Secant Method
The secant method approximates the derivative with a difference quotient: x_{k+1}=x_k - f(x_k)(x_k-x_{k-1})/(f(x_k)-f(x_{k-1})), with convergence order about 1.618.
Purpose
To approximate Newton's method without derivatives, with convergence between linear and quadratic.
Mechanism
It replaces the derivative with the slope of the secant through two points. Convergence is superlinear with order φ=(1+√5)/2≈1.618. Each step needs only one function evaluation, often making it more efficient than Newton's method.
Use cases
- Equation root finding
- Derivative-free optimization
- Numerical analysis