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International Congress on Mathematical Software

ICMS 2014: Mathematical Software – ICMS 2014 pp 718–724Cite as

From Calculus to Algorithms without Errors

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8592))

Abstract

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.

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References

  1. Brattka, V., Hertling, P.: Feasible real random access machines. Journal of Complexity 14(4), 490–526 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Brattka, V., Weihrauch, K.: Computability on Subsets of Euclidean Space I: Closed and Compact Subsets. Theoretical Computer Science 219, 65–93 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. 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. Ko, K.-I.: Computational Complexity of Real Functions. Birkhäuser (1991)

    Google Scholar 

  5. 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. 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. Linz, P.: A critique of numerical analysis. Bulletin of the American Mathematical Society 19(2), 407–416 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  8. Li, C., Pion, S., Yap, C.: Recent progress in exact geometric computation. J. Log. Algebr. Program. 64(1), 85–111 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Müller, N.T.: Uniform Computational Complexity of Taylor Series. In: Ottmann, T. (ed.) ICALP 1987. LNCS, vol. 267, pp. 435–444. Springer, Heidelberg (1987)

    Chapter  Google Scholar 

  10. 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. 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)

    Chapter  Google Scholar 

  12. 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. Pauly, A., Ziegler, M.: Relative Computability and Uniform Continuity of Relations. Journal of Logic and Analysis 5 (2013)

    Google Scholar 

  14. Specker, E.: Nicht konstruktiv beweisbare Sätze der Analysis. Journal of Symbolic Logic 14(3), 145–158 (1949)

    Article  MATH  MathSciNet  Google Scholar 

  15. 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. 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. Weihrauch, K.: Computable Analysis. Springer (2000)

    Google Scholar 

  18. Weihrauch, K.: Computational Complexity on Computable Metric Spaces. Mathematical Logic Quarterly 49(1), 3–21 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Weihrauch, K.: The Computable Multi-Functions on Multi-represented Sets are Closed under Programming. Journal of Universal Computer Science 14(6), 801–844 (2008)

    MATH  MathSciNet  Google Scholar 

  20. Yap, C.-K.: On Guaranteed Accuracy Computation. In: Geometric Computation, pp. 322–373. World Scientific Publishing (2004)

    Google Scholar 

  21. 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)

    Chapter  Google Scholar 

  22. Ziegler, M., Brattka, V.: Computability in Linear Algebra. Theoretical Computer Science 326, 187–211 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Weihrauch, K., Zhong, N.: Computability theory of generalized functions. Journal of the ACM 50(4), 469–505 (2003)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Universität Trier, Germany

    Norbert Müller

  2. TU Darmstadt, Germany

    Martin Ziegler

Authors
  1. Norbert Müller
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  2. Martin Ziegler
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Editor information

Editors and Affiliations

  1. Department of Mathematics, North Carolina State University, Box 8205, 27695, Raleigh, NC, USA

    Hoon Hong

  2. Courant Institute, Dept. of Computer Science, New York Institute, 251 Mercer Street, 10012, New York, NY, USA

    Chee Yap

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Müller, N., Ziegler, M. (2014). From Calculus to Algorithms without Errors. In: Hong, H., Yap, C. (eds) Mathematical Software – ICMS 2014. ICMS 2014. Lecture Notes in Computer Science, vol 8592. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44199-2_107

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  • DOI: https://doi.org/10.1007/978-3-662-44199-2_107

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-44198-5

  • Online ISBN: 978-3-662-44199-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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