Finding Roots with the Secant Method

Required Reading

Secant Method

To find a root of an equation $f$ using the secant method, choose two points, $x_0$ and $x_1$, such that $f(x_0)$ and $f(x_1)$ are near the root. Calculate the $x$-intercept of $\overleftrightarrow{f(x_0) f(x_1)}$.

Now, $x_1$ becomes $x_0$, and $x_0$ becomes the $x$-intercept previously calculated. The root is found iteratively by continuing this process until difference between $x_0$ and $x_1$ becomes sufficiently small.

Unlike the Newton-Raphson method, the secant method does not require a first derivative of $f$.

The secant method can be expressed generally, given $x_{n-1}$ and $x_{n}$,

\begin{equation} x_{n+1} = x_n - f(x_n)\frac{x_n - x_{n-1}}{f(x_n) - f(x_{n-1})}\text{.} \label{secant-method} \end{equation}

The image below may help to visualise the how the secant method finds the root.

Graphical example of the secant root finding method.