Approximately Singular Systems and Ill-Conditioned Polynomial Systems

  • Tateaki Sasaki
  • Daiju Inaba
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7442)


By “approximately singular system” we mean a system of multivariate polynomials the dimension of whose variety is increased by small amounts of perturbations. First, we give a necessary condition that the given system is approximately singular. Then, we classify polynomial systems which seems ill-conditioned to solve numerically into four types. Among these, the third one is approximately singular type. We give a simple well-conditioning method for the third type. We test the third type and its well-conditioned systems by various examples, from viewpoints of “global convergence”, “local convergence” and detail of individual computation. The results of experiments show that our well-conditioning method improves the global convergence largely.


approximate ideal approximately linear-dependent relation approximately singular system ill-conditioned polynomial system multivariate Newton’s method well-conditioning 


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  1. 1.
    Corless, R.M., Gianni, P.M., Trager, B.M.: A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots. In: Proceedings of ISSAC 1997 (Intn’l Symposium on Symbolic and Algebraic Computation), pp. 133–140. ACM Press (1997)Google Scholar
  2. 2.
    Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. Springer, New York (1997)CrossRefGoogle Scholar
  3. 3.
    Dayton, B.H., Zeng, Z.: Computing the multiplicity structure in solving polynomial systems. In: Proceedings of ISSAC 2005, pp. 116–123. ACM Press (2005)Google Scholar
  4. 4.
    Fortune, S.: Polynomial root finding using iterated eigenvalue computation. In: Proceedings of ISSAC 2001, pp. 121–128. ACM Press (2001)Google Scholar
  5. 5.
    Janovitz-Freireich, I., Rónyai, L., Szánto, A.: Approximate radical of ideals with clusters of roots. In: Proceedings of ISSAC 1997, pp. 146–153. ACM Press (2006)Google Scholar
  6. 6.
    Mantzaflaris, A., Mourrain, B.: Deflation and certified isolation of singular zeros of polynomial systems. In: Proceedings of ISSAC 2011, pp. 249–256. ACM Press (2011)Google Scholar
  7. 7.
    Noda, M.-T., Sasaki, T.: Approximate GCD and its application to ill-conditioned algebraic equations. J. Comput. App. Math. 38, 335–351 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ochi, M., Noda, M.-T., Sasaki, T.: Approximate GCD of multivariate polynomials and application to ill-conditioned system of algebraic equations. J. Inf. Proces. 14, 292–300 (1991)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Sasaki, T.: A theory and an algorithm of approximate Gröbner bases. In: Proceedings of SYNASC 2011 (Symbolic and Numeric Algorithms for Scientific Computing), pp. 23–30. IEEE Computer Society Press (2012)Google Scholar
  10. 10.
    Sasaki, T.: Proposal of singularization of approximately singular systems. Preprint of Univ. Tsukuba, 14 pages (May 2012)Google Scholar
  11. 11.
    Sasaki, T., Noda, M.-T.: Approximate square-free decomposition and root-finding of ill-conditioned algebraic equations. J. Inf. Proces. 12, 159–168 (1989)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Weispfenning, V.: Comprehensive Gröbner bases. J. Symb. Comp. 14, 1–29 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Wu, X., Zhi, L.: Determining singular solutions of polynomial systems via symbolic-numeric reduction to geometric involutive forms. J. Symb. Comput. 47, 227–238 (2012)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Tateaki Sasaki
    • 1
  • Daiju Inaba
    • 2
  1. 1.University of TsukubaTsukuba-cityJapan
  2. 2.Japanese Association of Mathematics CertificationTokyoJapan

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