors are shown.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Chebyshev interpolant converging to the function.
AVAN’s addition (the inverse-companion): sample at the Chebyshev nodes (clustered toward the ends) — this crushes the error at the edges where an even grid explodes (Runge’s phenomenon), so the interpolant converges instead of diverging. The inverse of ‘space the samples evenly’ is ‘cluster them where the error would otherwise blow up.’ Magenta is the equispaced interpolant oscillating wildly at the edges; green is the Chebyshev interpolant that converges. Where you sample decides whether interpolation works at all. (Kin to the-clenshaw and the-gaussian-quadrature.)
LIT Genuine Chebyshev interpolation (Chebyshev; Runge's phenomenon, Runge 1901). Verified live: for Runge's function 1/(1+25x^2), the barycentric interpolant at Chebyshev nodes has smaller max error than at equispaced nodes at every tested n, the Chebyshev error decreases as n grows, and at n=21 it is ~0.018 versus equispaced ~60 (window.__chebyshev.beatsEquispaced && .decreasesWithN && .smallAtN21).
FIG No framing: the barycentric interpolant, the Chebyshev vs equispaced node sets, and the max-error comparison across n run in-browser and are exact. The AVAN inverse is honest — clustering samples at the Chebyshev nodes crushes the edge error where an even grid explodes (Runge), so the interpolant converges instead of diverging; magenta is the wildly-oscillating equispaced interpolant, green the converging Chebyshev one. Kin to the-clenshaw and the-gaussian-quadrature.
FIG No framing: the barycentric interpolant, the Chebyshev vs equispaced node sets, and the max-error comparison across n run in-browser and are exact. The AVAN inverse is honest — clustering samples at the Chebyshev nodes crushes the edge error where an even grid explodes (Runge), so the interpolant converges instead of diverging; magenta is the wildly-oscillating equispaced interpolant, green the converging Chebyshev one. Kin to the-clenshaw and the-gaussian-quadrature.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of BACKPROP · David Lee Wise (ROOT0), with AVAN