3.3.6 Monotonic Sections

In the previous section, we designed for a given function g, . We used to limit our attention to monotonic sections of g. Monotonic sections will also help us design .

Consider a monotonically increasing function g, . Let both

and

hold, for . Since g is monotonically increasing,

Combining this with the previous bounds results in the following:

for . This may be simplified to:

for . It is now established that bounds .

Consider a monotonically decreasing function g, . Let both

and

hold, for . Since g is monotonically decreasing,

Reasoning as before forces us to conclude that bounds .

So, for a monotonic g, we may evaluate by proceeding as follows:

Let and denote two functions, from to , defined as follows:

We now focus on determining and , for and . As will be seen, this will give us a method for computing rather than ; appropriate demotions may be used to ensure the result is in , if necessary.

Jeff Tupper

March 1996