Bisection Method

Solution of Equations in One Variable
Root-finding problem is a process that involves finding a root, or solution, of an equation of the form f (x) = 0, for a given function f. A root of this equation is also called a zero of the function f.

1- Bisection Method
The first technique, based on the Intermediate Value Theorem, is called the Bisection, or Binary-search, method.

Theorem 1: Intermediate Value Theorem

If f ? C [a, b] and K is any number between f (a) and f (b), then there exists a number c in (a, b) for which f (c) = K. (see figure 1.)
Example 1: Show that ?f(x)=x?^5-2x^3+3x^2-1=0 has a solution in the interval [0, 1].
Solution: f(x) is continuous in the interval [0, 1].
f(0)= -1, f(1)=1.
The Intermediate Value Theorem implies that a number x exists, with 0 < x < 1, for which x^5-2x^3+3x^2-1=0.
The Intermediate Value Theorem is used to determine when solutions to certain problems exist. It does not give an efficient means for finding these solutions.