A new arithmetic for scientific computation
The paper summarizes an extensive research activity in computer arithmetic and scientific computation that went on during the last fifteen years. We also discuss the experience gained through various implementations of a new approach to arithmetic on diverse processors including microprocessors.
We begin with a complete listing of the spaces that occur in numerical computations. This leads to a new and general definition of computer arithmetic.
Then we discuss aspects of traditional computer arithmetic such as the definition of the basic arithmetic operations, the definition of the operations in product spaces and some consequences of these definitions for error analysis of numerical algorithms.
In contrast to this we then give the new defintion of computer arithmetic. The arithmetic operations are defined by a general mapping principle which is called a semimorphism. We discuss the properties of semimorphisms, show briefly how they can be obtained and mention the most important feartures of their implementation on computers.
Then we show that the new operations can not be properly addressed by existing programming languages. Correcting this limitation led to extensions of PASCAL and FORTRAN.
A demonstration of a computer that has been systematically equipped with the new arithmetic will be given. The new arithmetic turns out to be a key property for an automatic error control in numerical analysis. By means of a large number of examples we show that guaranteed bounds for the solution with maximum accuracy can be obtained. The computer even proves the existence and uniqueness of the solution within the calculated bounds. If there is no unique solution (e.g. in case of a singular matrix) the computer recognizes it. Toward the end of the paper we sketch how expressions or program parts can be evaluated with high accuracy.
KeywordsArithmetic Operation Interval Arithmetic Maximum Accuracy Computer Arithmetic Program Part
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- /1/.Apostolatos, N., Kulisch, U., Krawczyk, R., Lortz, B., Nickel, K., Wippermann, H.-W.: The Algorithmic Language TRIPLEX ALGOL 60, Num. Math. 11, 175–180 (1968)Google Scholar
- /2/.Bohlender, G.: Floating-point Computation of functions with maximum accuracy. IEEE Trans. Comp. C-26, Nr. 7, 621–632 (1977)Google Scholar
- /3/.Bohlender, G., Kaucher, E., Klatte, R., Kulisch, U., Miranker, W.L., Ullrich, Ch. und Wolff von Gudenberg, J.: FORTRAN for contemporary numerical computation, Report RC 8348, IBM Thomas J. Watson Research Center 1980 and Computing 26, 277–314 (1981)Google Scholar
- /4/.Kulisch, U.: An axiomatic approach to rounded computations, TS Report No. 1020. Mathematics Research Center, University of Wisconsin, Madison, Wisconsin, 1969 und Number. Math. 19, 1–17 (1971)Google Scholar
- /5/.Kulisch, U.: Interval arithmetic over completely ordered ringoids, TS Report No. 1105, Mathematics Research Center, University of Wisconsin, Madison, Wisconsin, 1970Google Scholar
- /6/.Kulisch, U.: Grundlagen des Numerischen Rechnens — Mathematische Begründung der Rechnerarithmetik. Reihe Informatik, Band 19, Wissenschaftsverlag des Bibliographischen Instituts Mannheim, 1976Google Scholar
- /7/.Kulisch, U., Miranker, W.L.: Computer Arithmetic in Theory and Practice, Academic Press, 1980Google Scholar
- /8/.Coonan, J. et al.: A proposed standard for floating-point arithmetic, SIGNUM newsletter, Oct. 1979Google Scholar
- /9/.INTEL 12 1586-001: The 8086 family user's manual, Numeric Supplement, July 1980Google Scholar
- /10/.Kulisch, U., Miranker, W.L. (Editors): A New Approach to Scientific Computation, Academic Press, 1983Google Scholar
- /11/.Kulisch, U., Miranker, W.L.: The Arithmetic of the Digital Computer, IBM Research Report RC 10580, 1984, to appear in SIAM ReviewsGoogle Scholar
- /12/.High Accuracy Arithmetic, Subroutine Library, IBM Program Description and User's Guide, Program Number 5664-185, 1984Google Scholar
- /13/.Böhm, H.: Berechnung von Polynomnullstellen und Auswertung arithmetischer Ausdrücke mit garantierter maximaler Genauigkeit. Dissertation, Universität Karlsruhe 1984Google Scholar