Re: Arguments for supporting Motion P1788/0023.01:NoMidRad
Svetoslav Markov wrote:
Yes, absolutely true. Just one typical
example, showing that in linear algebra
the mid-rad form is the natural presentation.
Write the two intervals A, B in mid-rad
form, then write down the condition that
the intersection of A and B is not empty and
you obtain the the basic relation used in Prager-
Oettli theorem.
This is about theory using _exact_ interval arithmetic,
where (for bounded intervals) midrad and infsup are
completely equivalent.
I never questioned the usefulness of midrad as a theoretical
tool for the analysis of problems and algorithms. I even use it
a lot in my book ''Interval methods for systems of equations''.
But the P1788 standard is about rigorous implementation of
interval _arithmetic_ in a finite precision environment.
We are _not_ standardizing theory.
Even the Oettli-Prager results cannot be efficiently implemented
rigorously not by using operations on single midrad intervals
(which P1788 could specify). An efficient implementation works
throughout with real vectors and matrices and directed rounding,
never forming the midrad interval datatype.
Thus a midrad specification in a future P1788 standard would not
help implementors at all.
The case would be slightly different for a standardization of
interval BLAS2 or BLAS3 functionality.
Here a number of algorithms using midrad input and output are useful
and hence should be standardized.
But to be efficient, the input/output would have to be not a matrix
of midrad intervals but a pair of two real matrices containing the
midpoint matrix and the radius matrix. Thus efficient interval BLAS
can be build upon an P1788 having infsup intervals without _any_ loss
of efficiency for the matrix midrad algorithms.
Therefore _this_ forum is the wrong one to push for the merits of
midrad as a tool in interval algorithms.
Many of the papers by Jiri Rohn
are actually formulated in mid-rad form.
Rohn's sign-accord algorithm for computing the hull of the solution
set of an interval linear system works only in exact arithmetic.
Although the algorithm is now almost 30 years old and constitutes
one of the fundamental achievements of Rohn, nobody has succeeded
in creating a rigorous finite-precision version of it.
This illustrates the difficulty of creating rigorous midrad algorithms
in finite precision arithmetic.
Arnold Neumaier