Find a root of f(x)=0 using the secant method — no derivative needed.
The secant method approximates Newton's method by replacing the derivative with a finite difference: f'(xₙ) ≈ [f(xₙ) − f(xₙ₋₁)]/(xₙ − xₙ₋₁). It requires two initial guesses rather than one, and converges at rate φ ≈ 1.618 (superlinear) — faster than bisection but slightly slower than Newton's quadratic rate. No derivative computation is needed.
Secant iteration
x(n+1) = x(n) - f(x(n)) · (x(n) - x(n-1)) / (f(x(n)) - f(x(n-1)))
Use secant when the derivative is expensive or unavailable. It needs one extra initial point but avoids derivative computation entirely.
Yes, if the two function values are nearly equal (horizontal secant line) or if the guesses are too far from the root. Bisection is safer as a fallback.