Robust Certified Numerical Homotopy Tracking
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We describe, for the first time, a completely rigorous homotopy (path-following) algorithm (in the Turing machine model) to find approximate zeros of systems of polynomial equations. If the coordinates of the input systems and the initial zero are rational our algorithm involves only rational computations, and if the homotopy is well posed an approximate zero with integer coordinates of the target system is obtained. The total bit complexity is linear in the length of the path in the condition metric, and polynomial in the logarithm of the maximum of the condition number along the path, and in the size of the input.
KeywordsSymbolic–numeric methods Polynomial systems Complexity Condition metric Homotopy method Rational computation Computer proof
Mathematics Subject Classification (2010)14Q20 65H20 68W30
In earlier stages of the ideas behind this work, we maintained many related conversations with Clement Pernet; thanks go to him for helpful discussions and comments. We also thank Gregorio Malajovich, Luis Miguel Pardo and Michael Shub for their questions and answers. Our beloved friend and colleague Jean Pierre Dedieu also inspired us in many occasions. The second author thanks Institut Mittag-Leffler for hosting him in the Spring semester of 2011. A part of this work was done while we were participating in several workshops related to Foundations of Computational Mathematics in the Fields Institute. We thank this institution for its kind support.
C. Beltrán partially supported by MTM2010-16051, Spanish Ministry of Science (MICINN).
A. Leykin partially supported by NSF grant DMS-0914802.
- 2.D.J. Bates, J.D. Hauenstein, A.J. Sommese, C.W. Wampler, Bertini: software for numerical algebraic geometry. Available at http://www.nd.edu/~sommese/bertini.
- 18.D.R. Grayson, M.E. Stillman, Macaulay 2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/.
- 19.J.D. Hauenstein, F. Sottile, alphacertified: certifying solutions to polynomial systems (2010). arXiv:1011.1091v1.
- 21.M.H. Kim, Computational complexity of the Euler type algorithms for the roots of complex polynomials. Ph.D. Thesis, The City University of New York, 1985. Google Scholar
- 22.T.L. Lee, T.Y. Li, C.H. Tsai, Hom4ps-2.0: A software package for solving polynomial systems by the polyhedral homotopy continuation method. Available at. http://hom4ps.math.msu.edu/HOM4PS_soft.htm.
- 25.G. Malajovich, On the complexity of path-following Newton algorithms for solving systems of polynomial equations with integer coefficients. Ph.D. Thesis, Univ. California, Berkley, 1993. Google Scholar
- 28.G. Malajovich, PSS—Polynomial System Solver version 3.0.5. Available at. http://www.labma.ufrj.br/~gregorio/software.php.
- 34.M. Shub, S. Smale, Complexity of Bezout’s theorem. V. Polynomial time, in Selected Papers of the Workshop on Continuous Algorithms and Complexity, Barcelona, 1993. Theoret. Comput. Sci., vol. 133 (1994), pp. 141–164. Google Scholar
- 37.J. van der Hoeven, Reliable homotopy continuation. Technical Report, HAL 00589948 (2011). Google Scholar