Polynomial-Time Solution of Initial Value Problems Using Polynomial Enclosures

  • Amin Farjudian
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7456)


Domain theory has been used with great success in providing a semantic framework for Turing computability, over both discrete and continuous spaces. On the other hand, classical approximation theory provides a rich set of tools for computations over real functions with (mainly) polynomial and rational function approximations.

We present a semantic model for computations over real functions based on polynomial enclosures. As an important case study, we analyse the convergence and complexity of Picard’s method of initial value problem solving in our framework. We obtain a geometric rate of convergence over Lipschitz fields and then, by using Chebyshev truncations, we modify Picard’s algorithm into one which runs in polynomial-time over a set of polynomial-space representable fields, thus achieving a reduction in complexity which would be impossible in the step-function based domain models.


computable analysis computational complexity initial value problem Picard’s method approximation theory Chebyshev series 


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  1. 1.
    Aberth, O.: The failure in computable analysis of a classical existence theorem for differential equations. Proceedings of the American Mathematical Society 30(1), 151–156 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. 3, pp. 1–168. Clarendon Press, Oxford (1994)Google Scholar
  3. 3.
    Achieser, N.I.: Theory of Approximation. Frederick Ungar Publishing Co. (1956)Google Scholar
  4. 4.
    Bernstein, S.N.: Sur les recherches récentes relatives à la meilleure approximation des fonctions continues par les polynômes. In: Proc. of 5th Inter. Math. Congress, vol. 1, pp. 256–266 (1912)Google Scholar
  5. 5.
    Bernstein, S.N.: Sur l’ordre de la meilleure approximation des fonctions continues par les polynômes de degré donné. Mem. Cl. Sci. Acad. Roy. Belg. 4, 1–103 (1912)Google Scholar
  6. 6.
    Bishop, E.: Foundations of Constructive Analysis. McGraw-Hill (1967)Google Scholar
  7. 7.
    Bournez, O., Graça, D.S., Pouly, A.: Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 170–181. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    Edalat, A.: Dynamical systems, measures and fractals via domain theory. Information and Computation 120(1), 32–48 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Edalat, A., Lieutier, A.: Domain theory and differential calculus (functions of one variable). In: Proceedings of 17th Annual IEEE Symposium on Logic in Computer Science (LICS 2002), Copenhagen, Denmark, pp. 277–286 (2002)Google Scholar
  10. 10.
    Edalat, A., Pattinson, D.: A domain-theoretic account of Picard’s theorem. LMS Journal of Computation and Mathematics 10, 83–118 (2007)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Escardó, M.H.: PCF extended with real numbers: a domain theoretic approach to higher order exact real number computation. Ph.D. thesis, Imperial College (1997)Google Scholar
  12. 12.
    Farjudian, A., Konečný, M.: Time Complexity and Convergence Analysis of Domain Theoretic Picard Method. In: Hodges, W., de Queiroz, R. (eds.) WoLLIC 2008. LNCS (LNAI), vol. 5110, pp. 149–163. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  13. 13.
    Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M.W., Scott, D.S.: Continuous Lattices and Domains. Encycloedia of Mathematics and its Applications, vol. 93. Cambridge University Press (2003)Google Scholar
  14. 14.
    Gutknecht, M.H., Trefethen, L.N.: Real polynomial Chebyshev approximation by the Carathéodory-Fejér method. SIAM Journal on Numerical Analysis 19(2), 358–371 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Jackson, D.: Über die Genauigkeit der Annäherung stetiger Funktionen durch ganze rationale Funktionen gegebenen Grades und trigonometrische Summen gegebener Ordnung. Ph.D. thesis, Göttingen (1911)Google Scholar
  16. 16.
    Kawamura, A.: Lipschitz Continuous Ordinary Differential Equations are Polynomial-Space Complete. In: CCC 2009: 24th Annual IEEE Conference on Computational Complexity, pp. 149–160 (2009)Google Scholar
  17. 17.
    Ko, K.I.: Complexity Theory of Real Functions. Birkhäuser, Boston (1991)zbMATHCrossRefGoogle Scholar
  18. 18.
    Konečný, M., Farjudian, A.: Compositional semantics of dataflow networks with query-driven communication of exact values. Journal of Universal Computer Science 16(18), 2629–2656 (2010)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Konečný, M., Farjudian, A.: Semantics of query-driven communication of exact values. Journal of Universal Computer Science 16(18), 2597–2628 (2010)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Lorentz, G.G.: Approximation of Functions. AMS Chelsea Publishing (1986)Google Scholar
  21. 21.
    Mhaskar, H.N., Pai, D.V.: Fundamentals of Approximation Theory. Narosa (2007)Google Scholar
  22. 22.
    Miller, R., Michel, A.: Ordinary Differential Equations. Academic Press (1982)Google Scholar
  23. 23.
    Müller, N., Moiske, B.: Solving initial value problems in polynomial time. In: Proc. 22nd JAIIO - PANEL 1993, Part 2, pp. 283–293 (1993)Google Scholar
  24. 24.
    Rivlin, T.J.: Chebyshev Polynomials: from Approximation Theory to Algebra and Number Theory, 2nd edn. Wiley, New York (1990)zbMATHGoogle Scholar
  25. 25.
    Weihrauch, K.: Computable Analysis, An Introduction. Springer (2000)Google Scholar
  26. 26.
    Yoshizawa, T., Hayashi, K.: On the uniqueness of solutions of a system of ordinary differential equations. Memoirs of the College of Science, University of Kyoto. Ser. A, Mathematics 26(1), 19–29 (1950)MathSciNetzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Amin Farjudian
    • 1
  1. 1.Division of Computer ScienceUniversity of Nottingham NingboChina

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