Secant method ($x_{k+1}=x_k-f(x_k)(x_k-x_{k-1})/(f(x_k)-f(x_{k-1}))$)

The secant method approximates a zero of a function by repeatedly taking the zero of a secant line through the two most recent sampled points. The formula follows by setting the line $f(x_k)+(x-x_k)(f(x_k)-f(x_{k-1}))/(x_k-x_{k-1})$ equal to zero. It requires distinct sampled positions and a nonzero difference of sampled values, unless a root has already been found. After initialization it needs one new function evaluation per step. Unlike bisection, it need not retain a bracket. For a sufficiently smooth function near a simple root, its usual local order of convergence is $(1+\sqrt{5})/2$.^[ref]