This method is based on intermediate value theorem which states that if a function f(x) is continuous between a and b and f(a) and f(b) are of opposite signs then there exists at least one root between a and b for definiteness.
Let f(a) be negative, and f(b) be positive (see figure below). Then the the root lies between a and b and let its approximate value be given by \[x_0\] = (a+b)/2.
If f(\[x_0\])=0, we conclude that x0 is root of the equation f(\[x_0\])=0 otherwise the root lies either \[x_0\] and b or between \[x_0\] and a depending on whether f(\[x_0\]) is negative or positive. We designate this new interval as [\[a_1 ,b_1\]] whose length is |b-a|/2
As before this is bisected at \[x_1\]and the new interval will be exactly half the length of the previous one. We repeat this process until the latest interval is as small as desired say k. it is clear that the interval will be [\[a_n,b_n\]] of length\[ \frac{|b-a|}{2^{n}}\]
We then have \[ \frac{|b-a|}{2^{n}}\] < which gives on simplification n \[\geqslant \frac{ln\frac{b-a}{k}}{ln2}\]
Boundary condition
1. b > a
2. k should be greater than 0
Results
for b=8 a=3 k=3 we get n=0.737
