From Calculus to Algorithms without Errors

  • Norbert Müller
  • Martin Ziegler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8592)


Using mathematics within computer software almost always includes the necessity to compute with real (or complex) numbers. However, implementations often just use the 64-bit double precision data type. This may lead to serious stability problems even for mathematically correct algorithms. There are many ways to reduce these software-induced stability problems, for example quadruple or multiple-precision data types, interval arithmetic, or even symbolic computation. We propagate Exact Real Arithmetic (ERA) as a both convenient and practically efficient framework for rigorous numerical algorithms.


Recursive Analysis Rigorous Numerics 


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  1. [BrHe98]
    Brattka, V., Hertling, P.: Feasible real random access machines. Journal of Complexity 14(4), 490–526 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  2. [BrWe99]
    Brattka, V., Weihrauch, K.: Computability on Subsets of Euclidean Space I: Closed and Compact Subsets. Theoretical Computer Science 219, 65–93 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  3. [KMRZ14]
    Kawamura, A., Müller, N., Rösnick, C., Ziegler, M.: Computational Benefit of Smoothness: Rigorous Parameterized Complexity Analysis in High-Precision Numerics of Operators on Gevrey’s Hierarchy (submitted)Google Scholar
  4. [Ko91]
    Ko, K.-I.: Computational Complexity of Real Functions. Birkhäuser (1991)Google Scholar
  5. [Ko98]
    Ko, K.-I.: Polynomial-Time Computability in Analysis. In: Ershov, Y.L., et al. (eds.) Handbook of Recursive Mathematics, vol. 2, pp. 1271–1317 (1998)Google Scholar
  6. [KrMa82]
    Kreisel, G., Macintyre, A.: Constructive Logic versus Algebraization I. In: Troelstra, van Dalen (eds.) Proc. L.E.J. Brouwer Centenary Symposium, pp. 217–260. North-Holland (1982)Google Scholar
  7. [Linz88]
    Linz, P.: A critique of numerical analysis. Bulletin of the American Mathematical Society 19(2), 407–416 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  8. [LPY05]
    Li, C., Pion, S., Yap, C.: Recent progress in exact geometric computation. J. Log. Algebr. Program. 64(1), 85–111 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  9. [Müll87]
    Müller, N.T.: Uniform Computational Complexity of Taylor Series. In: Ottmann, T. (ed.) ICALP 1987. LNCS, vol. 267, pp. 435–444. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  10. [Müll95]
    Müller, N.T.: Constructive Aspects of Analytic Functions. In: Proc. Workshop on Computability and Complexity in Analysis (CCA), vol. 190, pp. 105–114. InformatikBerichte FernUniversität Hagen (1995)Google Scholar
  11. [Müll01]
    Müller, N.T.: The iRRAM: Exact Arithmetic in C++. In: Blank, J., Brattka, V., Hertling, P. (eds.) CCA 2000. LNCS, vol. 2064, pp. 222–252. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  12. [MüKo10]
    Müller, N.T., Korovina, M.: Making big steps in trajectories. In: Proc. 7th Int. Conf. on Computability and Complexity in Analysis (CCA 2010). Electronic Proceedings in Theoretical Computer Science, vol. 24, pp. 106–119 (2010)Google Scholar
  13. [PaZi13]
    Pauly, A., Ziegler, M.: Relative Computability and Uniform Continuity of Relations. Journal of Logic and Analysis 5 (2013)Google Scholar
  14. [Spec49]
    Specker, E.: Nicht konstruktiv beweisbare Sätze der Analysis. Journal of Symbolic Logic 14(3), 145–158 (1949)CrossRefzbMATHMathSciNetGoogle Scholar
  15. [Spec59]
    Specker, E.: Der Satz vom Maximum in der rekursiven Analysis. In: Heyting, A. (ed.) Constructivity in Mathematics. Studies in Logic and The Foundations of Mathematics, pp. 254–265. North-Holland (1959)Google Scholar
  16. [Spec69]
    Specker, E.: The fundamental theorem of algebra in recursive analysis. In: Constructive Aspects of the Fundamental Theorem of Algebra, pp. 321–329. Wiley-Interscience (1969)Google Scholar
  17. [Weih00]
    Weihrauch, K.: Computable Analysis. Springer (2000)Google Scholar
  18. [Weih03]
    Weihrauch, K.: Computational Complexity on Computable Metric Spaces. Mathematical Logic Quarterly 49(1), 3–21 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  19. [Weih08]
    Weihrauch, K.: The Computable Multi-Functions on Multi-represented Sets are Closed under Programming. Journal of Universal Computer Science 14(6), 801–844 (2008)zbMATHMathSciNetGoogle Scholar
  20. [Yap04]
    Yap, C.-K.: On Guaranteed Accuracy Computation. In: Geometric Computation, pp. 322–373. World Scientific Publishing (2004)Google Scholar
  21. [YSS13]
    Yap, C., Sagraloff, M., Sharma, V.: Analytic Root Clustering: A Complete Algorithm Using Soft Zero Tests. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds.) CiE 2013. LNCS, vol. 7921, pp. 434–444. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  22. [ZiBr04]
    Ziegler, M., Brattka, V.: Computability in Linear Algebra. Theoretical Computer Science 326, 187–211 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  23. [ZhWe03]
    Weihrauch, K., Zhong, N.: Computability theory of generalized functions. Journal of the ACM 50(4), 469–505 (2003)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Norbert Müller
    • 1
  • Martin Ziegler
    • 2
  1. 1.Universität TrierGermany
  2. 2.TU DarmstadtGermany

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