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By Ulrich W. Kulisch, Willard L. Miranker

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This is achieved especially by the general function concept and the possibility of user defined operators with arbitrary result type. In the following the bounds of intervals are displayed with as many figures as is the negative of the exponent of the relative error between them. If, for instance, the first five figures of the left and right bound coincide, then for either bound six figures are displayed. Therefore the accuracy of the computed result can be recognized optically. ^Present address: IBM Deutschland, Entwicklung und Forschung, 7030 Boeblingen, West Germany A NEW APPROACH TO SCIENTIFIC COMPUTATION 27 Copyright © 1983 by Academic Press, Inc.

24 Ulrich Kulisch The following sentence is a golden rule in mathematics: "If a computation is correct one gets a correct result". In numerical analysis in general we cannot execute correct computations. Nevertheless, we are interested in the correct result. In the past mathematicians tried to close the gap between the result of a computation and the correct result of the problem by concepts like: stability of the process, the condition number of an algorithm or an error analysis made by hand. Computers, in the meantime, have reached speeds of several hundred million floating-point operations per second.

W. (1968). Implementierung eines ALGOL-60 Systems mit Schrankenzahlen, Electron Datenverarb. 10, 189-194. Wolff von Gudenberg, J. Evaluation of the standard functions in generalized comput­ er arithmetic. Wolff von Gudenberg, J. (1980). Einbettung allgemeiner Rechnerarithmetik in Ulrich Kulisch 26 [29] [30] [31] PASCAL mittels eines Operatorkonzeptes und Implementierung der Standardfunktionen mit optimaler Genauigkeit. Dissertation, Universitat Karlsruhe. , et al. (1979). A proposed standard for floating-point arithmetic, SIGNUM Newsletter.

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