Re: exact dot product
Dan (and P-1788),
If I may summarize (and Ulrich can correct me if I am wrong),
"complete arithmetic" Involves using a "long accumulator"
(something like 1000 bits for double precision), doing the
summation (e.g. for the dot product, but also for other operations)
in the long accumulator, then rounding. However, there are
other ways of obtaining a correctly rounded (or the slightly
less stringent faithfully rounded) dot product, and I am
hoping we can clarify exactly what the intent of the position
from motion 29 is in this regard.
Baker
On 05/20/2013 01:19 PM, Dan Zuras Intervals wrote:
Date: Sun, 19 May 2013 16:37:48 -0500
From: Ralph Baker Kearfott <rbk@xxxxxxxxxxxxx>
To: Dan Zuras Intervals <intervals08@xxxxxxxxxxxxxx>
CC: Ulrich Kulisch <ulrich.kulisch@xxxxxxx>, stds-1788@xxxxxxxxxxxxxxxxx
Subject: Re: exact dot product
Dan (and P-1788),
Was complete arithmetic (in addition to exact dot
product) also discussed in 754?
Baker
Baker,
I must confess I am not quite sure what you mean by "complete
arithmetic" in this context. But if it is an exact arithmetic
or an arithmetic with an exact part & a smaller unknown part,
then YES, it was discussed many times & in many different contexts.
Also, exact dot product in the context of exact products together
with correct sums which are exact enough to round correctly all
the time, these were discussed at length. Ulrich needed them for
his work & we were willing to provide them.
(Actually, in the discussion it often came up that someone wanted
NOT to provide them from time to time due to their difficulty.
But eventually a paper was published that put a bound on the extra
precision needed which pretty much killed the objection. Alas,
the paper hit the streets too late for us to change the text of
the 754 standard. It will happen to you too in some context or
another. Don't be in too much of a hurry.)
This notion of "complete arithmetic" was also often discussed in
the context of intervals. The notion was to compare two numbers
by having an exact part & a much smaller interval part such that
the comparison was clear once you subtracted out the exact parts.
That is, either it was away from zero or not by looking at what
amounted to only the interval parts. It was necessary to carry
out such a comparison by having either exact arithmetic or some
sort of arithmetic that accumulated its error in the interval
part. It was the only way to decide the answer.
Does this answer your "complete arithmetic" questions?
Dan
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Ralph Baker Kearfott, rbk@xxxxxxxxxxxxx (337) 482-5346 (fax)
(337) 482-5270 (work) (337) 993-1827 (home)
URL: http://interval.louisiana.edu/kearfott.html
Department of Mathematics, University of Louisiana at Lafayette
(Room 217 Maxim D. Doucet Hall, 1403 Johnston Street)
Box 4-1010, Lafayette, LA 70504-1010, USA
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