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Term Cancellations in Computing Floating-Point Gröbner Bases

  • Tateaki Sasaki
  • Fujio Kako
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6244)

Abstract

We discuss the term cancellation which makes the floating-point Gröbner basis computation unstable, and show that error accumulation is never negligible in our previous method. Then, we present a new method, which removes accumulated errors as far as possible by reducing matrices constructed from coefficient vectors by the Gaussian elimination. The method manifests amounts of term cancellations caused by the existence of approximate linearly dependent relations among input polynomials.

Keywords

Matrix Method Gaussian Elimination Basis Computation Intrinsic Error Border Basis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bodrato, M., Zanoni, A.: Intervals, syzygies, numerical Gröbner bases: a mixed study. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2006. LNCS, vol. 4194, pp. 64–76. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Chen, Y., Meng, X.: Border bases of positive dimensional polynomial ideals. In: Proceedings of SNC 2007, Symbolic Numeric Computation, London, Canada, pp. 65–71 (2007)Google Scholar
  3. 3.
    Fortuna, E., Gianni, P., Trager, B.: Degree reduction under specialization. J. Pure Appl. Algebra 164, 153–164 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gonzalez-Vega, L., Traverso, C., Zanoni, A.: Hilbert stratification and parametric Gröbner bases. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2005. LNCS, vol. 3718, pp. 220–235. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  5. 5.
    Kreuzer, M., Kehrein, A.: Computing border bases. J. Pure Appl. Alg. 200, 279–295 (2006)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Kalkbrener, M.: On the stability of Gröbner bases under specialization. J. Symb. Comput. 24, 51–58 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kako, F., Sasaki, T.: Proposal of “effective” floating-point number. Preprint of Univ. Tsukuba (May 1997) (unpublished)Google Scholar
  8. 8.
    Kondratyev, A., Stetter, H.J., Winkler, S.: Numerical computation of Gröbner bases. In: Proceedings of CASC 2004, Computer Algebra in Scientific Computing, St. Petersburg, Russia, pp. 295–306 (2004)Google Scholar
  9. 9.
    Mourrain, B.: A new criterion for normal form algorithms. LNCS, vol. 179, pp. 430–443. Springer, Heidelberg (1999)zbMATHGoogle Scholar
  10. 10.
    Mourrain, B.: Pythagore’s dilemma, symbolic-numeric computation, and the border basis method. In: Symbolic-Numeric Computations, Trends in Mathematics, pp. 223–243. Birkhäuser Verlag, Basel (2007)CrossRefGoogle Scholar
  11. 11.
    Nagasaka, K.: A study on gröbner basis with inexact input. In: Gerdt, V.P., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2009. LNCS, vol. 5743, pp. 248–258. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Sasaki, T.: A practical method for floating-point Gröbner basis computation. In: Proceedings of ASCM 2009, Asian Symposium on Computer Mathematics, Fukuoka, Japan, Math-for-industry series, vol. 22, pp. 167–176. Kyushu Univ. (2009)Google Scholar
  13. 13.
    Sasaki, T.: A subresultant-like theory for Buchberger’s procedure, 17 p. Preprint of Univ. Tsukuba (March 2010)Google Scholar
  14. 14.
    Sasaki, T., Kako, F.: Computing floating-point Gröbner base stably. In: Proceedings of SNC 2007, Symbolic Numeric Computation, London, Canada, pp. 180–189 (2007)Google Scholar
  15. 15.
    Sasaki, T., Kako, F.: Floating-point Gröbner basis computation with ill-conditionedness estimation. In: Kapur, D. (ed.) ASCM 2007. LNCS (LNAI), vol. 5081, pp. 278–292. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  16. 16.
    Shirayanagi, K.: An algorithm to compute floating-point Gröbner bases. In: Mathematical Computation with Maple V. Ideas and Applications, pp. 95–106. Birkhäuser, Basel (1993)CrossRefGoogle Scholar
  17. 17.
    Shirayanagi, K.: Floating point Gröbner bases. Mathematics and Computers in Simulation 42, 509–528 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Shirayanagi, K., Sweedler, M.: Remarks on automatic algorithm stabilization. J. Symb. Comput. 26, 761–765 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Stetter, H.J.: Stabilization of polynomial systems solving with Gröbner bases. In: Proceedings of ISSAC 1997, Intern’l Symposium on Symbolic and Algebraic Computation, pp. 117–124. ACM Press, New York (1997)CrossRefGoogle Scholar
  20. 20.
    Stetter, H.J.: Numerical Polynomial Algebra. SIAM Publ., Philadelphia (2004)CrossRefzbMATHGoogle Scholar
  21. 21.
    Stetter, H.J.: Approximate Gröbner bases – an impossible concept? In: Proceedings of SNC 2005, Symbolic-Numeric Computation, Xi’an, China, pp. 235–236 (2005)Google Scholar
  22. 22.
    Suzuki, A.: Computing Gröbner bases within linear algebra. In: Gerdt, V.P., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2009. LNCS, vol. 5743, pp. 310–321. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  23. 23.
    Traverso, C.: Syzygies, and the stabilization of numerical Buchberger algorithm. In: Proceedings of LMCS 2002, Logic, Mathematics and Computer Science, RISC-Linz, Austria, pp. 244–255 (2002)Google Scholar
  24. 24.
    Traverso, C., Zanoni, A.: Numerical stability and stabilization of Gröbner basis computation. In: Proceedings of ISSAC 2002, Intern’l Symposium on Symbolic and Algebraic Computation, pp. 262–269. ACM Press, New York (2002)CrossRefGoogle Scholar
  25. 25.
    Weispfenning, V.: Gröbner bases for inexact input data. In: Proceedings of CASC 2003, Computer Algebra in Scientific Computing, Passau, Germany, pp. 403–411 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Tateaki Sasaki
    • 1
  • Fujio Kako
    • 2
  1. 1.Professor emeritusUniversity of TsukubaTsukuba-cityJapan
  2. 2.Department of Info. and Comp. Sci.Nara Women’s UniversityNara-cityJapan

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