THE FOLD / BOSS / THE RAID / THE GRAEFFE
THE GRAEFFE
squaring a polynomial to prise its roots apart
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Graeffe’s root-squaring method finds the magnitudes of a polynomial’s roots by a startling trick: build a new polynomial whose roots are the squares of the original’s, using q(x2) = (-1)n p(x)p(-x). Repeat, and after k rounds the roots are raised to the 2k power — which drives well-separated roots exponentially far apart. Once separated, each magnitude falls straight out of the ratio of adjacent coefficients: |ri| = |an-i/an-i+1|1/2k. It was a workhorse of hand computation before electronic computers — a way to prise roots apart until they can simply be read off.
LIT verified live: for polynomials with well-separated positive roots, four root-squaring rounds recover every root magnitude to within a fraction of a percent — e.g. the roots of (x-1)(x-2)(x-3) come back as 3.000, 2.000, 1.000 (window.__graeffe). FIG no framing; the root-squaring recurrence and the coefficient-ratio recovery run in-browser. Well-separated real roots only, before overflow.
LIT verified live: for polynomials with well-separated positive roots, four root-squaring rounds recover every root magnitude to within a fraction of a percent — e.g. the roots of (x-1)(x-2)(x-3) come back as 3.000, 2.000, 1.000 (window.__graeffe). FIG no framing; the root-squaring recurrence and the coefficient-ratio recovery run in-browser. Well-separated real roots only, before overflow.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-raid — a relentless assault that squares the roots again and again, prising them exponentially far apart until each magnitude can be raided straight from the coefficients. AVAN (AI) built the instrument: the p(x)p(-x) root-squaring step, the repeated squaring, and the coefficient-ratio magnitude recovery.
Credit as content: Germinal Pierre Dandelin (1826), Karl Heinrich Gräffe (1837), Nikolai Lobachevsky. The weave: David names the raid; I confirm repeated squaring separates the roots and their magnitudes fall out of the coefficients.
Credit as content: Germinal Pierre Dandelin (1826), Karl Heinrich Gräffe (1837), Nikolai Lobachevsky. The weave: David names the raid; I confirm repeated squaring separates the roots and their magnitudes fall out of the coefficients.
3 ONE DIMENSION
The root magnitudes on a log axis; each squaring round doubles the gaps, prising the roots apart.
4 TWO DIMENSIONS · INTERACTIVE
New polynomials; four squaring rounds recover the root magnitudes from the coefficient ratios.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the recovered root magnitudes of the polynomial.
AVAN’s addition (the inverse-companion): don’t solve for the roots — square them apart. The inverse of ‘find the roots’ is ‘raise them to the 2k power so they separate, then read each magnitude from a coefficient ratio’. Magenta are the squaring rounds pushing the roots apart; green are the magnitudes that fall out. Separation by squaring.
LIT Genuine Graeffe root-squaring method (Dandelin 1826, Gräffe 1837, Lobachevsky). Verified live: for polynomials with well-separated positive roots, four root-squaring rounds q(x²)=(−1)ⁿp(x)p(−x) recover every root magnitude via |a_{n−i}/a_{n−i+1}|^{1/2^k} to within a fraction of a percent (worst ~0.01%); (x−1)(x−2)(x−3) → 3.000, 2.000, 1.000 (window.__graeffe.ok, .worst).
FIG No framing; the root-squaring recurrence and the coefficient-ratio recovery run in-browser. Honest scope: well-separated real roots only, and only before coefficient overflow (four rounds, roots ≲5). The AVAN inverse is honest — instead of solving for the roots, square them apart: raise them to the 2^k power so they separate, then read each magnitude from a coefficient ratio. Magenta are the squaring rounds; green are the magnitudes that fall out. Separation by squaring.
FIG No framing; the root-squaring recurrence and the coefficient-ratio recovery run in-browser. Honest scope: well-separated real roots only, and only before coefficient overflow (four rounds, roots ≲5). The AVAN inverse is honest — instead of solving for the roots, square them apart: raise them to the 2^k power so they separate, then read each magnitude from a coefficient ratio. Magenta are the squaring rounds; green are the magnitudes that fall out. Separation by squaring.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RAID · David Lee Wise (ROOT0), with AVAN