# bisection.method()

## 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()
``````

plot of chunk demo-a

``````xx\$root  # solution
``````
``````##  2
``````
``````## a cubic curve
f = function(x) x^3 - 7 * x - 10
xx = bisection.method(f, c(-3, 5))
``````

plot of chunk demo-b

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