The bisection method is based on the mean value theorem and assumes that f (a) and f (b) have opposite signs. Basically, the method involves repeatedly halving the subintervals of [a, b] and in each step, locating the half containing the solution, m.
Below is an example of approximation of the root of the function f (x) = 10x ^ 2:
interval a: -2 interval b: 5
n | a | b | c | f(a) | f(b) | f(c)
1 | -2.00000 | 5.00000 | 1.50000 | 6.00000 | -15.00000 | 7.75000
2 | 1.50000 | 5.00000 | 3.25000 | 7.75000 | -15.00000 | -0.56250
3 | 1.50000 | 3.25000 | 2.37500 | 7.75000 | -0.56250 | 4.35938
4 | 2.37500 | 3.25000 | 2.81250 | 4.35938 | -0.56250 | 2.08984
5 | 2.81250 | 3.25000 | 3.03125 | 2.08984 | -0.56250 | 0.81152
6 | 3.03125 | 3.25000 | 3.14062 | 0.81152 | -0.56250 | 0.13647
7 | 3.14062 | 3.25000 | 3.19531 | 0.13647 | -0.56250 | -0.21002
8 | 3.14062 | 3.19531 | 3.16797 | 0.13647 | -0.21002 | -0.03603
9 | 3.14062 | 3.16797 | 3.15430 | 0.13647 | -0.03603 | 0.05041
10 | 3.15430 | 3.16797 | 3.16113 | 0.05041 | -0.03603 | 0.00724
Result:
Approximate root: 3.16113
Iterations performed: 10
Error: 0.00724