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Next: 3.2.7 Charts Up: 3.2 Constant Interval Arithmetic Previous: 3.2.5 Optimality

3.2.6 Piecewise Models

Any function tex2html_wrap_inline34165 may be cut into sections where each section fits into one monotonicity class:    

math12495

A model of a function tex2html_wrap_inline34165 may be built up in pieces. To determine tex2html_wrap_inline34159 , for tex2html_wrap_inline34383 , a proper cover tex2html_wrap_inline34445 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:

math12502

A cover is proper if it cannot be trivially shrunken:

math12509

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 tex2html_wrap_inline34445 of j is found,

math12517

Since tex2html_wrap_inline34475 , tex2html_wrap_inline34477 is monotonic; hence tex2html_wrap_inline34479 is simpler to evaluate than tex2html_wrap_inline31397 . The union of two intervals is an interval which includes the two given intervals:

math12527

math12536

Often, we form a set tex2html_wrap_inline34483   from which proper covers of j may be easily formed, for any tex2html_wrap_inline34383 . We will not mandate a particular choice of tex2html_wrap_inline34489 ; there will be a natural choice for each g we consider. Using several tex2html_wrap_inline34479 , with tex2html_wrap_inline34495 , we may then evaluate tex2html_wrap_inline34159 for any tex2html_wrap_inline34383 using the above strategy.


next up previous notation contents
Next: 3.2.7 Charts Up: 3.2 Constant Interval Arithmetic Previous: 3.2.5 Optimality
Jeff TupperMarch 1996