complex interval multiplication

[David G. Hough at validgh]

On the subject of real interval multiplication, I concluded that hardware
support would be helpful
in the form of min/max functions and functions which computed 
a product and returned it rounded both to left and to right.   Four such
multiplications and six such min/max would suffice to compute interval
products without any explicit conditionals, and the bounds would be as
good as possible.    That's 10 ops instead of 1.   Since addition is 2 ops
instead of 1, for linear algebra, the real interval slowdown over real scalar
appears to be around 12:2 or 6:1.

Then I started wondering how to support complex interval multiplication, and
stumbled upon something that I had heard about in another context but failed
to connect to this.    The "usual formula" for complex multiplication,
applied in the usual way to intervals, causes an inherent growth in
intervals due to geometry, related to the problems of using interval methods
for orbit problems.

Anyway scalar complex multiplication takes six ops, and simple complex
interval multiplication takes four real interval multiplications and two
real interval additions, for a total of 4*10+2*2 = 44 ops. 
None of the real products appears to be reusable, alas.
Since complex interval addition requires four ops, complex interval 
linear algebra looks to be 48:8 or 6:1 over complex scalar as well.

Now as to geometry, consider multiplying these complex intervals

	([1,1],[1,1]) * ([1-e,1+e],[1-e,1+e])

which leads in a straightforward way to an enclosing product

	([-2e,+2e],[2-2e,2+2e])

which is too big by a factor of two in area; the actual mapping of the
endpoints of the interval in question leads to the smaller box whose
endpoints are the midpoints of sides of the larger one:

	(0, 2+-2e) and (+-2e,2)

but the axes of this box are not parallel to the coordinate axes; the larger
box is the smallest such containing the smaller.   And so the interval has
doubled in area due to geometry.

This is somewhat discouraging since it suggests that complex 
interval methods may run out of steam rather early, either due to geometry
or due to much slower interpretive methods that avoid geometric problems.
How has this worked out in practice?

For those who are new to the list, here's what I wrote up a few years
ago on orbit problems: