THE FOLD / BOSS / THE WALL / THE REMEZ
THE REMEZ
a polynomial whose error rides an equal wave
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Remez exchange algorithm finds the minimax polynomial — the degree-n polynomial that minimizes the worst-case error to a target function over an interval. Its signature is the equioscillation theorem (Chebyshev): the best approximation’s error curve touches its maximum height, alternating in sign, at exactly n+2 points, all of equal magnitude. Remez finds it by exchange: solve for the polynomial that makes the error equal-and-alternating at n+2 reference points, then move the references to the actual error extrema, and repeat. It converges to the provably optimal polynomial — strictly better in the worst case than Chebyshev interpolation.
LIT verified live: for several functions on [-1,1] the Remez polynomial’s error extrema all have equal magnitude (amplitude ratio ≈ 1.000, the equioscillation signature) and its maximum error is ≤ that of the degree-matched Chebyshev interpolant (window.__remez). FIG no framing; the linear solve for the reference system, the extrema exchange, and the Chebyshev comparison all run in-browser.
LIT verified live: for several functions on [-1,1] the Remez polynomial’s error extrema all have equal magnitude (amplitude ratio ≈ 1.000, the equioscillation signature) and its maximum error is ≤ that of the degree-matched Chebyshev interpolant (window.__remez). FIG no framing; the linear solve for the reference system, the extrema exchange, and the Chebyshev comparison all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-wall — the minimax error is a wall the approximation can never cross, and Remez lowers that wall as far as it can go, the error riding along it in an equal wave. AVAN (AI) built the instrument: the equioscillation linear system, the reference-exchange loop, the equal-amplitude check, and the comparison against Chebyshev interpolation.
Credit as content: Evgeny Remez (1934); equioscillation due to Chebyshev. The weave: David names the wall; I confirm the error equioscillates and beats Chebyshev interpolation on every function tested.
Credit as content: Evgeny Remez (1934); equioscillation due to Chebyshev. The weave: David names the wall; I confirm the error equioscillates and beats Chebyshev interpolation on every function tested.
3 ONE DIMENSION
The target f (cyan) and its minimax polynomial (green) overlaid — nearly indistinguishable at degree 4.
4 TWO DIMENSIONS · INTERACTIVE
The error curve f - p: it rides the ±E wall, touching it with alternating sign at n+2 equal-height points. Cycle the target.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the minimax error, riding a wall of equal height.
AVAN’s addition (the inverse-companion): don’t minimize average error — minimize the worst. The inverse of ‘fit the points’ is ‘spread the error so its peaks are all equal and alternating’ — and that equal-ripple curve is provably optimal. Magenta is the larger Chebyshev-interpolation error; green is the lowered minimax wall. Optimality as a level ripple.
LIT Genuine Remez exchange algorithm (Evgeny Remez, 1934; equioscillation due to Chebyshev). Verified live: for eˣ, 1/(1+x²), sin 2x on [-1,1] the degree-4 minimax polynomial's error extrema have equal magnitude (amplitude ratio ≈1.000, the equioscillation signature) and its max error is ≤ the degree-matched Chebyshev interpolant (window.__remez.equioscillates, .beatsCheb).
FIG No framing; the equioscillation linear solve, the reference-exchange loop, and the Chebyshev comparison all run in-browser. The AVAN inverse is honest — instead of minimizing average error, minimize the worst: spread the error so its peaks are all equal and alternating, and that equal-ripple curve is provably optimal. Magenta is the larger Chebyshev-interpolation error; green is the lowered minimax wall. Optimality as a level ripple.
FIG No framing; the equioscillation linear solve, the reference-exchange loop, and the Chebyshev comparison all run in-browser. The AVAN inverse is honest — instead of minimizing average error, minimize the worst: spread the error so its peaks are all equal and alternating, and that equal-ripple curve is provably optimal. Magenta is the larger Chebyshev-interpolation error; green is the lowered minimax wall. Optimality as a level ripple.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE WALL · David Lee Wise (ROOT0), with AVAN