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Re: Please listen to Ulrich here...

On 8/27/2013 3:32 AM, Ulrich Kulisch wrote:
Am 24.08.2013 21:49, schrieb G. William (Bill) Walster:
Ian,

As far as I can tell the only time when a case can be made that EDP is 
essential for interval computations is when all interval inputs are 
degenerate and therefore infinitely precise. Otherwise, with interval 
bounds on the accuracy of typical measured data, I don't see the 
requirement for EDP.  I continue to wait to see even one practical 
example thereof.

Cheers,

Bill
Bill,
 
the dot product is a fundamental arithmetic operation in the vector and matrix spaces as well as in mathematics.
No problem with this.  I would go further and say that COMPUTING the dot product is an important
and popular task.  See for example, BLAS,  in www.netlib.org/blas  which has several routines.

Computing an EDP is simple, error free and fast.
If it requires special hardware, it would be essentially unobtainable for almost everyone.
Of course it doesn't REQUIRE special hardware.

If it works only for IEEE754 single/double it seems to be very restrictive for people who are using
multiple-precision numbers, a tool that seems to be of growing importance and has a special
relevance for interval arithmetic and reliable computing generally.


If in a particular application the active part of the long accumulator is supported by hardware

Since, for 99.9999+% of computers, it is not supported by hardware, this is not relevant.
it comes with utmost speed. Not a conventional computation of the dot product in floating-point arithmetic nor computing a correctly or otherwise rounded dot product can reach this speed.
The speed in hardware is irrelevant.  Repeatedly bringing this up suggests that you have
inadequate real arguments.

If EDP is so important, why is it not in BLAS?  The search function in netlib seems to be broken,
so I was not able to see if some other package in netlib includes it.  It is true that there are separate
libraries available though.

Let me now comment on your mail. I wonder why you care so much finding a small corner of applications where the EDP might not be useful.
It seems to me that applications which REQUIRE something like the EDP occupy a small corner. 
Applications which COULD use EDP are perhaps a broader section.
 Some of those applications could be more effectively implemented without using EDP, where
for example only a small amount of additional precision is needed.

These are typical applications of interval arithmetic: Verified solution of systems of linear or nonlinear equations, guaranteed evaluation of polynomials or of arithmetic expressions, computing enclosures of the solution of an ordinary or a partial differential equation, and others. In the vast majority of these applications the data are floating-point numbers or how you call it degenerate intervals.
I think it is generally a mistake to view numerical scientific data which is entered into the computer and stored
as a double-precision floating-point number as known to one part in 2^53.  Much less to claim that
such numbers are single points (exact rational numbers) on the real line.  There are arguments to
be made that -- in the absence of other information -- all floating-point numbers are exact rational
numbers from a subset of the real line.  I suspect that it is rare in scientific computation when this
view is dominant.  It appears to be the assumption in the distinct pure-mathematics computing / "pure" numerical
analysis world in which EDP has been developed. 
Interval arithmetic allows computing close bounds for the solution and even to prove the existence of a solution within the computed bounds.

In these and other cases computing these results follows a general pattern: First an approximate solution is computed, then bounds for the expected exact solution are established (in case of a partial differential equation this may require computing highly accurate eigen- or singular values, for instance). Then a mathematical fixed-point theorem (Banach, Brower, or Schauder, depending on the problem) is applied which verifies the existence of a solution within these bounds. Finally, if necessary, an interative refinement is applied to improve the result. In all these steps an EDP is repeatedly applied and it often is essential for successs.
I do not see evidence that EDP is ever essential.  Is there some example that cannot be done by
some other multiple-precision technique? Since multiple-precision can, in some implementations,
provide a broader exponent range, it would seem to be superior to CA, and so there would be
examples in which EDP fails.

RJF