THE FOLD / CHEAT / THE SPEEDRUN / THE DURAND-KERNER
THE DURAND-KERNER
all polynomial roots at once — estimates that repel into place
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Newton’s method finds one root at a time. Durand–Kerner (Weierstrass) finds all n roots of a degree-n polynomial simultaneously, iterating each estimate zᵢ by zᵢ ← zᵢ − p(zᵢ) / ∏j≠i(zᵢ − zⱼ).
The denominator divides out the influence of the other roots, so the estimates repel each other toward distinct roots. Started from evenly-spread complex guesses, it converges (usually quadratically) to all roots at once — no deflation, no root-by-root sequencing.
LIT verified live: over 100 polynomials built from known integer roots, Durand–Kerner converges so that |p(z)| < 10⁻⁴ at every returned root (window.__durandkerner). FIG no framing; exact complex iteration to machine precision.
The denominator divides out the influence of the other roots, so the estimates repel each other toward distinct roots. Started from evenly-spread complex guesses, it converges (usually quadratically) to all roots at once — no deflation, no root-by-root sequencing.
LIT verified live: over 100 polynomials built from known integer roots, Durand–Kerner converges so that |p(z)| < 10⁻⁴ at every returned root (window.__durandkerner). FIG no framing; exact complex iteration to machine precision.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-speedrun — all the roots in one parallel sweep instead of one-at-a-time. Durand–Kerner is the root-finding speedrun. AVAN (AI) built the instrument: the complex-arithmetic iteration, the mutual-repulsion update, the residual check at every root.
Credit as content: Karl Weierstrass (1891); rediscovered by Émile Durand (1960) and Immo Kerner (1966). The weave: David names the speedrun; I let n complex estimates repel one another through the polynomial, converging to all roots at once, and confirm each makes the polynomial vanish.
Credit as content: Karl Weierstrass (1891); rediscovered by Émile Durand (1960) and Immo Kerner (1966). The weave: David names the speedrun; I let n complex estimates repel one another through the polynomial, converging to all roots at once, and confirm each makes the polynomial vanish.
3 ONE DIMENSION
Each estimate is nudged by the polynomial’s value divided by its distance to all the others — so the estimates push apart, each sliding toward its own root, none colliding.
4 TWO DIMENSIONS · INTERACTIVE
A polynomial’s roots in the complex plane. Iterate and watch the estimates spiral in from a circle to all roots at once; the residual |p(z)| at each drops to zero.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the n estimates converging in parallel across the complex plane, repelling into their roots.
AVAN’s addition (the inverse-companion): the method treats the roots as mutually defining. Each estimate’s update divides by its distance to all the others, so the n estimates form a coupled system that self-organises — you cannot find one without implicitly accounting for all. The inverse of ‘a single root’ is ‘the whole root-set as one coupled fixed point,’ reached exactly when every numerator p(zᵢ) hits zero at once. Magenta is the one-at-a-time Newton path (deflate, repeat); green is the n estimates converging together, repelling into place. Solve the system as a whole and the roots find each other — a polynomial’s factorisation emerging all at once.
LIT Genuine Durand-Kerner / Weierstrass method (Weierstrass 1891; Durand 1960; Kerner 1966). Verified live: iterating n complex estimates by z_i -= p(z_i)/prod_{j!=i}(z_i-z_j) from spread initial guesses converges so that the residual |p(z)| < 1e-4 at every returned root, across 100 polynomials built from known integer roots (window.__durandkerner.allRootsFound).
FIG No framing: the complex iteration, the mutual-repulsion update, and the residual check at every root run in-browser. The AVAN inverse is honest — each update divides by distance to all other estimates, making the roots a coupled system that self-organizes to a joint fixed point (all p(z_i)->0 together); magenta is the one-at-a-time Newton path, green the parallel convergence.
FIG No framing: the complex iteration, the mutual-repulsion update, and the residual check at every root run in-browser. The AVAN inverse is honest — each update divides by distance to all other estimates, making the roots a coupled system that self-organizes to a joint fixed point (all p(z_i)->0 together); magenta is the one-at-a-time Newton path, green the parallel convergence.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SPEEDRUN · David Lee Wise (ROOT0), with AVAN