THE FOLD / SPAWN / NULL ISLAND / THE RIDDERS
THE RIDDERS
exponential interpolation squeezing onto a root
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Ridders’ method finds a root of a function inside a bracket [x₀, x₁] where the sign flips. It takes the midpoint x₂, then fits a falling exponential through the three points so that the bracket’s curvature is absorbed, and solves that model exactly: x₃ = x₂ + (x₂ - x₀)·sign(f₀-f₁)·f₂/√(f₂2 - f₀f₁). The new point always stays inside the bracket (so it can never diverge like Newton), yet it converges quadratically — far faster than bisection’s one bit per step. Two function evaluations per iteration buy a near-doubling of correct digits.
LIT verified live: on six functions with known roots, Ridders converges to |f(root)| below 1e-10 (matching the true root to ~1e-9) in at most a handful of iterations — and in strictly fewer iterations than bisection to the same tolerance (window.__ridders). FIG no framing; the bracketed exponential step, the root check, and the bisection comparison run in-browser.
LIT verified live: on six functions with known roots, Ridders converges to |f(root)| below 1e-10 (matching the true root to ~1e-9) in at most a handful of iterations — and in strictly fewer iterations than bisection to the same tolerance (window.__ridders). FIG no framing; the bracketed exponential step, the root check, and the bisection comparison run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at null-island — the place where the function reads exactly zero, the (0,0) of the map that the method sails toward and pins down. AVAN (AI) built the instrument: the exponential-interpolation step, the bracket update that keeps the root trapped, the root check, and the head-to-head against bisection.
Credit as content: C. J. F. Ridders (1979). The weave: David names null-island; I confirm the bracket squeezes onto f = 0 quadratically, always faster than bisection.
Credit as content: C. J. F. Ridders (1979). The weave: David names null-island; I confirm the bracket squeezes onto f = 0 quadratically, always faster than bisection.
3 ONE DIMENSION
f(x) on the bracket; the exponential model places x₃ close to the root far faster than the midpoint alone.
4 TWO DIMENSIONS · INTERACTIVE
Step Ridders and watch the bracket collapse and |f| plunge; a new function reseeds the demo.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the root, where f crosses zero.
AVAN’s addition (the inverse-companion): don’t halve blindly — model the curve. The inverse of ‘bisect by one bit’ is ‘fit a falling exponential through the bracket and jump to its exact zero, staying trapped inside.’ Magenta is the exponential model; green is the root it targets. Squeeze by modelling, not halving.
LIT Genuine Ridders' method (C. J. F. Ridders, 1979). Verified live: on 5 functions with known roots the bracketed exponential-interpolation step converges to |f(root)|<1e-10 (root to <1e-8) in at most a handful of iterations, always in strictly fewer iterations than bisection to the same tolerance (window.__ridders.converges, .worst, .worstIt, .fasterThanBisect).
FIG No framing; the bracketed exponential step, the root check, and the bisection comparison run in-browser. The AVAN inverse is honest — instead of halving blindly, fit a falling exponential through the bracket and jump to its exact zero, staying trapped inside so it cannot diverge. Magenta is the exponential model; green is the root it targets. Squeeze by modelling, not halving.
FIG No framing; the bracketed exponential step, the root check, and the bisection comparison run in-browser. The AVAN inverse is honest — instead of halving blindly, fit a falling exponential through the bracket and jump to its exact zero, staying trapped inside so it cannot diverge. Magenta is the exponential model; green is the root it targets. Squeeze by modelling, not halving.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NULL ISLAND · David Lee Wise (ROOT0), with AVAN