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3.3.9 Examples with a Piecewise Linear Function

Consider the floor function, tex2html_wrap_inline34653 , which is a piecewise constant function. We previously stated that

math18477

another possibility is to let

math18483

With either sectioning, the function is seen to be piecewise linear, and is monotonically increasing:

math18489

Consider the following diagram:

figure18494

This type of diagram will be used throughout this section. The light grey region will often represent g(m), as it does in the rightmost diagram:

math18633

In the leftmost diagram, the region represents m; the two diagrams together illustrate how tex2html_wrap_inline35733 was determined.

The evaluation of tex2html_wrap_inline35733 ,

math18641

proceeds as follows, using the first sectioning:

math18647

with

math18669

math18674

Perusal of the following figure may ease the comprehension of the preceding evaluation. The grey region represents g(m), while tex2html_wrap_inline35733 is displayed as the upper and lower solid lines.

figure18679

The evaluation of tex2html_wrap_inline35733 ,

math18641

proceeds as follows, using the second sectioning:

math18758

with

math18779

math18784

The following figure graphically illustrates portions of the preceding evaluation.

figure18788

Which of the two methods is used depends on the values of

math18846

The same method need not be used for both bounds. With a reasonable choice of tex2html_wrap_inline35695 , an optimal bound is easily computed.


next up previous notation contents
Next: 3.3.10 Concave Up Functions Up: 3.3 Linear Interval Arithmetic Previous: 3.3.8 Example with a Linear
Jeff TupperMarch 1996