Abstract
In this paper we consider the computational complexity of solving initial-value problems defined with analytic ordinary differential equations (ODEs) over unbounded domains of ℝn and ℂn, under the Computable Analysis setting. We show that the solution can be computed in polynomial time over its maximal interval of definition, provided it satisfies a very generous bound on its growth, and that the function admits an analytic extension to the complex plane.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ko, K.I.: Computational Complexity of Real Functions. Birkhäuser, Basel (1991)
Shannon, C.E.: Mathematical theory of the differential analyzer. J. Math. Phys. 20, 337–354 (1941)
Bush, V.: The differential analyzer. A new machine for solving differential equations. J. Franklin Inst. 212, 447–488 (1931)
Graça, D.S., Costa, J.F.: Analog computers and recursive functions over the reals. J. Complexity 19(5), 644–664 (2003)
Graça, D., Zhong, N., Buescu, J.: Computability, noncomputability and undecidability of maximal intervals of IVPs. Trans. Amer. Math. Soc. 361(6), 2913–2927 (2009)
Collins, P., Graça, D.S.: Effective computability of solutions of differential inclusions — the ten thousand monkeys approach. Journal of Universal Computer Science 15(6), 1162–1185 (2009)
Pour-El, M.B., Richards, J.I.: A computable ordinary differential equation which possesses no computable solution. Ann. Math. Logic 17, 61–90 (1979)
Demailly, J.-P.: Analyse Numérique et Equations Différentielles. Presses Universitaires de Grenoble (1991)
Smith, W.D.: Church’s thesis meets the N-body problem. Applied Mathematics and Computation 178(1), 154–183 (2006)
Perko, L.: Differential Equations and Dynamical Systems, 3rd edn. Springer, Heidelberg (2001)
Ruohonen, K.: An effective Cauchy-Peano existence theorem for unique solutions. Internat. J. Found. Comput. Sci. 7(2), 151–160 (1996)
Kawamura, A.: Lipschitz continuous ordinary differential equations are polynomial-space complete. In: 2009 24th Annual IEEE Conference on Computational Complexity, pp. 149–160. IEEE, Los Alamitos (2009)
Müller, N., Moiske, B.: Solving initial value problems in polynomial time. In: Proc. 22 JAIIO - PANEL 1993, Part 2, pp. 283–293 (1993)
Müller, N.T., Korovina, M.V.: Making big steps in trajectories. Electr. Proc. Theoret. Comput. Sci. 24, 106–119 (2010)
Birkhoff, G., Rota, G.C.: Ordinary Differential Equations, 4th edn. John Wiley, Chichester (1989)
Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)
Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proc. London Math. Soc. (Ser.2–42), 230–265 (1936)
Grzegorczyk, A.: On the definitions of computable real continuous functions. Fund. Math. 44, 61–71 (1957)
Lacombe, D.: Extension de la notion de fonction récursive aux fonctions d’une ou plusieurs variables réelles III. C. R. Acad. Sci. Paris 241, 151–153 (1955)
Weihrauch, K.: Computable Analysis: an Introduction. Springer, Heidelberg (2000)
Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Springer, Heidelberg (1989)
Ko, K.I., Friedman, H.: Computational complexity of real functions. Theoret. Comput. Sci. 20, 323–352 (1982)
Müller, N.T.: Uniform computational complexity of taylor series. In: Ottmann, T. (ed.) ICALP 1987. LNCS, vol. 267, pp. 435–444. Springer, Heidelberg (1987)
Graça, D.S., Campagnolo, M.L., Buescu, J.: Computability with polynomial differential equations. Adv. Appl. Math. 40(3), 330–349 (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag GmbH Berlin Heidelberg
About this paper
Cite this paper
Bournez, O., Graça, D.S., Pouly, A. (2011). Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains. In: Murlak, F., Sankowski, P. (eds) Mathematical Foundations of Computer Science 2011. MFCS 2011. Lecture Notes in Computer Science, vol 6907. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22993-0_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-22993-0_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22992-3
Online ISBN: 978-3-642-22993-0
eBook Packages: Computer ScienceComputer Science (R0)