The Bisection Method for Root finding on an Interval

A root of a function is an input value that produces an output of zero. When a function is plotted, the roots are points where the function crosses the x-axis. Usually, we can use Vieta’s formulas to find roots of polynomial equations. But how does the computer find the roots? Some scientists have implemented several numerical root- finding methods to solve the problem.

The numerical root finding methods use iteration, producing a sequence of numbers that hopefully converge towards a limit which is a root.

In this article, the animation package’s function bisection.method() is applied to show you one of the numercical root-finding methods – the Bisection Method.

Basically, the bisection method finding the root from both sides by making use of the Intermediate Value Theorem. In essence, this theorem says that if $f$ is a continous function on $[a, b]$ and the sign of $f(a)$ is different from the sign of $f(b)$, then there must be some point $c$, in the interval such that $f(c)=0$, and is thus a root of the function.

Here is an example of Bisection Method, the continuous function is $f(x)=x{^3}-7x-10$, the interval is $[-3,5]$. The root on the interval is 3.184.

library(animation)
ani.options(interval = 2)
par(mar = c(4, 4.5, 2, 1.5))
f = function(x) x^3 - 7 * x - 10
xx = bisection.method(f, c(-3, 5))

xx$root

## [1] 3.184