Alan Mathison Turing is revered among computer scientists for laying down the foundations of theoretical computer science via the introduction of the Turing machine, an abstract model of computation upon which, an elegant notion of cost and a theory of complexity can be developed. In this paper we argue that the contribution of Turing to “the other side of computer science”, namely the domain of numerical computations as pioneered by Newton, Gauss, &c, and carried out today in the name of numerical analysis, is of an equally foundational nature.


Turing Machine Complexity Class Polynomial System Theoretical Computer Science Input Gate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Beltrán, C., Pardo, L.M.: Smale’s 17th problem: average polynomial time to compute affine and projective solutions. J. Amer. Math. Soc. 22(2), 363–385 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beltrán, C., Pardo, L.M.: Fast linear homotopy to find approximate zeros of polynomial systems. Found. Comput. Math. 11(1), 95–129 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, Heidelberg (1998)CrossRefzbMATHGoogle Scholar
  4. 4.
    Blum, L., Shub, M., Smale, S.: On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bulletin of the Amer. Math. Soc. 21, 1–46 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bürgisser, P., Cucker, F.: Condition. Forthcoming bookGoogle Scholar
  6. 6.
    Bürgisser, P., Cucker, F.: Exotic quantifiers, complexity classes, and complete problems. Found. Comput. Math. 9, 135–170 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bürgisser, P., Cucker, F.: On a problem posed by Steve Smale. In: To Appear at Annals of Mathematics (2011)Google Scholar
  8. 8.
    Cook, S.: The complexity of theorem proving procedures. In: 3rd Annual ACM Symp. on the Theory of Computing, pp. 151–158 (1971)Google Scholar
  9. 9.
    Edelman, A.: Eigenvalues and condition numbers of random matrices. SIAM J. of Matrix Anal. and Applic. 9, 543–556 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Edmonds, J.: Paths, trees, and flowers. Canadian J. of Mathematics 17, 449–467 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Goldstine, H.H.: A History of Numerical Analysis from the 16th through the 19th Century. Springer, Heidelberg (1977)CrossRefzbMATHGoogle Scholar
  12. 12.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press (1972)Google Scholar
  13. 13.
    Koiran, P.: The real dimension problem is NPIR-complete. J. of Complexity 15, 227–238 (1999)CrossRefzbMATHGoogle Scholar
  14. 14.
    Levin, L.: Universal sequential search problems. Probl. Pered. Inform., IX 3, 265–266 (1973) (in Russian); (English translation in Problems of Information Trans. 9,3; corrected translation in [18]) Google Scholar
  15. 15.
    Renegar, J.: Is it possible to know a problem instance is ill-posed? J. of Complexity 10, 1–56 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Smale, S.: Complexity theory and numerical analysis. In: Iserles, A. (ed.) Acta Numerica, pp. 523–551. Cambridge University Press (1997)Google Scholar
  17. 17.
    Smale, S.: Mathematical problems for the next century. In: Arnold, V., Atiyah, M., Lax, P., Mazur, B. (eds.) Mathematics: Frontiers and Perspectives, pp. 271–294. AMS (2000)Google Scholar
  18. 18.
    Trakhtenbrot, B.A.: A survey of russian approaches to perebor (brute-force search) algorithms. Annals of the History of Computing 6, 384–400 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proc. London Math. Soc., Ser. 2 42, 230–265 (1936)zbMATHGoogle Scholar
  20. 20.
    Turing, A.M.: Rounding-off errors in matrix processes. Quart. J. Mech. Appl. Math. 1, 287–308 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    von Neumann, J., Goldstine, H.H.: Numerical inverting matrices of high order. Bulletin of the Amer. Math. Soc. 53, 1021–1099 (1947)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    von Neumann, J., Goldstine, H.H.: Numerical inverting matrices of high order, II. Proceedings of the Amer. Math. Soc. 2, 188–202 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Wilkinson, J.: Some comments from a numerical analyst. Journal ACM 18, 137–147 (1971)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Felipe Cucker
    • 1
  1. 1.Department of MathematicsCity University of Hong KongHong KongP.R. of China

Personalised recommendations