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 $n\ge 2$, 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.