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## 3.2.6 Piecewise Models

Any function may be cut into sections where each section fits into one monotonicity class:

A model of a function may be built up in pieces. To determine , for , a proper cover of j is found. The cover C is a set of sets. If C covers j, then every point in j is in a member of C:

A cover is proper if it cannot be trivially shrunken:

Given a cover C of j it is trivial to construct a proper cover C' of j, simply by discarding members of C which do not overlap j. After a proper cover of j is found,

Since , is monotonic; hence is simpler to evaluate than . The union of two intervals is an interval which includes the two given intervals:

Often, we form a set   from which proper covers of j may be easily formed, for any . We will not mandate a particular choice of ; there will be a natural choice for each g we consider. Using several , with , we may then evaluate for any using the above strategy.

Next: 3.2.7 Charts Up: 3.2 Constant Interval Arithmetic Previous: 3.2.5 Optimality
 Jeff Tupper March 1996