Interval arithmetic's power to bound ranges of functions has
enabled successful application to solution of
nonlinear systems and to global optimization. Consider
global search algorithms
for nonlinear systems of the form 1, that is,
If 
, that is, if an interval evaluation of
one of the components reveals that it is non-zero, then the box 
need not be considered further. Similarly, in global optimization
problems of the form 2,
a box can be removed from consideration if the lower bound of
an interval evaluation 
is greater than some previously
computed point value 
, or if an interval value
or 
reveals that a constraint
cannot be satisfied within .
These elementary techniques, based on bounds on the ranges of the functions in the problem, form the basis of the rejection steps in Algorithm 1 of §4. The verification steps are done with interval Newton methods, introduced in §6.