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Foundations of Computational Mathematics

, Volume 13, Issue 2, pp 253–295 | Cite as

Robust Certified Numerical Homotopy Tracking

  • Carlos Beltrán
  • Anton Leykin
Article

Abstract

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.

Keywords

Symbolic–numeric methods Polynomial systems Complexity Condition metric Homotopy method Rational computation Computer proof 

Mathematics Subject Classification (2010)

14Q20 65H20 68W30 

Notes

Acknowledgements

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.

References

  1. 1.
    D.J. Bates, C. Peterson, A.J. Sommese, C.W. Wampler, Numerical computation of the genus of an irreducible curve within an algebraic set, J. Pure Appl. Algebra 215(8), 1844–1851 (2011). MathSciNetzbMATHCrossRefGoogle Scholar
  2. 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.
  3. 3.
    W. Baur, V. Strassen, The complexity of partial derivatives, Theor. Comput. Sci. 22(3), 317–330 (1983). MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    C. Beltrán, A continuation method to solve polynomial systems, and its complexity, Numer. Math. 117(1), 89–113 (2011). MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    C. Beltrán, A. Leykin, Certified numerical homotopy tracking, Exp. Math. 21(1), 69–83 (2012). MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    C. Beltrán, L.M. Pardo, On Smale’s 17th problem: a probabilistic positive solution, Found. Comput. Math. 8(1), 1–43 (2008). MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    C. Beltrán, L.M. Pardo, Smale’s 17th problem: average polynomial time to compute affine and projective solutions, J. Am. Math. Soc. 22, 363–385 (2009). zbMATHCrossRefGoogle Scholar
  8. 8.
    C. Beltrán, L.M. Pardo, Fast linear homotopy to find approximate zeros of polynomial systems, Found. Comput. Math. 11(1), 95–129 (2011). MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    S. Billey, R. Vakil, Intersections of Schubert varieties and other permutation array schemes, in Algorithms in Algebraic Geometry, ed. by A. Dickenstein, F.O. Schreyer, A.J. Sommese. The IMA Vol. Math. Appl., vol. 146 (Springer, New York, 2008), pp. 21–54. CrossRefGoogle Scholar
  10. 10.
    L. Blum, M. Shub, S. Smale, On a theory of computation and complexity over the real numbers; NP completeness, recursive functions and universal machines, Bull. Am. Math. Soc. 21, 1–46 (1989). MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    L. Blum, F. Cucker, M. Shub, S. Smale, Complexity and Real Computation (Springer, New York, 1998). CrossRefGoogle Scholar
  12. 12.
    P. Bürguisser, F. Cucker, On a problem posed by Steve Smale, Ann. Math. 174, 1785–1836 (2011). CrossRefGoogle Scholar
  13. 13.
    D. Castro, K. Hägele, J.E. Morais, L.M. Pardo, Kronecker’s and Newton’s approaches to solving: a first comparison, J. Complex. 17(1), 212–303 (2001). zbMATHCrossRefGoogle Scholar
  14. 14.
    D. Castro, J.L. Montaña, L.M. Pardo, J. San Martín, The distribution of condition numbers of rational data of bounded bit length, Found. Comput. Math. 2(1), 1–52 (2002). MathSciNetzbMATHGoogle Scholar
  15. 15.
    T.H. Cormen, C.E. Leiserson, R.L. Rivest, Introduction to Algorithms (MIT Press, Cambridge, 1990). zbMATHGoogle Scholar
  16. 16.
    J.-P. Dedieu, G. Malajovich, M. Shub, Adaptative step size selection for homotopy methods to solve polynomial equations, IMA J. Numer. Anal. (2010). doi: 10.1093/imanum/drs007. Google Scholar
  17. 17.
    J.D. Dixon, Exact solution of linear equations using p-adic expansions, Numer. Math. 40(1), 137–141 (1982). MathSciNetzbMATHCrossRefGoogle Scholar
  18. 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. 19.
    J.D. Hauenstein, F. Sottile, alphacertified: certifying solutions to polynomial systems (2010). arXiv:1011.1091v1.
  20. 20.
    R.B. Kearfott, Z. Xing, An interval step control for continuation methods, SIAM J. Numer. Anal. 31(3), 892–914 (1994). MathSciNetzbMATHCrossRefGoogle Scholar
  21. 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. 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.
  23. 23.
    A. Leykin, Numerical algebraic geometry for Macaulay2, J. Softw. Algebr. Geom. 3, 5–10 (2011). MathSciNetGoogle Scholar
  24. 24.
    A. Leykin, F. Sottile, Galois groups of Schubert problems via homotopy computation, Math. Comput. 78(267), 1749–1765 (2009). MathSciNetzbMATHCrossRefGoogle Scholar
  25. 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
  26. 26.
    G. Malajovich, On generalized Newton algorithms: quadratic convergence, path-following and error analysis, Theor. Comput. Sci. 133, 65–84 (1994). MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    G. Malajovich, Condition number bounds for problems with integer coefficients, J. Complex. 16(3), 529–551 (2000). MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    G. Malajovich, PSS—Polynomial System Solver version 3.0.5. Available at. http://www.labma.ufrj.br/~gregorio/software.php.
  29. 29.
    C.H. Papadimitriou, Computational Complexity (Addison-Wesley, Reading, 1994). zbMATHGoogle Scholar
  30. 30.
    M. Shub, Some remarks on Bezout’s theorem and complexity theory, in From Topology to Computation: Proceedings of the Smalefest, ed. by M.W. Hirsch, J.E. Marsden, M. Shub (Springer, New York, 1993), pp. 443–455. CrossRefGoogle Scholar
  31. 31.
    M. Shub, Complexity of Bézout’s theorem. VI: Geodesics in the condition (number) metric, Found. Comput. Math. 9(2), 171–178 (2009). MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    M. Shub, S. Smale, Complexity of Bézout’s theorem. II. Volumes and probabilities, in Computational Algebraic Geometry, ed. by Fr. Eyssette, A. Galligo. Progr. Math., vol. 109 (Birkhäuser, Boston, 1993), pp. 267–285. CrossRefGoogle Scholar
  33. 33.
    M. Shub, S. Smale, Complexity of Bézout’s theorem. I. Geometric aspects, J. Am. Math. Soc. 6(2), 459–501 (1993). MathSciNetzbMATHGoogle Scholar
  34. 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
  35. 35.
    S. Smale, The fundamental theorem of algebra and complexity theory, Bull. Am. Math. Soc. 4(1), 1–36 (1981). MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    S. Smale, Newton’s method estimates from data at one point, in The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics (Springer, New York, 1986), pp. 185–196. CrossRefGoogle Scholar
  37. 37.
    J. van der Hoeven, Reliable homotopy continuation. Technical Report, HAL 00589948 (2011). Google Scholar
  38. 38.
    J. Verschelde, Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation, ACM Trans. Math. Softw. 25(2), 251–276 (1999). Available at http://www.math.uic.edu/~jan. zbMATHCrossRefGoogle Scholar

Copyright information

© SFoCM 2013

Authors and Affiliations

  1. 1.Departamento de Matemáticas, Estadística y ComputaciónUniversidad de CantabriaSantanderSpain
  2. 2.School of MathematicsGeorgia TechAtlantaUSA

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