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Am 24.08.2013 21:49, schrieb G. William
(Bill) Walster:
Bill,Ian, As far as I can tell the only time when a case can be made that EDP is essential for interval computations is when all interval inputs are degenerate and therefore infinitely precise. Otherwise, with interval bounds on the accuracy of typical measured data, I don't see the requirement for EDP. I continue to wait to see even one practical example thereof. Cheers, Bill the dot product is a fundamental arithmetic operation in the vector and matrix spaces as well as in mathematics. Computing an EDP is simple, error free and fast. If in a particular application the active part of the long accumulator is supported by hardware it comes with utmost speed. Not a conventional computation of the dot product in floating-point arithmetic nor computing a correctly or otherwise rounded dot product can reach this speed. Let me now comment on your mail. I wonder why you care so much finding a small corner of applications where the EDP might not be useful. These are typical applications of interval arithmetic: Verified solution of systems of linear or nonlinear equations, guaranteed evaluation of polynomials or of arithmetic expressions, computing enclosures of the solution of an ordinary or a partial differential equation, and others. In the vast majority of these applications the data are floating-point numbers or how you call it degenerate intervals. Interval arithmetic allows computing close bounds for the solution and even to prove the existence of a solution within the computed bounds. In these and other cases computing these results follows a general pattern: First an approximate solution is computed, then bounds for the expected exact solution are established (in case of a partial differential equation this may require computing highly accurate eigen- or singular values, for instance). Then a mathematical fixed-point theorem (Banach, Brower, or Schauder, depending on the problem) is applied which verifies the existence of a solution within these bounds. Finally, if necessary, an interative refinement is applied to improve the result. In all these steps an EDP is repeatedly applied and it often is essential for successs. This can already be seen in case of computing a verified solution of a system of linear equations. Let me now consider problems with non degenerate intervals in the data. If the bounds are small the computation follows exactly the pattern described above. There is no reason why less care in computing the approximate solution, in establishing bounds for the exact solution or in the verification step is necessary. If the bounds for the data grow, the problem may gradually degenerate into an NP-hard problem. You probably know that I held patents on the EDP in Europe, the US, and in Japan between 1981 and 2007. I paid quite some fees for keeping them alife over all these years. The XSC-languages Pascal-XSC, ACRITH-XSC of IBM, and C-XSC all provide CA and the EDP. The verification methods developed in these languages make heavily use of these tools. See the language descriptions and the corresponding toolbox volumes. References are listed in my mail of August 7, 2013. The patents you obtained in the US a few years ago do not make use of CA and the EDP. I am absolutely convinced that the methods which use CA and the EDP are superior to those which do not use these. With best regards Ulrich -- 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 |