Secant Method

Numerical Analysis · Axiom Academy

Newton's method without derivatives - using two points to approximate the tangent Newton's method uses the tangent line (derivative). The secant method uses a secant line through the two most recent points instead. This line approximates the tangent for nearby points. The secant line through (xₙ₋₁, f(xₙ₋₁)) and (xₙ, f(xₙ)) has slope: Replace f'(xₙ) in Newton's formula with the secant slope approximation: Example: Find √2 by solving x² - 2 = 0, starting with x₀ = 1, x₁ = 2. The secant method has superlinear convergence with order φ ≈ 1.618 (the golden ratio!). It's between linear (order 1) and quadratic (order 2). 5. Secant vs Newton vs Bisection Lower cost per iteration than Newton Can diverge with poor initial guesses Division by zero if f(xₙ) = f(xₙ₋₁) Slower than Newton near the root

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