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Re: exact dot product

Baker and Dan,
if you have my book "Computer Arithmetic and Validity" (2008, or second edition 2013), see Chapter 8: "Scalar products and complete arithmetic" and section 9.7.2 "Multiple precision interval arithmetic". 
It is really not at all complicated to realize these things.
I attach a copy of the preface and the contents of the second edition of the book, pages ix - xxii. 

Best wishes
Ulrich



Am 20.05.2013 20:19, schrieb Dan Zuras Intervals:

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



--
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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