Itnuit Janovitz-Freireich
ABSTRACT
We present a method based on Dickson's lemma to compute the "approximate radical" of a zero dimensional ideal I in C[x1, . . . , xm] which has zero clusters: the approximate radical ideal has exactly one root in each cluster for sufficiently small clusters. Our method is "global" in the sense that it does not require any local approximation of the zero clusters: it reduces the problem to the computation of the numerical nullspace of the so called "matrix of traces", a matrix computable from the generating polynomials of I. To compute the numerical nullspace of the matrix of traces we propose to use Gauss elimination with pivoting, and we prove that if I has k distinct zero clusters each of radius at most ε in the ∞-norm, then k steps of Gauss elimination on the matrix of traces yields a submatrix with all entries asymptotically equal to ε2. We also prove that the computed approximate radical has one root in each cluster with coordinates which are the arithmetic mean of the cluster, up to an error term asymptotically equal to ε2. In the univariate case our method gives an alternative to known approximate square-free factorization algorithms which is simpler and its accuracy is better understood.
- Z. Bai, J. Demmel, and A. McKenney. On the conditioning of the nonsymmetric eigenproblem: Theory and software. Technical report, Knoxville, TN, USA, 1989. Google Scholar
Digital Library
- E. Briand and L. Gonzalez-Vega. Multivariate Newton sums: Identities and generating functions. Communications in Algebra, 30(9):4527--4547, 2001.Google ScholarCross Ref
- J. Cardinal and B. Mourrain. Algebraic approach of residues and applications. In J. Reneger, M. Shub, and S. Smale, editors, Proceedings of AMS-Siam Summer Seminar on Math. of Numerical Analysis (Park City, Utah, 1995), volume 32 of Lectures in Applied Mathematics, pages 189--219, 1996.Google Scholar
- E. Cattani, A. Dickenstein, and B. Sturmfels. Computing multidimensional residues. In Algorithms in algebraic geometry and applications (Santander, 1994), volume 143 of Progr. Math., pages 135--164. Birkhäuser, Basel, 1996. Google ScholarDigital Library
- E. Cattani, A. Dickenstein, and B. Sturmfels. Residues and resultants. J. Math. Sci. Univ. Tokyo, 5(1):119--148, 1998.Google Scholar
- M. Chardin. Multivariate subresultants. Journal of Pure and Applied Algebra, 101:129--138, 1995.Google ScholarCross Ref
- R. M. Corless. Gröbner bases and matrix eigenproblems. ACM SIGSAM Bulletin, 30(4):26--32, 1996. Google ScholarDigital Library
- R. M. Corless, P. M. Gianni, and B. M. Trager. A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots. In ISSAC '97, pages 133--140, 1997. Google ScholarDigital Library
- R. M. Corless, P. M. Gianni, B. M. Trager, and S. M. Watt. The singular value decomposition for polynomial systems. In ISSAC '95, pages 195--207, 1995. Google ScholarDigital Library
- B. H. Dayton and Z. Zeng. Computing the multiplicity structure in solving polynomial systems. In ISSAC '05, pages 116--123, 2005. Google ScholarDigital Library
- J. Demmel. Accurate singular value decompositions of structured matrices. SIMAX, 1999. Google ScholarDigital Library
- J. Demmel and P. Koev. Accurate SVD's of polynomial vandermonde matrices involving orthonormal polynomials. Linear Algebra Applications, to appear, 2005.Google Scholar
- G. M. Díaz-Toca and L. González-Vega. An explicit description for the triangular decomposition of a zero-dimensional ideal through trace computations. In Symbolic computation: solving equations in algebra, geometry, and engineering (South Hadley, MA, 2000), volume 286 of Contemp. Math., pages 21--35. AMS, 2001.Google Scholar
- L. Dickson. Algebras and Their Arithmetics. University of Chicago Press, 1923.Google Scholar
- K. Friedl and L. Rónyai. Polynomial time solutions of some problems of computational algebra. In STOC '85, pages 153--162. ACM Press, 1985. Google ScholarDigital Library
- G. H. Golub and C. F. Van Loan. Matrix computations. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, third edition, 1996.Google Scholar
- V. Hribernig and H. J. Stetter. Detection and validation of clusters of polynomial zeros. J. Symb. Comput., 24(6):667--681, 1997. Google ScholarDigital Library
- W. Kahan. Numerical linear algebra. Canadian Mathematical Bulletin, (9):757--801, 1966.Google Scholar
- E. Kaltofen. On computing determinants of matrices without divisions. In P. S. Wang, editor, ISSAC'92, pages 342--349, New York, N. Y., 1992. ACM Press. Google ScholarDigital Library
- E. Kaltofen and J. May. On approximate irreducibility of polynomials in several variables. In ISSAC '03, pages 161--168. 2003. Google ScholarDigital Library
- K. H. Ko, T. Sakkalis, and N. M. Patrikalakis. Nonlinear Polynomial Systems: Multiple Roots and their Multiplicities. Proceedings of the Shape Modeling International 2004, 2004. Google ScholarDigital Library
- D. Lazard. Resolution des systemes d'equations algebriques. Theoret. Comp. Sci., 15(1), 1981. French, English summary.Google Scholar
- G. Lecerf. Quadratic Newton iterarion for systems with multiplicity. Foundations of Computational Mathematics, (2):247--293, 2002.Google Scholar
- A. Leykin, J. Verschelde, and A. Zhao. Evaluation of Jacobian matrices for Newton's method with deflation to approximate isolated singular solutions of polynomial systems. In D. Wang and L. Zhi, editors, SNC 2005 Proceedings. International Workshop on Symbolic-Numeric Computation., pages 19--28, 2005.Google Scholar
- D. Manocha and J. Demmel. Algorithms for Intersecting Parametric and Algebraic Curves II: Multiple Intersections. Graphical Models and Image Processing, 57(2):81--100, March 1995. Google ScholarDigital Library
- M. G. Marinari, T. Mora, and H. M. Möller. Gröbner duality and multiplicities in polynomial system solving. In ISSAC '95, pages 167--179, 1995. Google ScholarDigital Library
- H. M. Möller and H. J. Stetter. Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems. Numerische Matematik, 70:311--329, 1995. Google ScholarDigital Library
- A. P. Morgan, A. J. Sommese, and C. W. Wampler. Computing singular solutions to nonlinear analytic systems. Numer. Math., 58(7):669--684, 1991.Google Scholar
- A. P. Morgan, A. J. Sommese, and C. W. Wampler. Computing singular solutions to polynomial systems. Adv. Appl. Math., 13(3):305--327, 1992. Google ScholarDigital Library
- A. P. Morgan, A. J. Sommese, and C. W. Wampler. A power series method for computing singular solutions to nonlinear analytic systems. Numer. Math., 63(3):391--409, 1992.Google ScholarDigital Library
- S. Moritsugu and K. Kuriyama. A linear algebra method for solving systems of algebraic equations. In RISC-Linz Report Series, volume 35, 1997.Google Scholar
- S. Moritsugu and K. Kuriyama. On multiple zeros of systems of algebraic equations. In ISSAC '99, pages 23--30, 1999. Google ScholarDigital Library
- B. Mourrain. Generalized normal forms and polynomial system solving. In ISSAC '05, pages 253--260, 2005. Google ScholarDigital Library
- T. Ojika. Modified deflation algorithm for the solution of singular problems. I. A system of nonlinear algebraic equations. J. Math. Anal. Appl., 123(1):199--221, 1987.Google ScholarCross Ref
- T. Ojika. Modified deflation algorithm for the solution of singular problems. II. Nonlinear multipoint boundary value problems. J. Math. Anal. Appl., 123(1):222--237, 1987.Google ScholarCross Ref
- T. Ojika, S. Watanabe, and T. Mitsui. Deflation algorithm for the multiple roots of a system of nonlinear equations. J. Math. Anal. Appl., 96(2):463--479, 1983.Google ScholarCross Ref
- R. S. Pierce. Associative algebras, volume 88 of Graduate Text in Mathematics. Springer-Verlag, 1982.Google Scholar
- T. Sasaki and M.-T. Noda. Approximate square-free decomposition and root-finding of ill-conditioned algebraic equations. J. Inform. Process., 12(2):159--168, 1989. Google ScholarDigital Library
- E. Schost. Personal communication, 2005.Google Scholar
- V. Shoup. Efficient computation of minimal polynomials in algebraic extensions of finite fields. In ISSAC '99, pages 53--58, 1999. Google ScholarDigital Library
- H. J. Stetter. Analysis of zero clusters in multivariate polynomial systems. In ISSAC '96, pages 127--136, 1996. Google ScholarDigital Library
- H. J. Stetter. Numerical Polynomial Algebra. Society for Industrial and Applied Mathematics, 2004. Google ScholarDigital Library
- A. Szanto. Solving over-determined systems by subresultant methods. Preprint, 2001.Google Scholar
- K. Yokoyama, M. Noro, and T. Takeshima. Solutions of systems of algebraic equations and linear maps on residue class rings. J. Symb. Comput., 14(4):399--417, 1992. Google ScholarDigital Library
- Z. Zeng. A method computing multiple roots of inexact polynomials. In ISSAC '03, pages 266--272, 2003. Google ScholarDigital Library
Index Terms
- Approximate radical of ideals with clusters of roots
Recommendations
Moment matrices, trace matrices and the radical of ideals
ISSAC '08: Proceedings of the twenty-first international symposium on Symbolic and algebraic computationLet f1,..., fs be a system of polynomials in K[x1,..., xm] generating a zero-dimensional ideal I , where K is an arbitrary algebraically closed field. Assume that the factor algebra A = K[x1 , . . . , xm]/I is Gorenstein and that we have a bound delta > ...
The subresultant and clusters of close roots
ISSAC '03: Proceedings of the 2003 international symposium on Symbolic and algebraic computationThis paper investigates the subresultant of univariate polynomials from the viewpoint of close roots. First, we derive formulas which express the subresultant and its cofactors in the root-differences. Then, we consider the case that the given ...
Comments