Re: IEEEP1788

[Ulrich Kulisch <ulrich.kulisch@xxxxxxx>]

Am 09.06.2015 um 14:23 schrieb Vincent Lefevre:

> Well, correctly-rounded dot product should have been included in IEEE 754 seven years ago. But this is not the exact dot product.

> So, if I summarize, you wanted to standardize a particular implementation (the one with a long accumulator) of a particular complex building block of a particular arithmetic.

Dear all:

While I was typing this note I got a mail from Baker Kearfott:
IEEE 1788 is now approved.
Of course, I congratulate all those who worked and fought for it, although it does not really make me happy.

So I was wondering for a while whether I should send you the following note. But soon I came to the conclusion that the fight for the mathematical truth will go on. So I am asking everybody: please read the following text!
The fact that Baker now already pleads for a reveision of IEEE 754 shows that the mathemetical insight might be increasing.

In numerical analysis the dot product is ubiquitous. It appears in vector and matrix arithmetic and in residual refinement or defect correction techniques for interval and floating-point problems and in many other places.

The fastest way computing a dot product is to compute it exactly. By pipelining it can be computed in the time the processor needs to read the data, i.e., no other method computing a dot product can be faster, not a  conventional computation in floating-point arithmetic and not a correctly rounded dot product. So by the way the exact dot product is a mean to speed up computing. In fact, computing the exact dot product is faster by at least one magnitude than any of the known methods for computing a correctly rounded dot product. For more details see my book: Computer Arithmetic and Validity.

Unwillingness to study and accept these facts is a tragic disservice of many P1788 members to interval arithmetic.

Interval arithmetic is the tool for computing bounds for the solution of numerical problems. Naive interval arithmetic delivers huge bounds in general. The exact dot product is the most appropriate tool for computing close bounds. For more details on the matter and some sample applications see my book Computer Arithmetic and Validity, in particular chapter 8: Scalar products and complete arithmetic and chapter 9: Principles of verified computing.

Computing the dot product exactly is not a new invention, It has been available in Pascal-XSC since 1980 and in the IBM products ACRITH and ACRITH-XSC since 1987 and in C-XSC since 1993. Many detailed applications are discussed in the toolbox volumes for these languages. All these books are available via:
http://www.xsc.de/.

Best wishes
Ulrich

-- 
Karlsruher Institut für Technologie (KIT)
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Prof. Ulrich Kulisch
KIT Distinguished Senior Fellow 

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