Bisection method

The bisection method is a way to estimate solutions for single equations. When we solve one equation, this method can help us to get a number that is very close to the real solution.

Think about a continuous function F(x). If the curve of F(x) passes through the x-axis, it means that there must be one specific x number which makes F(x) = 0, in other words, a solution. But how to judge whether the curve passes the x-axis or not? We pick one section with lower boundary a1 and upper boundary b1. Then we calculate F(a) and F(b). If F(b) has different signal with F(a) (or F(a)*F(b) < 0), the F(x) curve will pass through the x-axis in this section, which means F(x) must have a solution in this section. In this way, we can estimate a range of the solution.

To make our estimation more precise, we need to shorten the range which contains the solution. We divide this section into two equal parts: ( a , a/2+b/2) and (a/2+b/2, b). Then we apply the same method to judge whether there is a solution in each section. If there is no solutions in one section( F(a)*F(a/2+b/2) > 0 or F(a/2+b/2)*F(b)> 0), we ignore this section. If there is a solution in one section (F(a)*F(a/2+b/2) < 0 or F(a/2+b/2)*F(b) < 0), we continue to shorten the range until the range is smaller than the error of estimation we expected. Finally, we take the middle point of the final range as a solution of F(x).