3.2.25 Examples with a Binary Function

 

Consider the multiplication function, g(x,y) = xy,

An example evaluation follows:

For interval arguments that lie within one of the sections of , two applications of are used. For interval arguments which span two sections of , four applications of g are used, as was done in the above example evaluation. Since g is continuous, we may reduce the number of applications of .

Consider evaluating g(x,y), for the situation depicted below.

The argument, (x,y), is covered by two sections of :

The sections share a common boundary, and points on the boundary are not strict local maxima or minima of g. This is true above, as both horizontal arrows point similarly, while the common boundary is vertical. With such a situation,

It follows that , , so:

The bounds of may therefore be computed with two applications of g, rather than four. With the common binary functions, this situation always occurs when an argument spans two sections, unless the function is discontinuous at the common boundary. If the function is discontinuous at the common boundary, then the function is -bumpy. Similar reasoning may be used when the argument spans four sections, to reduce the number of applications of g from eight to four. Concern for efficiency is focussed on evaluation of when is small.

Jeff Tupper

March 1996