Gradient descent only feels the slope; Newton’s method also feels the CURVATURE, and leaps almost straight to the target. To find where a function hits zero, follow its tangent line to the axis, and repeat — each step roughly DOUBLES the correct digits. It finds √2 to machine precision in five steps. Fast, but it needs second-derivative information and can be unstable. Slide the steps.
Newton’s method finds a root of f by following the tangent: x ← x − f(x)/f′(x). Near a simple root it converges QUADRATICALLY — the number of correct digits roughly doubles each step — far faster than linear methods. For optimization it uses curvature: θ ← θ − H⁻¹∇L (the Hessian H), stepping toward the minimum in one leap on a quadratic bowl. Finding √2 via x−(x²−2)/(2x) reaches machine precision in ~5 steps. A fail-loud self-check throws unless it converges to √2 in under 8 steps. ◆ real numerical optimization, node-verified.
Quadratic convergence holds only NEAR a simple root with a good start; far away it can overshoot, cycle, or diverge, and it needs f′ (or the Hessian, expensive in high dimensions — hence quasi-Newton methods like BFGS). The √2 convergence is exact.