Abstract
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.
Work supported by Japan Society for the Promotion of Science under Grants 23500003 and 08039686.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. Springer, New York (1997)
Dayton, B.H., Zeng, Z.: Computing the multiplicity structure in solving polynomial systems. In: Proceedings of ISSAC 2005, pp. 116–123. ACM Press (2005)
Fortune, S.: Polynomial root finding using iterated eigenvalue computation. In: Proceedings of ISSAC 2001, pp. 121–128. ACM Press (2001)
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)
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)
Noda, M.-T., Sasaki, T.: Approximate GCD and its application to ill-conditioned algebraic equations. J. Comput. App. Math. 38, 335–351 (1991)
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)
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)
Sasaki, T.: Proposal of singularization of approximately singular systems. Preprint of Univ. Tsukuba, 14 pages (May 2012)
Sasaki, T., Noda, M.-T.: Approximate square-free decomposition and root-finding of ill-conditioned algebraic equations. J. Inf. Proces. 12, 159–168 (1989)
Weispfenning, V.: Comprehensive Gröbner bases. J. Symb. Comp. 14, 1–29 (1992)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sasaki, T., Inaba, D. (2012). Approximately Singular Systems and Ill-Conditioned Polynomial Systems. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2012. Lecture Notes in Computer Science, vol 7442. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32973-9_26
Download citation
DOI: https://doi.org/10.1007/978-3-642-32973-9_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32972-2
Online ISBN: 978-3-642-32973-9
eBook Packages: Computer ScienceComputer Science (R0)