◄ WORLD V · SONNY 5DART 171 · a helldive at the net

HALLEY’S METHOD one order past Newton; triple the digits each step

Newton’s method uses the slope to jump toward a root and doubles the correct digits each step (quadratic). Halley’s method also uses the curvature (the second derivative), bending the step to fit — and triples the correct digits each step (cubic). For a smooth function near a simple root it reaches full precision in a couple of iterations where bisection needs fifty. Yes, that Halley — the comet astronomer.

THE TECHNIQUE uses f, f', and f''; cubic convergence

Find √2 from a start of 1.0. Watch the correct digits roughly triple each iteration — three steps to full f64 precision, against Newton’s ~five and bisection’s ~fifty: live demo







  
HISTORY & CREDIT Edmond Halley, 1694

  
“Higher order always means faster in practice.” — not simply. Halley converges in fewer iterations than Newton, but each iteration costs a second derivative; whether it wins on total work depends on how expensive f'' is. More order, more cost per step — a real trade, not a free lunch. cited

  
1694 · Edmond Halley — Methodus nova accurata & facilis… (Phil. Trans., May 1694; the familiar English title “A new, exact, and easy method…” is the 1809 translation): the cubic-convergence root method, by the astronomer of the comet.
context · it is the second member of the Householder family (Newton is the first) — each member uses one more derivative for one higher order of convergence (Newton converges quadratically, Halley cubically).
modern · Halley’s iteration still appears in high-precision library routines (reciprocals, roots) where a few cubic steps beat many quadratic ones.

  
The step is x − 2ff’ / (2f’² − ff’’): drop the ff’’ curvature term and it collapses back to Newton’s x − f/f’ — Halley is Newton plus one correction. Halley, 1694






  
RECOMMEND FOR I-13 cubic convergence, computed

  
Halley reaches √2 to full f64 precision in three iterations on the canonical compiler:

  
$ i13 run halley.i13   # sqrt2 from x=1, Halley step
h1 = 1.4                    -- 2 correct digits
h2 = 1.4142131979695431     -- ~7 correct digits
h3 = 1.414213562373095      -- full f64 precision (digits roughly tripled each step)

  
Recommend: Halley’s method is LIT and the fast root-finder for I-13 — verified it reaches √2 = 1.4142135624 from x=1 in three native-recursion steps (1.4 → 1.41421320 → 1.41421356), the correct digits tripling each time. Where the transcendental library (dart 033) uses Newton for sqrt, Halley converges in fewer steps when f'' is cheap (it is, for roots and reciprocals). It degrades gracefully to Newton (drop the curvature term) and pairs with bisection (170) as the fast half of a hybrid. Pure f64.