applicability of interval arithmetic

[David G. Hough on validgh]

Jim Demmel reminds us that given uncertain input data,

> subsequent output intervals must expand by the condition number of
> the problem solved by B. Subsequent subroutine calls
> magnify the interval widths by further condition number
> factors. This is mathematically inescapable...

And then goes on to conclude:

> Any interval arithmetic
> system with just single and double (and even quad) precision
> will be limited to a fairly small number of applications.

I question that conclusion.  Inasmuch most physical input data is known
to single precision, at best --- most of the PERFECT club benchmarks 
produce acceptable results in IEEE single precision --- I think it's
likely that quadruple precision interval arithmetic can be used to produce
satisfactory results for many realistic applications, since quadruple precision
has enough bits to lose 20 to ill condition factors of 2**20 several times
in the course of a complete computation.

And even variable-precision interval arithmetic is no panacea in the general
worst case.  I recall a lecture by Kahan about twenty years ago about
finding the interval bounding the solution of a differential equation.
Any intervals bounding the solution that were constrained to be parallel
to the coordinate axes - in other words, any conventional intervals of
whatever precision - were doomed to grow in width exponentially over
time because the natural axes of the solution were rotating.  The interval
method that worked for that problem required describing the interval
solution in terms of a coordinate system that rotated over time, and
describing the interval solution as the interior of an ellipsoid rather
than the interior of a parallelopiped.

That's the kind of insight that won't occur in a naive application of
interval arithmetic, which is why its widespread applicability will 
depend on availability of high-quality mathematical software.  Not to
mention efficient hardware implementations, but that's another story.
I should investigate: how well can the complex multiply-add hardware
I outlined a few weeks ago be adapted to also compute (real)
interval addition, subtraction, and multiplication?  Can "microcoded"
RISC hardware do that faster than software?