Applying the bisection method to a function begins by selecting a lower bound and an upper bound . The values and should have opposite signs, guaranteeing that a root of lies in the interval by the Intermediate Value Theorem.
The method proceeds by considering , the midpoint of the interval. If by chance , then is the desired root. Otherwise, has the same sign as either or , so the process can be repeated with replacing or respectively.
Bisection halves the width of the interval containing the root at each step. In binary, this corresponds to gaining one digit of precision per iteration. While finding a root exactly at the midpoint of any step is rare, the endpoints of the intervals produced are guaranteed to converge to a root.
The bisection method will not find roots of even multiplicity because does not change sign at such roots. One way to correct the omission is to use the bisection method on the derivative , since a root of multiplicity of is a root of multiplicity of .