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IEEE754R
Ulrich Kulisch February 10, 2006
Dear colleagues
My thanks to everybody who sent me comments on my letter of January
28, 2006 to stds-754@xxxxxxxx and to Nelson Beebe. In the responses,
the wordings accurate sum and dot product, exact sum and dot product,
and faithfully rounded sum and dot product are used more or less
synonymously. For me the meaning of the last differs from that of the
others. An accurate or exact dot product means that the result is
computed to the fullest possible accuracy. Not a single bit is lost.
Here is why I see the issue in this way:
A. The most natural way to accumulate numbers is fixed-point
accumulation. It is simple, error free and fast. This is also true for
the accumulation of floating-point numbers and of their products. If
the result register is wide enough it can be done without exception.
The result is exact. Not a single bit is lost. The arithmetic to
achieve this is much the same as that of a conventional CPU.
Fixed-point accumulation of floating-point sums and dot products can
be realized in hardware at low cost. And it is very very fast. If
supported by a vectorizing compiler it would boost both the speed of a
computation and the accuracy of the result.
Fully accurate sums and dot products improve many applications. As a
byproduct, multiple precision real and interval arithmetic can be
done at very high speed. With operator overloading they are very easy
to use. With a long precision interval arithmetic, for instance,
highly accurate enclosures of orbits of dynamical systems have been
obtained for considerable long durations. Iterated defect correction
is another important class of applications. The method can be applied
to compute enclosures of arithmetic expressions or of polynomials with
very high accuracy. The result is an enclosure as a long precision
interval. Verified solution of badly conditioned systems of linear
equations by use of the Rump-operator is another large class of
important applications. Finally, of course, a faithfully rounded
result can be obtained. All these and other applications of an
accurate dot product come with very high speed. They can be considered
as top-down approaches of a fully accurate dot product.
B. In contrast to this the Rump-Ogita-Oishi method is a bottom-up
approach. I mentioned in my mail of January 26, 2006 that I very much
admire this methods. It achieves faithfully rounded sums and dot
products just by using conventional floating-point arithmetic. The
methods are fast in comparison with other software methods. This
certainly is a great achievement of our field. Applications of these
methods are inherently a subset of the applications of A.
I do not see any reason why we need the methods B. as justification
for accurate sums and dot products in the next arithmetic standard.
Fixed-point accumulation of sums and dot products is the additive
equivalent of fast multiplication techniques, for instance by an adder
tree. The advantages, the cost and gain in speed of both techniques
are similar. I do not see any reason why mathematicians should
hesitate to require this mode of operation from a new arithmetic
standard. If we do not require it we will never get it. We would not
have got floating-point operations with directed roundings if IEEE 754
hadnt required it.
With best regards
Ulrich Kulisch
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Ulrich Kulisch Privat:
Institut für Angewandte Mathematik Im Eichbäumle 37
Universität Karlsruhe 76139 Karlsruhe
76128 Karlsruhe, Germany Te.: (+49) 721 686263
Tel.: (+49) 721 608 2680 oder (+49) 721 608 4202
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