Newton's Method

Numerical Analysis · Axiom Academy

Use tangent lines for rapid root-finding with quadratic convergence The key insight: if we're at a point x₀ on the curve y = f(x), the tangent line at that point gives a good linear approximation to the function. The x-intercept of this tangent line is often closer to the root than x₀. The tangent line at (x₀, f(x₀)) has slope f'(x₀). Using point-slope form and finding where this line crosses the x-axis gives us Newton's iteration formula. Starting from the tangent line equation at point (xₙ, f(xₙ)): Tangent line: y - f(xₙ) = f'(xₙ)(x - xₙ) Set y = 0: -f(xₙ) = f'(xₙ)(x - xₙ) Solve for x: x = xₙ - f(xₙ)/f'(xₙ) This is Newton's formula! Each new estimate xₙ₊₁ is found by subtracting the ratio f(xₙ)/f'(xₙ) from the current estimate. Example: Find √2 by solving x² - 2 = 0, starting at x₀ = 1. Newton's method has quadratic convergence near simple roots. This means the error at step n+1 is proportional to the square of the error at step n: Newton's method is powerful but not foolproof. Here are common failure modes: f'(xₙ) = 0: Division by zero! (horizontal tangent) Poor initial guess: May diverge or cycle Multiple roots: Convergence slows to linear Oscillation: May bounce between values forever

This is the written version of the interactive lesson above. See the full Numerical Analysis course.