THE FOLD / CHEAT / THE SPEEDRUN / THE HALLEY
THE HALLEY
cubic-convergence root finding via the second derivative
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Halley’s method finds a root of a function even faster than Newton’s. Newton follows the tangent line; Halley also uses the second derivative (the curvature), fitting a better local model, so the error is roughly cubed each step instead of squared — about 3× the correct digits per iteration versus Newton’s 2×. The step is x − 2ff′ / (2f′² − ff″).
LIT verified live: over 200 cases Halley converges to the true cube root, and reaches the tolerance in no more iterations than Newton (usually fewer) — e.g. ∛50 in 3 Halley steps versus 4 Newton (window.__halley). FIG no framing; genuine cubic convergence.
LIT verified live: over 200 cases Halley converges to the true cube root, and reaches the tolerance in no more iterations than Newton (usually fewer) — e.g. ∛50 in 3 Halley steps versus 4 Newton (window.__halley). FIG no framing; genuine cubic convergence.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-speedrun — reach the root in the fewest possible steps by using more of the local shape. Halley is Newton, sped up by curvature. AVAN (AI) built the instrument: the Halley iteration, the Newton comparison, the root and iteration-count checks.
Credit as content: Edmond Halley (1694), of comet fame. The weave: David names the speedrun; I fold the second derivative into each step and confirm the root is correct and reached in no more steps than Newton.
Credit as content: Edmond Halley (1694), of comet fame. The weave: David names the speedrun; I fold the second derivative into each step and confirm the root is correct and reached in no more steps than Newton.
3 ONE DIMENSION
Newton follows the straight tangent; Halley follows a curve that bends toward the root using the second derivative — so it overshoots less and homes in cubically.
4 TWO DIMENSIONS · INTERACTIVE
Solve x³ = a; watch Halley’s iterates converge, and compare the step count to Newton’s on the same start.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the cubic-convergence path to the root.
AVAN’s addition (the inverse-companion): use the curvature too. Halley’s step incorporates the second derivative, fitting a better local model, so the error cubes each step instead of squaring — roughly 3× the correct digits per iteration versus Newton’s 2×. The inverse of ‘follow the tangent line’ is ‘follow a curve that bends toward the root.’ Magenta is Newton’s quadratic convergence; green is Halley’s cubic. Second-order information triples the digit yield.
LIT Genuine Halley's method (Halley 1694). Verified live: the curvature-aware iteration converges to the true cube root (|x^3-a|<1e-6) and reaches tolerance in no more iterations than Newton's method on the same start, across 200 cases (window.__halley.converges && .noMoreThanNewton); cbrt(50) in 3 Halley steps vs 4 Newton.
FIG No framing: the Halley iteration, the Newton comparison, and the root + iteration-count checks run in-browser and hold. The AVAN inverse is honest — incorporating the second derivative fits a better local model so the error cubes each step (cubic vs Newton's quadratic convergence), tripling the digit yield; magenta is Newton's quadratic path, green Halley's cubic. Second-order information for fewer steps.
FIG No framing: the Halley iteration, the Newton comparison, and the root + iteration-count checks run in-browser and hold. The AVAN inverse is honest — incorporating the second derivative fits a better local model so the error cubes each step (cubic vs Newton's quadratic convergence), tripling the digit yield; magenta is Newton's quadratic path, green Halley's cubic. Second-order information for fewer steps.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SPEEDRUN · David Lee Wise (ROOT0), with AVAN