Newtons Method Calculator

Newton's Method (Newton-Raphson method) is an iterative algorithm for finding roots of a differentiable function f(x) — that is, values of x where f(x) = 0. Starting from an initial guess x₀, it repeatedly applies the update formula to converge to a root. When it converges, it converges quadratically — the number of correct decimal places roughly doubles with each iteration.

1. 1xₙ₊₁ = xₙ − f(xₙ) / f'(xₙ)
2. 2f'(x) is approximated numerically: f'(x) ≈ [f(x+h) − f(x−h)] / 2h, h = 10⁻⁷
3. 3Converges when |xₙ₊₁ − xₙ| < 10⁻¹⁰
4. 4May fail if f'(x₀) ≈ 0 or if starting guess is far from a root
5. 5Multiple roots require different initial guesses