The bisection method
The bisection method is considered the simplest one-dimensional root-finding algorithm. The general interest is to find the value 
of a continuous function 
such that 
.
Suppose we know the two points of an interval 
and 
, where 
, and that 
and 
lie along the continuous function, taking the midpoint of this interval as 
, where 
, the bisection method then evaluates this value as f(c).
Let's illustrate the setup of points along a nonlinear function with the following graph:
Since the value of f(a) is negative and f(b) is positive, the bisection method assumes that the root lies somewhere between a and b and gives .
If 
or is very close to zero by some predetermined error tolerance value, then a root is declared as found. If 
, then we may conclude that a root exists along the interval and , or interval and otherwise.
On the next evaluation, is replaced as either or accordingly. With the new interval shortened, the bisection method repeats with the same evaluation to determine...
