Here is a very common mathematical situation: let f be a continuous monotone function (that is, increasing or decreasing) defined on an interval with endpoints a and b, such that f (a) and f (b) have opposite signs. Since f is continuous, its graph must cross the x-axis (by the intermediate value theorem). Moreover, since it is monotone (in this case, increasing), the equation f (x) = 0 has exactly one solution. This solution is called a root of the equation and is denoted by α.
In some cases—for example, when f (x) is a quadratic polynomial—there are formulas, familiar to high-school students, that give the value of α. In general, however, there is no exact method for finding this value for an arbitrary function f (x). There are, however, iterative methods for finding an approximate value of α. Let's look at the best-known ones.
Bisecting the interval
------------------------
The first method is called the bisection method. Behind this rather technical term lies a very simple idea: divide the interval [a, b] into two equal parts and determine which one contains the root α. This new interval will be half the length of the original interval and will contain α. By repeating the process enough times, we obtain an approximate value of α to any desired accuracy, since the interval's length is halved at each step. In fact, we obtain two approximate values: a lower approximation and an upper approximation.
