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Rounding Near Zero

In: Advanced Arithmetic for the Digital Computer

Author

Listed:
  • Ulrich W. Kulisch

    (Universität Karlsruhe, Institut für Angewandte Mathematik)

Abstract

Summary This paper deals with arithmetic on a discrete subset S of the real numbers ℝ and with floating-point arithmetic in particular. We assume that arithmetic on S is defined by semimorphism. Then for any element a ∈ S the element −a ∈ S is an additive inverse of a, i.e. a⊕(-a) = 0. The first part of the paper describes a necessary and sufficient condition under which -a is the unique additive inverse of a in S. In the second part this result is generalized. We consider algebraic structures M which carry a certain metric, and their semimorphic images on a discrete subset N of M. Again, a necessary and sufficient condition is given under which elements of N have a unique additive inverse. This result can be applied to complex floating-point numbers, real and complex floating-point intervals, real and complex floating-point matrices, and real and complex floatingpoint interval matrices.

Suggested Citation

  • Ulrich W. Kulisch, 2002. "Rounding Near Zero," Springer Books, in: Advanced Arithmetic for the Digital Computer, chapter 2, pages 71-79, Springer.
    • Handle: RePEc:spr:sprchp:978-3-7091-0525-2_2
    DOI: 10.1007/978-3-7091-0525-2_2
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