Newton’s method is fast but needs the derivative — what if you don’t have it? Draw a straight line through your last TWO guesses instead of the tangent, and follow IT to zero. No calculus required, and it still converges almost as fast (order 1.618, the golden ratio). It’s Newton’s practical cousin, used when the derivative is unknown or expensive. Slide to step and watch the secant chase the root.
The secant method finds a root using a finite-difference stand-in for the derivative: from the last two iterates it draws the SECANT line and takes its zero, xₙ₊₁=xₙ−f(xₙ)(xₙ−xₙ₋₁)/(f(xₙ)−f(xₙ₋₁)). No derivative is needed (unlike [[the-newton-method|Newton]]), and convergence is SUPERLINEAR with order φ≈1.618 (the golden ratio) — slower than Newton’s quadratic but with one cheaper evaluation per step. It underlies Broyden’s method and many derivative-free solvers. A fail-loud self-check throws unless it reaches √2 (with a divide-by-zero guard on convergence) to 1e−10. ◆ real numerical methods, node-verified.
The secant can diverge without a bracket, and the denominator f(xₙ)−f(xₙ₋₁) → 0 near convergence (guarded here). Superlinear φ-order and the derivative-free step are exact.