Abstract
Computation of Gröbner bases of polynomial systems with coefficients of floating-point numbers has been a serious problem in computer algebra for many years; the computation often becomes very unstable and people did not know how to remove the instability. Recently, the present authors clarified the origin of instability and presented a method to remove the instability. Unfortunately, the method is very time-consuming and not practical. In this paper, we first investigate the instability much more deeply than in the previous paper, then we give a theoretical analysis of the term cancellation which causes loss of accuracy in various cases. On the basis of this analysis, we propose a practical method for computing Gröbner bases with coefficients of floating-point numbers. The method utilizes multiple precision floating-point numbers, and it removes the drawbacks of the previous method almost completely. Furthermore, we present a practical method of estimating the ill-conditionedness of the input system.
Work supported in part by Japan Society for the Promotion of Science under Grants 19300001.
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References
Sasaki, T., Kako, F.: Floating-point Gröber Basis Computation with Ill-conditionedness Estimation. Technical Report of Univ. of Tsukuba, in (December 2007), http://www.math.tsukuba.ac.jp/~sasaki/papers/ASCM2007
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)
Collins, J.E.: Subresultant and reduced polynomial remainder sequence. J. ACM 14, 128–142 (1967)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. Springer, New York (1997)
Fortuna, E., Gianni, P., Trager, B.: Degree reduction under specialization. J. Pure Appl. Algebra 164, 153–164 (2001)
Gonzalez-Vega, L., Traverso, C., Zanoni, A.: Hilbert stratification and parametric Gröber bases. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2005. LNCS, vol. 3718, pp. 220–235. Springer, Heidelberg (2005)
Kako, F., Sasaki, T.: Proposal of “effective” floating-point number. Preprint of Univ. Tsukuba (May 1997) (unpublished)
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)
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)
Sasaki, T., Kako, F.: Computing floating-point Gröbner base stably. In: Proceedings of SNC 2007 (Symbolic Numeric Computation), London, Canada, pp. 180–189 (2007)
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)
Shirayanagi, K.: Floating point Gröbner bases. Mathematics and Computers in Simulation 42, 509–528 (1996)
Shirayanagi, K., Sweedler, M.: Remarks on automatic algorithm stabilization. J. Symb. Comput. 26, 761–765 (1998)
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)
Stetter, H.J.: Numerical Polynomial Algebra. SIAM Publ., Philadelphia (2004)
Stetter, H.J.: Approximate Gröbner bases – an impossible concept? In: Proceedings of SNC 2005 (Symbolic-Numeric Computation), Xi’an, China, pp. 235–236 (2005)
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)
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)
Weispfenning, V.: Gröbner bases for inexact input data. In: Proceedings of CASC 2003 (Computer Algebra in Scientific Computing), Passau, Germany, pp. 403–411 (2003)
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Sasaki, T., Kako, F. (2008). Floating-Point Gröbner Basis Computation with Ill-conditionedness Estimation. In: Kapur, D. (eds) Computer Mathematics. ASCM 2007. Lecture Notes in Computer Science(), vol 5081. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87827-8_23
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DOI: https://doi.org/10.1007/978-3-540-87827-8_23
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