Secant method

This article is about a root-finding algorithm. See all root-finding algorithms

Definition

The secant method is a root-finding algorithm that makes successive point estimates for the value of a root of a continuous function. In general, the secant method is not guaranteed to converge towards a root, but under some conditions, it does. A slight variant of this method, called the false position method, functions very similarly to the bisection method.

Iterative process

The secant method requires two initial guesses for the root, say $x_{0}$ and $x_{1}$. For $n\geq 2$, we define $x_{n}$ as the following affine combination of the previous two guesses $x_{n-1}$ and $x_{n-2}$

$x_{n}={\frac {x_{n-2}f(x_{n-1})-x_{n-1}f(x_{n-2})}{f(x_{n-1})-f(x_{n-2})}}$

Geometrically, this can be interpreted as follows: we make a line through the points $(x_{n-2},f(x_{n-2}))$ and $(x_{n-1},f(x_{n-1}))$ in the $(x,f(x))$-plane, and define $x_{n}$ as the $x$-coordinate of the intersection of this line with the $x$-axis.