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Dear colleagues:
Let me comment a little on [7]. Compared with P1788 the book takes a more general approach to interval arithmetic. It does not just consider arithmetic for intervals over the real numbers. It also defines and studies arithmetic for intervals in the usual product spaces of computation like complex numbers, and for intervals of vectors and matrices over the real and complex numbers. All these interval operations are defined by a general mapping principle (semimorphism) which produces the best possible answer. This immediately leads to the requirement to compute dot products of two floating-point vectors with just one rounding (correctly rounded). However, a look into the computing history (before the electronic age) shows that the simplest and fastest way for computing a dot product is to compute it exactly in fixed point arithmetic§. This is an essential mean to speed up computing. Complete arithmetic does just this. It is fast and exact vector processing. In addition to this it turns out that the exact dot product is a most general tool for obtaining high accuracy in interval computations, a feature not yet considered in P1788. There is nothing that can be or needs to be standardized in complete arithmetic. Repeating the empty argument "complete arithmetic needs a standard of its own" again and again, has distracted the attention of many members of P1788. It hurt its development. In an early stage of the P1788 development complete arithmetic was listed very early under basic arithmetic operations. This would give it the necessary emphasis. I have no doubts that manufacturers would react and provide it well implmented on modern processors. The technology allows this easily. For more details see [7]. With best wishes Ulrich § No intermediate results after multiplications and additions need to be stored and read in again for the next operation. No intermediate roundings and normalizations have to be performed. No intermediate overflow or underflow can occur. No error analysis is necessary. The result is always exact. It is independent of the order in which the summands are added. Rounding is only done, if required, at the very end of the accumulation. Am 06.05.2014 01:20, schrieb Jürgen Wolff von Gudenberg: is now online! -- Karlsruher Institut für Technologie (KIT) Institut für Angewandte und Numerische Mathematik D-76128 Karlsruhe, Germany Prof. Ulrich Kulisch KIT Distinguished Senior Fellow 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 |