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Am 27.08.2013 17:37, schrieb Richard
Fateman:
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.
Computer technology is so powerful today that I am absolutely
convinced that we shall get in hardware if we require it. My
experience is that hardware designers are very openminded to new
ideas.
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.
You should ask the BLAS people.not me.
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
The advantage of the EDP is that it eliminates cancellation from
accumulations. If multiple precision is implemented in software
nobody uses it. So lets assume that both are implemented by
hardware. Then implementing the EDP is much simpler and it runs much
faster than a quadruple precision arithmetic. You have to provide an
extemely high precision until multiple precision arithmetic can
compete with the EDP in double precision with respect to
cancellation.
If the exponent range does not suffice you can add an exponent part
to the EDP and use it as a movable window. Such systems are also
available. See my book or [5] on the poster.
Best regards
Ulrich
--
Karlsruher Institut für Technologie (KIT)
Institut für Angewandte und Numerische Mathematik
D-76128 Karlsruhe, Germany
Prof. Ulrich Kulisch
Telefon: +49 721 608-42680
Fax: +49 721 608-46679
E-Mail: ulrich.kulisch@xxxxxxx
www.kit.edu
www.math.kit.edu/ianm2/~kulisch/
KIT - Universität des Landes Baden-Württemberg
und nationales Großforschungszentrum in der
Helmholtz-Gesellschaft
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