Newton-Raphson Method

Newton–Raphson method for finding roots

Newton–Raphson solves f(x) = 0 by iterating x_{n+1} = x_n − f(x_n)/f'(x_n). Each step replaces f by its tangent at x_n and jumps to where that tangent crosses zero. Convergence is quadratic: the number of correct digits roughly doubles per iteration — extremely fast when it works. Requirements: f must be differentiable and f' should not be near zero at the iterate; a bad initial guess can diverge or cycle. Example — compute √2 by solving x²−2=0, giving x_{n+1} = (x_n + 2/x_n)/2. From x₀=1: 1.5 → 1.4167 → 1.41422 → 1.41421356… (8 digits in 4 iterations). When f' is expensive, quasi-Newton methods like BFGS approximate it from past steps.

Applications and context

Newton–Raphson powers spreadsheet solvers (Excel Solver and Goal Seek), CAD/CAM (curve intersections, IK in robotics), IRR in finance, machine learning optimization (Newton steps and quasi-Newton variants), and computer graphics (ray marching, distance fields). It's the default root-finder when you can supply a derivative cheaply.

FAQ

What if Newton diverges? Bracket the root first with bisection, then switch to Newton once you're close — the basis of Brent's method.

How do I stop? Standard tests: |x_{n+1} − x_n| < tol, |f(x_n)| < tol, or a max-iterations cap (usually 50 is plenty thanks to quadratic convergence).

What if I don't have f'? Use the secant method (estimates f' from the last two iterates) — superlinear convergence (~1.618) but no derivative needed; or autodiff if you have source code for f.