# The Bisection Method for root-finding on an interval

### Yihui Xie & Lijia Yu / 2017-04-04

This is a visual demonstration of finding the root of an equation $$f(x) = 0$$ on an interval using the Bisection Method.

Suppose we want to solve the equation $$f(x) = 0$$. Given two points a and b such that $$f(a)$$ and $$f(b)$$ have opposite signs, we know by the intermediate value theorem that $$f$$ must have at least one root in the interval $$[a, b]$$ as long as $$f$$ is continuous on this interval. The bisection method divides the interval in two by computing $$c = (a + b) / 2$$. There are now two possibilities: either $$f(a)$$ and $$f(c)$$ have opposite signs, or $$f(c)$$ and $$f(b)$$ have opposite signs. The bisection algorithm is then applied recursively to the sub-interval where the sign change occurs.

During the process of searching, the mid-point of subintervals are annotated in the graph by both texts and blue straight lines, and the end-points are denoted in dashed red lines. The root of each iteration is also plotted in the right margin of the graph.

library(animation)
ani.options(nmax = 30)

## default example
xx = bisection.method()

xx\$root  # solution

##  2

## a cubic curve
f = function(x) x^3 - 7 * x - 10
xx = bisection.method(f, c(-3, 5))

## interaction: use your mouse to select the two end-points
if (interactive()) bisection.method(f, c(-3, 5), interact = TRUE)