Subdivision methods
[221,333,286,392,402,133]
are generally efficient (in finding simple intersections) and stable.
Therefore, they are the most frequently used methods in practice. As
we will see, they can be combined with interval methods to numerically
guarantee that certain subdomains do not contain solutions. Interval
Newton methods
[273,131,191,27,159,158]
are a promising class of subdivision methods. However, the
subdivision methods are not as general as algebraic methods, since
they are only capable of isolating zero-dimensional solutions.
Furthermore, although the chances, that all roots have been found,
increase as the resolution tolerance is lowered, there is no certainty
that each root has been extracted/isolated. Subdivision methods
typically do not provide a guarantee as to how many roots there may be
in the remaining subdomains. However, if these subdomains are very
small, the existence of a (single) root within these subdomains is a
typical assumption. Lastly, subdivision techniques provide no
explicit information about root multiplicities without additional
computation. Despite these drawbacks, subdivision methods are very
useful in practice and are further described below.
