Although interval arithmetic is increasingly used in combination with computer algebra and other methods, both approaches | symbolic-algebraic and interval-arithmetic | are used separately.Implementing symbolic interval arithmetic seems not suitable due to the exponential growth in the \size" of the end-points.In this paper we propose a methodology for \true" symbolic-algebraic manipulations on symbolic-numerical interval expressions involving interval variables instead of symbolic intervals.Due to the better algebraic properties, resembling to classical analysis, and the containment of classical interval arithmetic as a special case, we consider the algebraic extension of conventional interval arithmetic as an appropriate environment for solving interval algebraic problems.Based on the distributivity relations, a general framework for simpli cation of symbolicnumerical expressions involving intervals is given and some of the wider implications of the theory pertaining to interval algebraic problems are discussed.

Article Numerical stability of geometric algorithms Share on Authors: T. Ottmann Institut für Angewandte Informatik, und Formale Beschreibungsverfahren, Universität Karlsruhe, Postfach 6980, D-7500 Karlsrube, West Germany Institut für Angewandte Informatik, und Formale Beschreibungsverfahren, Universität Karlsruhe, Postfach 6980, D-7500 Karlsrube, West GermanyView Profile , G. Theimt Institut für Angewandte Informatik, und Formale Beschreibungsverfahren, Universität Karlsruhe, Postfach 6980, D-7500 Karlsrube, West Germany Institut für Angewandte Informatik, und Formale Beschreibungsverfahren, Universität Karlsruhe, Postfach 6980, D-7500 Karlsrube, West GermanyView Profile , C. Ullrich Institut für Angewandte Mathematik, Universität Karlsruhe, Postfach 6980, D-7500 Karlsruhe, West Germany Institut für Angewandte Mathematik, Universität Karlsruhe, Postfach 6980, D-7500 Karlsruhe, West GermanyView Profile Authors Info & Claims SCG '87: Proceedings of the third annual symposium on Computational geometryOctober 1987 Pages 119–125https://doi.org/10.1145/41958.41970Online:01 October 1987Publication History 38citation416DownloadsMetricsTotal Citations38Total Downloads416Last 12 Months3Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

Usually, higher programming languages provide a floating point arithmetic without specifying the accuracy of the operations. In contrast, ADA defines the operations by means of model numbers and rounding (see [2], [11]). Nevertheless, this definition is not strong enough to satisfy the modern requirement of maximum accuracy in all spaces of scientific computation (see [7]-[9]). By this we mean that no computer representable element lies between the actual and the computer generated result of an operation. The object of this paper is to specify the new arithmetic in ADA for all usual numerical data types. The new arithmetic is made available in a convenient operator form.

In ordered sets it is possible to show under certain assumptions two basic theorems concerning the cycle length of sequences of iterates generated by monotone operators. These results are applied to different iterative methods, where the conclusions are valid for the sequences of iterates produced by the numerical computations only, if the used computer arithmetic is properly implemented.