# 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()
```

```
xx$root # solution
```

```
## [1] 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)
```