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On Interval Methods with Zero Rewriting and Exact Geometric Computation

Conference paper
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Part of the Lecture Notes in Computer Science book series (LNCS, volume 10693)

Abstract

We oppose interval-symbol methods with zero rewriting developed by Shirayanagi and Sekigawa [14, 31, 32, 33] to the exact geometric computation paradigm [17, 37], especially to exact decisions computation via lazy adaptive evaluation with expression-dags, in doing so carving out analogies and disparities.

Keywords

Interval-symbol method Exact geometric computation Robustness and precision issues Verified numerical computing 

References

  1. 1.
    Benouamer, M.O., Jaillon, P., Michelucci, D., Moreau, J.M.: A lazy exact arithmetic. In: Proceedings of the 11th Symposium on Computer Arithmetic, pp. 242–249. IEEE (1993)Google Scholar
  2. 2.
    Boost C++ Libraries. http://www.boost.org/
  3. 3.
    Burnikel, C., Fleischer, R., Mehlhorn, K., Schirra, S.: Efficient exact geometric computation made easy. In: Proceedings of the 15th Symposium on Computational Geometry, pp. 341–350. ACM (1999)Google Scholar
  4. 4.
    Burnikel, C., Funke, S., Mehlhorn, K., Schirra, S., Schmitt, S.: A separation bound for real algebraic expressions. Algorithmica 55(1), 14–28 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    CGAL: Computational Geometry Algorithms Library. http://www.cgal.org/
  6. 6.
    Emiris, I.Z., Mourrain, B., Tsigaridas, E.P.: The DMM bound: multivariate (aggregate) separation bounds. In: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, ISSAC 2010, pp. 243–250. ACM, New York (2010). http://doi.acm.org/10.1145/1837934.1837981
  7. 7.
    Fortune, S., van Wyk, C.J.: Static analysis yields efficient exact integer arithmetic for computational geometry. ACM Trans. Graph. 15(3), 223–248 (1996)CrossRefGoogle Scholar
  8. 8.
    Funke, S., Mehlhorn, K.: LOOK: a lazy object-oriented kernel design for geometric computation. Comput. Geom. Theor. Appl. 22(1–3), 99–118 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Funke, S., Mehlhorn, K., Näher, S.: Structural filtering: a paradigm for efficient and exact geometric programs. Comput. Geom. Theor. Appl. 31(3), 179–194 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Halperin, D.: Controlled perturbation for certified geometric computing with fixed-precision arithmetic. In: Fukuda, K., Hoeven, J., Joswig, M., Takayama, N. (eds.) ICMS 2010. LNCS, vol. 6327, pp. 92–95. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-15582-6_19 CrossRefGoogle Scholar
  11. 11.
    Halperin, D., Leiserowitz, E.: Controlled perturbation for arrangements of circles. In: Proceedings of the 19th Symposium on Computational Geometry, pp. 264–273. ACM (2003)Google Scholar
  12. 12.
    Halperin, D., Shelton, C.R.: A perturbation scheme for spherical arrangements with application to molecular modeling. Comput. Geom. Theor. Appl. 10, 273–287 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Karamcheti, V., Li, C., Pechtchanski, I., Yap, C.K.: A core library for robust numeric and geometric computation. In: Proceedings of the 15th Symposium on Computational Geometry, pp. 351–359. ACM (1999)Google Scholar
  14. 14.
    Katayama, A., Shirayanagi, K.: A new idea on the interval-symbol method with correct zero rewriting for reducing exact computations. ACM Commun. Comput. Algebra 50(4), 176–178 (2016). http://doi.acm.org/10.1145/3055282.3055295 CrossRefzbMATHGoogle Scholar
  15. 15.
    Kettner, L., Welzl, E.: One sided error predicates in geometric computing. In: Proceedings of the 15th IFIP World Computer Congress, Fundamentals - Foundations of Computer Science, pp. 13–26 (1998)Google Scholar
  16. 16.
    LEDA: Library of Efficient Data Types and Algorithms. http://www.algorithmic-solutions.com/
  17. 17.
    Li, C., Pion, S., Yap, C.K.: Recent progress in exact geometric computation. J. Logic Algebr. Program. 64(1), 85–111 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mehlhorn, K., Osbild, R., Sagraloff, M.: A general approach to the analysis of controlled perturbation algorithms. Comput. Geom. 44(9), 507–528 (2011). http://www.sciencedirect.com/science/article/pii/S0925772111000460 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mignotte, M.: Identification of algebraic numbers. J. Algorithms 3, 197–204 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mörig, M.: Algorithm engineering for expression dag based number types. Ph.D. thesis, Otto-von-Guericke-Universität Magdeburg (2015)Google Scholar
  21. 21.
    MPFR: A multiple precision floating-point library. http://www.mpfr.org/
  22. 22.
    Mulmuley, K.: Computational Geometry - An Introduction Through Randomized Algorithms. Prentice Hall, Englewood Cliffs (1994)zbMATHGoogle Scholar
  23. 23.
    Pion, S., Fabri, A.: A generic lazy evaluation scheme for exact geometric computations. Sci. Comput. Program. 76(4), 307–323 (2011)CrossRefGoogle Scholar
  24. 24.
    Pion, S., Yap, C.K.: Constructive root bound for \(k\)-ary rational input numbers. Theor. Comput. Sci. 369(1–3), 361–376 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Schirra, S.: Much ado about zero. In: Albers, S., Alt, H., Näher, S. (eds.) Efficient Algorithms. LNCS, vol. 5760, pp. 408–421. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-03456-5_27 CrossRefGoogle Scholar
  26. 26.
    Sekigawa, H.: Zero determination of algebraic numbers using approximate computation and its application to algorithms in computer algebra. Ph.D. thesis, University of Tokyo (2004)Google Scholar
  27. 27.
    Shewchuk, J.R.: Adaptive precision floating-point arithmetic and fast robust geometric predicates. Discrete Comput. Geom. 18(3), 305–363 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Shirayanagi, K., Sweedler, M.: A theory of stabilizing algebraic algorithms. Technical report, Mathematical Sciences Institute, Cornell University (1995)Google Scholar
  29. 29.
    Shirayanagi, K., Sweedler, M.: Remarks on automatic algorithm stabilization. J. Symb. Comput. 26(6), 761–765 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Shirayanagi, K.: Floating point Gröbner bases. Math. Comput. Simul. 42(4–6), 509–528 (1996).  https://doi.org/10.1016/S0378-4754(96)00027-4 MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Shirayanagi, K., Sekigawa, H.: A new method of reducing exact computations to obtain exact results. ACM Commun. Comput. Algebra 43(3/4), 102–104 (2009). http://doi.acm.org/10.1145/1823931.1823950 zbMATHGoogle Scholar
  32. 32.
    Shirayanagi, K., Sekigawa, H.: Reducing exact computations to obtain exact results based on stabilization techniques. In: Proceedings of Numeric Computation 2009, pp. 191–198 (2009). http://doi.acm.org/10.1145/1577190.1577219
  33. 33.
    Shirayanagi, K., Sekigawa, H.: Interval-symbol method with correct zero rewriting: reducing exact computations to obtain exact results. In: Proceedings of the 18th Asian Technology Conference in Mathematics, pp. 226–235 (2013)Google Scholar
  34. 34.
    Sugihara, K.: Computational geometry in the human brain. In: Akiyama, J., Ito, H., Sakai, T. (eds.) JCDCGG 2013. LNCS, vol. 8845, pp. 145–160. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-13287-7_13 Google Scholar
  35. 35.
    Sugihara, K., Iri, M.: A robust topology-oriented incremental algorithm for Voronoi diagrams. Int. J. Comput. Geom. Appl. 4(2), 179–228 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Sugihara, K., Iri, M., Inagaki, H., Imai, T.: Topology-oriented implementation - an approach to robust geometric algorithms. Algorithmica 27(1), 5–20 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Yap, C.K.: Towards exact geometric computation. Comput. Geom. Theor. Appl. 7(1–2), 3–23 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Yap, C.K.: Robust geometric computation. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, Chap. 41, 2nd edn., pp. 927–952. CRC, Boca Raton (2004)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Simulation and Graphics, Faculty of Computer ScienceUniversity of MagdeburgMagdeburgGermany

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