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Next: 3.2.16 Partial Functions Up: 3.2 Constant Interval Arithmetic Previous: 3.2.14 Examples with Piecewise Monotonic

3.2.15 Periodic Functions

Consider the piecewise monotonic function tex2html_wrap_inline34867 . With thin arguments, the evaluation of tex2html_wrap_inline34159 proceeds as follows:

math13718

with

math13748

With thick arguments, the evaluation of tex2html_wrap_inline34159 is displeasing:

math13763

with

math13815

Although the result returned is optimal, the amount of work performed to determine the result may be reduced.

We will cut the function tex2html_wrap_inline34165 into sections where each section attains the extreme values of g:  

math13826

math13835

where

math13841

When evaluating tex2html_wrap_inline34159 , we may simply return tex2html_wrap_inline34911 if any of the aforementioned sections lie within j:

math13851

As with the previous sectioning scheme, there will often be a preferred sectioning, denoted by tex2html_wrap_inline34915 , which we will use to check containment.

With our example function g, tex2html_wrap_inline34867 ,

math13860

so the previous evaluation may be shortened. It may now proceed as follows:

math13866

with

math13882

This rejection test may be performed with a single subtraction, to find the width of the argument. Another quick rejection test is possible, by allowing another class of intervals into tex2html_wrap_inline34915 :

math13886


next up previous notation contents
Next: 3.2.16 Partial Functions Up: 3.2 Constant Interval Arithmetic Previous: 3.2.14 Examples with Piecewise Monotonic
Jeff TupperMarch 1996