====== The Bisection Method for Root-finding on an Interval ======

In mathematics, the bisection method is a root-finding algorithm which works by repeatedly dividing an interval in half and then selecting the subinterval in which a root exists. 

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. 

The function ''bisection.method()'' gives a visual demonstration of this process of finding the root of an equation $f(x) = 0$.

===== Animation =====

<ani bm http://i288.photobucket.com/albums/ll181/xieyihui/bisection_method/ png 11 400 2|The Bisection Method for Root-finding on an Interval>
The bisection method is a root-finding algorithm which works by repeatedly dividing an interval in half and then selecting the subinterval in which a root exists.
</ani>

===== R code =====

<code r>
ani.start(nmax = 50, ani.height = 400, ani.width = 600, interval = 2,
    title = "The Bisection Method for Root-finding on an Interval",
    description = "The bisection method is a root-finding algorithm 
    which works by repeatedly dividing an interval in half and then 
    selecting the subinterval in which a root exists.")
par(mar = c(4, 4, 1, 1))
bisection.method(main = "")
ani.stop()
</code>