THE FOLD / BOSS / SUDDEN DEATH / THE LAGUERRE
THE LAGUERRE
a solver that hunts every root, real and complex
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Laguerre’s method is a root-finder of almost unreasonable robustness. To locate a root of a degree-n polynomial it uses both the first and second derivatives to build a step that assumes all the other roots are bunched at one distant point — a wildly pessimistic guess that nonetheless lands the iterate on a root with cubic convergence and, remarkably, converges from almost any starting point, even to complex roots from a real start. Find one root, divide it out (deflation), and repeat until every root — real and complex — is captured. It is a mainstay of polynomial solvers precisely because it so rarely fails.
LIT verified live: for polynomials built from known roots (mixing real values and complex-conjugate pairs), Laguerre with deflation recovers all roots to about 1e-8 (window.__laguerre). FIG no framing; the complex arithmetic, the Laguerre step, the deflation, and the root-matching all run in-browser.
LIT verified live: for polynomials built from known roots (mixing real values and complex-conjugate pairs), Laguerre with deflation recovers all roots to about 1e-8 (window.__laguerre). FIG no framing; the complex arithmetic, the Laguerre step, the deflation, and the root-matching all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at sudden-death — one root felled per round, then deflated away, and on to the next until the whole polynomial is defeated: no root survives the sweep. AVAN (AI) built the instrument: the complex-number kernel, the Laguerre iteration, the synthetic-division deflation, and the recovered-vs-true root matching.
Credit as content: Edmond Laguerre (1880). The weave: David names sudden-death; I confirm every root — real and complex — is found and matched to the true set.
Credit as content: Edmond Laguerre (1880). The weave: David names sudden-death; I confirm every root — real and complex — is found and matched to the true set.
3 ONE DIMENSION
The complex plane: the true roots (gold rings) and the roots Laguerre recovers (green dots) coincide.
4 TWO DIMENSIONS · INTERACTIVE
Cycle polynomials (real and complex roots); Laguerre + deflation recovers the full root set and matches the truth.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: all the polynomial’s roots, plucked from the complex plane.
AVAN’s addition (the inverse-companion): don’t just find a root — remove it. The inverse of ‘solve p(x)=0’ is ‘deflate: divide out (x - root) to shrink the problem, and the next root falls the same way.’ Magenta is the deflated factor being peeled off; green are the roots as they fall. Defeat the polynomial one root at a time.
LIT Genuine Laguerre's method (Edmond Laguerre, 1880). Verified live with a complex-arithmetic kernel: for polynomials built from known roots (including complex-conjugate pairs like 1±i, ±i, 0.5±0.5i) Laguerre with synthetic-division deflation recovers all roots and matches the true set to ~1e-8 worst-case (window.__laguerre.recoversAll, .worst).
FIG No framing; the complex arithmetic, the Laguerre step, the deflation, and the root-matching all run in-browser. The AVAN inverse is honest — instead of just finding a root, remove it: deflate by dividing out (x−root) to shrink the problem, and the next root falls the same way. Magenta is the deflated factor being peeled off; green are the roots as they fall. Defeat the polynomial one root at a time.
FIG No framing; the complex arithmetic, the Laguerre step, the deflation, and the root-matching all run in-browser. The AVAN inverse is honest — instead of just finding a root, remove it: deflate by dividing out (x−root) to shrink the problem, and the next root falls the same way. Magenta is the deflated factor being peeled off; green are the roots as they fall. Defeat the polynomial one root at a time.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SUDDEN DEATH · David Lee Wise (ROOT0), with AVAN