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Parameterized Complexity for Uniform Operators on Multidimensional Analytic Functions and ODE Solving

  • Akitoshi Kawamura
  • Florian Steinberg
  • Holger Thies
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10944)

Abstract

Real complexity theory is a resource-bounded refinement of computable analysis and provides a realistic notion of running time of computations over real numbers, sequences, and functions by relying on Turing machines to handle approximations of arbitrary but guaranteed absolute error. Classical results in real complexity show that important numerical operators can map polynomial time computable functions to functions that are hard for some higher complexity class like \(\mathsf {NP}\) or \(\mathsf {\# P}\). Restricted to analytic functions, however, those operators map polynomial time computable functions again to polynomial time computable functions. Recent work by Kawamura, Müller, Rösnick and Ziegler discusses how to extend this to uniform algorithms on one-dimensional analytic functions over simple compact domains using second-order and parameterized complexity. In this paper, we extend some of their results to the case of multidimensional analytic functions. We further use this to show that the operator mapping an analytic ordinary differential equations to its solution is computable in parameterized polynomial time. Finally, we discuss how the theory can be used as a basis for verified exact numerical computation with analytic functions and provide a prototypical implementation in the iRRAM C++ framework for exact real arithmetic.

Notes

Acknowledgements

This work was supported by JSPS KAKENHI Grant Numbers JP18H03203 and JP18J10407 and by the Japan Society for the Promotion of Science (JSPS), Core-to-Core Program (A. Advanced Research Networks).

References

  1. 1.
    Boehm, H.J., Cartwright, R., Riggle, M., O’Donnell, M.J.: Exact real arithmetic: a case study in higher order programming. In: Proceedings of the 1986 ACM Conference on LISP and Functional Programming, pp. 162–173. ACM (1986)Google Scholar
  2. 2.
    Bournez, O., Graça, D.S., Pouly, A.: On the complexity of solving initial value problems. In: ISSAC 2012-Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation, pp. 115–121. ACM, New York (2012).  https://doi.org/10.1145/2442829.2442849
  3. 3.
    Brattka, V., Hertling, P., Weihrauch, K.: A tutorial on computable analysis. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) New Computational Paradigms: Changing Conceptions of What is Computable, pp. 425–491. Springer, New York (2008).  https://doi.org/10.1007/978-0-387-68546-5_18CrossRefzbMATHGoogle Scholar
  4. 4.
    Brent, R.P., Kung, H.T.: Fast algorithms for manipulating formal power series. J. ACM (JACM) 25(4), 581–595 (1978)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chang, Y., Corliss, G.: ATOMFT: solving ODEs and DAEs using Taylor series. Comput. Math. Appl. 28(10–12), 209–233 (1994)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Friedman, H.: The computational complexity of maximization and integration. Adv. Math. 53(1), 80–98 (1984)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Geuvers, H., Niqui, M., Spitters, B., Wiedijk, F.: Constructive analysis, types and exact real numbers. Math. Struct. Comput. Sci. 17(1), 3–36 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    van der Hoeven, J.: Relax, but don’t be too lazy. J. Symb. Comput. 34(6), 479–542 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    van der Hoeven, J.: On effective analytic continuation. Math. Comput. Sci. 1, 111–175 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kawamura, A.: Lipschitz continuous ordinary differential equations are polynomial-space complete. Comput. Complex. 19(2), 305–332 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kawamura, A., Cook, S.: Complexity theory for operators in analysis. ACM Trans. Comput. Theory 4(2), 5:1–5:24 (2012)CrossRefGoogle Scholar
  12. 12.
    Kawamura, A., Müller, N., Rösnick, C., Ziegler, M.: Computational benefit of smoothness: parameterized bit-complexity of numerical operators on analytic functions and Gevrey’s hierarchy. J. Complex. 31(5), 689–714 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ko, K.I.: On the computational complexity of ordinary differential equations. Inf. Control 58(1–3), 157–194 (1983)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ko, K.I.: Complexity theory of real functions: Progress in Theoretical Computer Science. Birkhäuser Boston Inc., Boston (1991)CrossRefGoogle Scholar
  15. 15.
    Ko, K.I., Friedman, H.: Computational complexity of real functions. Theor. Comput. Sci. 20(3), 323–352 (1982)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ko, K.I., Friedman, H.: Computing power series in polynomial time. Adv. Appl. Math. 9(1), 40–50 (1988)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Moiske, B., Müller, N.: Solving initial value problems in polynomial time. In: Proceedings of the 22th JAIIO-PANEL, vol. 93, pp. 283–293 (1993)Google Scholar
  18. 18.
    Müller, N.T.: Uniform computational complexity of Taylor series. In: Ottmann, T. (ed.) ICALP 1987. LNCS, vol. 267, pp. 435–444. Springer, Heidelberg (1987).  https://doi.org/10.1007/3-540-18088-5_37CrossRefGoogle Scholar
  19. 19.
    Müller, N.T.: Constructive aspects of analytic functions. In: Proceedings of Workshop on Computability and Complexity in Analysis, InformatikBerichte, vol. 190, pp. 105–114. FernUniversität Hagen (1995)Google Scholar
  20. 20.
    Müller, N.T.: The iRRAM: exact arithmetic in C++. In: Blanck, J., Brattka, V., Hertling, P. (eds.) CCA 2000. LNCS, vol. 2064, pp. 222–252. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-45335-0_14CrossRefGoogle Scholar
  21. 21.
    Pour-El, M.B., Richards, J.I.: Computability in analysis and physics: Perspectives in Mathematical Logic. Springer-Verlag, Berlin (1989)CrossRefGoogle Scholar
  22. 22.
    Turing, A.: On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc. 2(42), 230–265 (1936).  https://doi.org/10.1112/plms/s2-42.1.230MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Weihrauch, K.: Computable Analysis. Springer, Heidelberg (2000).  https://doi.org/10.1007/978-3-642-56999-9CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Akitoshi Kawamura
    • 1
  • Florian Steinberg
    • 2
  • Holger Thies
    • 3
  1. 1.Kyushu UniversityFukuokaJapan
  2. 2.InriaRocquencourtFrance
  3. 3.University of TokyoTokyoJapan

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