Charts
charts may be constructed
for general k. The utility of such charts
is somewhat limited, however. For constant
bounds,
is quite useful
in bounding the derivative of g(x);
for linear bounds,
is again
quite useful in bounding the second derivative of g(ax+b);
for quadratic bounds,
does not allow us to bound the third derivative
of 
.
Additional knowledge of 
allows us to bound the third derivative of
in some cases.
As the derivative analysis is not as simple as before,
such a simple design aid is no longer sufficient.
This does not preclude the construction of quadratic
interval routines; it just shows that such construction
cannot be guided
by charts alone. A similar
situation was encountered while constructing
linear interval operators for binary functions.
| Jeff Tupper | March 1996 |