Foundations of Computational Mathematics

, Volume 17, Issue 5, pp 1265–1292 | Cite as

A Deterministic Algorithm to Compute Approximate Roots of Polynomial Systems in Polynomial Average Time

Article

Abstract

We describe a deterministic algorithm that computes an approximate root of n complex polynomial equations in n unknowns in average polynomial time with respect to the size of the input, in the Blum–Shub–Smale model with square root. It rests upon a derandomization of an algorithm of Beltrán and Pardo and gives a deterministic affirmative answer to Smale’s 17th problem. The main idea is to make use of the randomness contained in the input itself.

Keywords

Polynomial system Homotopy continuation Complexity  Smale’s 17th problem Derandomization 

Mathematics Subject Classification

Primary 68Q25 Secondary 65H10 65H20 65Y20 

References

  1. 1.
    Michael Shub. Complexity of Bezout’s theorem. VI. Geodesics in the condition (number) metric. In: Found. Comput. Math. 9.2 (2009), pp. 171–178. doi:10.1007/s10208-007-9017-6.
  2. 2.
    Michael Shub and Steve Smale. Complexity of Bézout’s theorem. I. Geometric aspects. In: J. Amer. Math. Soc. 6.2 (1993), pp. 459–501. doi:10.2307/2152805.
  3. 3.
    Michael Shub and Steve Smale. Complexity of Bezout’s theorem. II. Volumes and probabilities. In: Computational algebraic geometry (Nice, 1992). Vol. 109. Progr. Math. Birkhäuser Boston, Boston, MA, 1993, pp. 267–285.Google Scholar
  4. 4.
    Michael Shub and Steve Smale. Complexity of Bezout’s theorem. IV. Probability of success; extensions. In: SIAM J. Numer. Anal. 33.1 (1996), pp. 128–148. doi:10.1137/0733008.
  5. 5.
    Michael Shub and Steve Smale. Complexity of Bezout’s theorem. V. Polynomial time. In: Theoret. Comput. Sci. 133.1 (1994). Selected papers of the Workshop on Continuous Algorithms and Complexity (Barcelona, 1993), pp. 141–164. doi:10.1016/0304-3975(94)90122-8.
  6. 6.
    Steve Smale. Newton’s method estimates from data at one point. In: The merging of disciplines: new directions in pure, applied, and computational mathematics (Laramie, Wyo., 1985). Springer, New York, 1986, pp. 185–196. doi:10.1007/978-1-4612-4984-9_13.
  7. 7.
    Steve Smale. Mathematical problems for the next century. In: The Mathematical Intelligencer 20.2 (1998), pp. 7–15. doi:10.1007/BF03025291.
  8. 8.
    Lenore Blum, Michael Shub, and Steve Smale. On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. In: Bull. Amer. Math. Soc. N.S. 21.1 (1989), pp. 1–46. doi:10.1090/S0273-0979-1989-15750-9.
  9. 9.
    Carlos Beltrán and Luis Miguel Pardo. Fast linear homotopy to find approximate zeros of polynomial systems. In: Found. Comput. Math. 11.1 (2011), pp. 95–129. doi:10.1007/s10208-010-9078-9.
  10. 10.
    Peter Bürgisser and Felipe Cucker. On a problem posed by Steve Smale. In: Ann. of Math. (2) 174.3 (2011), pp. 1785–1836. doi:10.4007/annals.2011.174.3.8.
  11. 11.
    Peter Bürgisser and Felipe Cucker. Condition. The geometry of numerical algorithms. Vol. 349. Grundlehren der Mathematischen Wissenschaften. Springer Berlin Heidelberg, 2013. doi:10.1007/978-3-642-38896-5.
  12. 12.
    Carlos Beltrán and Luis Miguel Pardo. Smale’s 17th problem: average polynomial time to compute affine and projective solutions. In: J. Amer. Math. Soc. 22.2 (2009), pp. 363–385. doi:10.1090/S0894-0347-08-00630-9.
  13. 13.
    Daniel Spielman and Shang-Hua Teng. Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. In: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing. ACM, New York, 2001, 296–305 (electronic). doi:10.1145/380752.380813.
  14. 14.
    Irénée Briquel, Felipe Cucker, Javier Peña, and Vera Roshchina. Fast computation of zeros of polynomial systems with bounded degree under finite-precision. In: Math. Comp. 83.287 (2014), pp. 1279–1317. doi:10.1090/S0025-5718-2013-02765-2.
  15. 15.
    Lenore Blum, Felipe Cucker, Michael Shub, and Steve Smale. Complexity and real computation. Springer-Verlag, New York, 1998. doi:10.1007/978-1-4612-0701-6.
  16. 16.
    Michael Shub. Some remarks on Bezout’s theorem and complexity theory. In: From Topology to Computation: Proceedings of the Smalefest. Springer, New York, 1993. Chap. 40, pp. 443–455. doi:10.1007/978-1-4612-2740-3_40.
  17. 17.
    Masaaki Sibuya. A method for generating uniformly distributed points on N-dimensional spheres. In: Ann. Inst. Statist. Math. 14 (1962), pp. 81–85. doi:10.1007/BF02868626.
  18. 18.
    Diego Armentano, Carlos Beltrán, Peter Bürgisser, Felipe Cucker, and Michael Shub. A stable, polynomial-time algorithm for the eigenpair problem. 2015. arXiv:1505.03290.
  19. 19.
    Diego Armentano and Felipe Cucker. A randomized homotopy for the Hermitian eigenpair problem. In: Found. Comput. Math. 15.1 (2015), pp. 281–312. doi:10.1007/s10208-014-9217-9.
  20. 20.
    William Kahan. Accurate eigenvalues of a symmetric tri-diagonal matrix. Tech. rep. CS41. Stanford University, 1966.Google Scholar
  21. 21.
    Walter Baur and Volker Strassen. The complexity of partial derivatives. In: Theoretical Computer Science 22.3 (1983), pp. 317 –330. doi:10.1016/0304-3975(83)90110-X.
  22. 22.
    Diego Armentano, Carlos Beltrán, Peter Bürgisser, Felipe Cucker, and Michael Shub. Condition length and complexity for the solution of polynomial systems. In: Found. Comput. Math. (2016). doi:10.1007/s10208-016-9309-9.
  23. 23.
    Carlos Beltrán. A continuation method to solve polynomial systems and its complexity. In: Numer. Math. 117.1 (2011), pp. 89–113. doi:10.1007/s00211-010-0334-3.

Copyright information

© SFoCM 2016

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

Personalised recommendations