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On Approximate Linearized Triangular Decompositions

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Symbolic-Numeric Computation

Part of the book series: Trends in Mathematics ((TM))

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Abstract

In this paper, we describe progress on the development of algorithms for triangular decomposition of approximate systems.

We begin with the treatment of linear, homogeneous systems with positive-dimensional solution spaces, and approximate coefficients. We use the Singular Value Decomposition to decompose such systems into a stable form, and discuss condition numbers for approximate triangular decompositions. Results from the linear case are used as the foundation of a discussion on the fully nonlinear case. We introduce linearized triangular sets, and show that we can obtain useful stability information about sets corresponding to different variable orderings. Examples are provided, experiments are described, and connections with the works of Sommese, Verschelde, and Wampler are made.

This work is supported by NSERC, MITACS, and Maplesoft, Canada.

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References

  1. D. N. Bernstein. The Number of Roots of a System of Equations. Functional Anal. Appl., 9(3): 183–185. Translated from Funktsional. Anal. i Prilozhen., 9(3): 1–4, 1975.

    Article  MATH  Google Scholar 

  2. L. Blum, F. Cucker, M. Shub, and S. Smale. Complexity and Real Computation. Springer-Verlag, New York, 1997.

    MATH  Google Scholar 

  3. G. Chèze. Absolute Polynomial Factorization in Two Variables and the Knapsack Problem. J. Gutierrez, editor, Proc. ISSAC 2004, pages 87–94. ACM Press, 2004.

    Google Scholar 

  4. G. Chèze and A. Galligo. Four Lectures on Polynomial Absolute Factorization. Solving Polynomial Equations: Foundations, Algorithms, and Applications, vol. 14 of Algorithms and Computation in Mathematics, pages 339–392. Springer-Verlag, 2005.

    Article  Google Scholar 

  5. S.-C. Chou. Mechanical Geometry Theorem Proving. D. Reidel Publ. Comp., Dordrecht, 1988.

    Google Scholar 

  6. R. M. Corless, A. Galligo, I. S. Kotsireas, and S. M. Watt. A Geometric-Numeric Algorithm for Factoring Multivariate Polynomials. Proc. ISSAC 2002, pages 37–45. ACM Press, 2002.

    Google Scholar 

  7. D. Cox, J. Little, and D. O’Shea. Ideals, Varieties, and Algorithms. Springer-Verlag, New York, 2nd edition, 1997.

    Google Scholar 

  8. R. M. Corless, P. M. Gianni, B. M. Trager, and S. M. Watt. The Singular Value Decomposition for Polynomial Systems. Proc. ISSAC’ 95, pages 96–103, ACM Press, 1995.

    Google Scholar 

  9. X. Dahan, M. Moreno Maza, É. Schost, W. Wu, and Y. Xie. Equiprojectable Decompositions of Zero-Dimensional Varieties. Proc. ICPSS, pages 69–71. University of Paris 6, France, 2004.

    Google Scholar 

  10. X. Dahan, M. Moreno Maza, É. Schost, W. Wu, and Y. Xie. Lifting Techniques for Triangular Decompositions. Proc. ISSAC 2005, pages 108–115. ACM Press, 2005.

    Google Scholar 

  11. X. Dahan, X. Jin, M. Moreno Maza and É. Schost. Change of Ordering for Regular Chains in Positive Dimension. Accepted by Maple Conference, Canada, 2006.

    Google Scholar 

  12. X. Dahan and É. Schost. Sharp Estimates for Triangular Sets. Proc. ISSAC 2004, pages 103–110. ACM Press, 2004.

    Google Scholar 

  13. C. Eckart and G. Young. The Approximation of one Matrix by Another of Lower Rank. Psychometrika, vol. 1: 211–218, 1936.

    Article  Google Scholar 

  14. M. V. Foursov and M. Moreno Maza. On Computer-Assisted Classification of Coupled Integrable Equations. J. Symb. Comput., 33: 647–660, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. Galligo and D. Rupprecht. Irreducible Decomposition of Curves. J. Symb. Comput., 33(5): 661–677, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  16. X.-S. Gao and Y. Luo. A Characteristic Set Algorithm for Difference Polynomial Systems. Proc. ICPSS, pages 28–30. University of Paris 6, France, 2004.

    Google Scholar 

  17. P. Gianni, B. Trager, and G. Zacharias. Gröbner Bases and Primary Decomposition of Polynomial Ideals. J. Symb. Comput., 6(2–3): 149–167, 1988.

    MathSciNet  Google Scholar 

  18. M. Giusti and J. Heintz. La Détermination de la Dimension et des Points Isolés d’une Variété Algébrique Peuvent S’effectuer en Temps Polynomial. D. Eisenbud and L. Robbiano, editors, Computational Algebraic Geometry and Commutative Algebra, Cortona 1991, volume XXXIV of Symposia Mathematica, pages 216–256. Cambridge University Press, 1993.

    Google Scholar 

  19. M. Giusti and J. Heinz. Kronecker’s Smart, Little Black Boxes, London Mathematical Society Lecture Note Series, vol. 284, pages 69–104. Cambridge University Press, 2001.

    Google Scholar 

  20. M. Giusti, G. Lecerf, and B. Salvy. A Gröbner Free Alternative for Polynomial System Solving. J. Complexity, 17(1): 154–211, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  21. G. Golub and C. V. Loan. Matrix Computations. John Hopkins University Press, 3rd edition, 1996.

    Google Scholar 

  22. M. Kalkbrener. A Generalized Euclidean Algorithm for Computing Triangular Representations of Algebraic Varieties. J. Symb. Comput., 15: 143–167, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  23. I. A. Kogan and M. Moreno Maza. Computation of Canonical Forms for Ternary Cubics. Teo Mora, editor, Proc. ISSAC 2002, pages 151–160. ACM Press, 2002.

    Google Scholar 

  24. D. Lazard. Solving Zero-Dimensional Algebraic Systems. J. Symb. Comput., 13: 117–133, 1992.

    MATH  MathSciNet  Google Scholar 

  25. G. Lecerf. Computing the Equidimensional Decomposition of an Algebraic Closed Set by Means of Lifting Fibers. J. Complexity, 19(4): 564–596, 2003.

    Article  MathSciNet  Google Scholar 

  26. F. Lemaire, M. Moreno Maza, and Y. Xie. The RegularChains Library. Proc. Maple Conference 2005, pages 355–368, 2005. Distributed software with Maple 10, Maplesoft, Canada.

    Google Scholar 

  27. A. Leykin and J. Verschelde. Phcmaple: A Maple Interface to the Numerical Homotopy Algorithms in PHCpack. Proc. ACA’ 04, pages 139–147, University of Texas at Beaumont, USA, 2004.

    Google Scholar 

  28. M. Moreno Maza. On Triangular Decompositions of Algebraic Varieties. Technical Report 4/99, NAG, UK. Presented at the MEGA-2000 Conference, Bath, UK, 2000. http://www.csd.uwo.ca/~moreno.

    Google Scholar 

  29. M. Moreno Maza, G. Reid, R. Scott, and W. Wu. On Approximate Triangular Decompositions I: Dimension Zero. Proc. SNC 2005, pages 250–275, 2005.

    Google Scholar 

  30. T. Sasaki. Approximate Multivariate Polynomial Factorization Based on Zero-Sum Relations. B. Mourrain, editor, Proc. ISSAC 2001, pages 284–291. ACM Press, 2001.

    Google Scholar 

  31. É. Schost. Complexity Results for Triangular Sets. J. Symb. Comput., 36(3–4): 555–594, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  32. A. J. Sommese and J. Verschelde. Numerical Homotopies to Compute Generic Points on Positive Dimensional Algebraic Sets. J. Complexity, 16(3): 572–602, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  33. A. J. Sommese, J. Verschelde, and C. W. Wampler. Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components. SIAM J. Numer. Anal., 38(6): 2022–2046, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  34. A. J. Sommese, J. Verschelde, and C. W. Wampler. Using Monodromy to Decompose Solution Sets of Polynomial Systems into Irreducible Components. C. Ciliberto, F. Hirzebruch, R. Miranda, and M. Teicher, editors, Application of Algebraic Geometry to Coding Theory, Physics and Computation, pages 297–315. Kluwer Academic Publishers, 2001.

    Google Scholar 

  35. A. J. Sommese, J. Verschelde, and C. W. Wampler. Symmetric Functions Applied to Decomposing Solution Sets of Polynomial Systems. SIAM J. Numer. Anal., 40(6): 2026–2046, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  36. A. J. Sommese, J. Verschelde, and C. W. Wampler. Numerical Irreducible Decomposition Using PHCpack. Algebra, Geometry, and Software Systems, pages 109–130. Springer-Verlag, 2003.

    Google Scholar 

  37. A. J. Sommese and C. W. Wampler. Numerical Algebraic Geometry. J. Renegar, M. Shub, and S. Smale, editors, Proc. AMS-SIAM Summer Seminar in Applied Mathematics, vol. 32 of Lectures in Applied Mathematics, 1995.

    Google Scholar 

  38. H. J. Stetter. The Nearest Polynomial with a Given Zero, and Similar Problems. ACM SIGSAM Bulletin, 33(4): 2–4, 1999.

    Article  MATH  Google Scholar 

  39. L. N. Trefethen and D. Bau. Numerical Linear Algebra. Society for Industrial & Applied Mathematics, Philadelphia, 3rd edition, 1997.

    Google Scholar 

  40. B. L. Van der Waerden. Modern Algebra. Frederick Ungar Publishing Co., New York, 2nd edition, 1953.

    Google Scholar 

  41. J. Verschelde. PHCpack: A General-Purpose Solver for Polynomial Systems by Homotopy Continuation. ACM Transactions on Mathematical Software, 25(2): 251–276, 1999.

    Article  MATH  Google Scholar 

  42. D. Wang. Elimination Methods. Springer-Verlag, Wein, New York, 2001.

    MATH  Google Scholar 

  43. W.-T. Wu. On Zeros of Algebraic Equations — An Application of Ritt Principle. Kexue Tongbao, 31(1): 1–5, 1986.

    Google Scholar 

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Author information

Authors and Affiliations

  1. ORCCA, Department of Computer Science, University of Western Ontario, London, Canada

    Marc Moreno Maza

Authors
  1. Marc Moreno Maza
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  2. Greg J. Reid
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  3. Robin Scott
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  4. Wenyuan Wu
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Editor information

Editors and Affiliations

  1. School of Science, Beihang University, 37 Xueyuan Road, Beijing, 100083, China

    Dongming Wang

  2. Laboratoire d’Informatique de Paris 6, Université Pierre et Marie Curie - CNRS, 8, rue due Capitaine Scott, 75015, Paris, France

    Dongming Wang

  3. Key Laboratory of Mathematics Mechanization, Academy of Mathematics and System Sciences, Beijing, 100080, China

    Lihong Zhi

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© 2007 Birkhäuser Verlag Basel/Switzerland

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Moreno Maza, M., Reid, G.J., Scott, R., Wu, W. (2007). On Approximate Linearized Triangular Decompositions. In: Wang, D., Zhi, L. (eds) Symbolic-Numeric Computation. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7984-1_17

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