Abstract
Many ordinary differential equations (ODEs) do not have a closed solution, therefore approximating them is an important problem in numerical analysis. This work formalizes a method to approximate solutions of ODEs in Isabelle/HOL.
We formalize initial value problems (IVPs) of ODEs and prove the existence of a unique solution, i.e. the Picard-Lindelöf theorem. We introduce generic one-step methods for numerical approximation of the solution and provide an analysis regarding the local and global error of one-step methods.
We give an executable specification of the Euler method as an instance of one-step methods. With user-supplied proofs for bounds of the differential equation we can prove an explicit bound for the global error. We use arbitrary-precision floating-point numbers and also handle rounding errors when we truncate the numbers for efficiency reasons.
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
Boldo, S., Clément, F., Filliâtre, J.-C., Mayero, M., Melquiond, G., Weis, P.: Formal Proof of a Wave Equation Resolution Scheme: The Method Error. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 147–162. Springer, Heidelberg (2010)
Bornemann, V., Deuflhard, P.: Scientific computing with ordinary differential equations (2002)
Haftmann, F., Nipkow, T.: Code Generation via Higher-Order Rewrite Systems. In: Blume, M., Kobayashi, N., Vidal, G. (eds.) FLOPS 2010. LNCS, vol. 6009, pp. 103–117. Springer, Heidelberg (2010)
Haftmann, F., Wenzel, M.: Local Theory Specifications in Isabelle/Isar. In: Berardi, S., Damiani, F., de’Liguoro, U. (eds.) TYPES 2008. LNCS, vol. 5497, pp. 153–168. Springer, Heidelberg (2009)
Harrison, J.: Theorem Proving with the Real Numbers. Ph.D. thesis (1996)
Harrison, J.: A HOL Theory of Euclidean Space. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 114–129. Springer, Heidelberg (2005)
Hölzl, J.: Proving inequalities over reals with computation in Isabelle/HOL. In: Reis, G.D., Théry, L. (eds.) Programming Languages for Mechanized Mathematics Systems (ACM SIGSAM PLMMS 2009), pp. 38–45 (2009)
Immler, F., Hölzl, J.: Ordinary Differential Equations. Archive of Formal Proofs (April 2012), http://afp.sf.net/entries/Ordinary_Differential_Equations.shtml , Formal proof development
Krebbers, R., Spitters, B.: Computer Certified Efficient Exact Reals in COQ. In: Davenport, J.H., Farmer, W.M., Urban, J., Rabe, F. (eds.) MKM 2011 and Calculemus 2011. LNCS (LNAI), vol. 6824, pp. 90–106. Springer, Heidelberg (2011)
Krebbers, R., Spitters, B.: Type classes for efficient exact real arithmetic in COQ. CoRR abs/1106.3448 (2011)
Melquiond, G.: Proving Bounds on Real-Valued Functions with Computations. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 2–17. Springer, Heidelberg (2008)
Muñoz, C., Lester, D.R.: Real Number Calculations and Theorem Proving. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 195–210. Springer, Heidelberg (2005)
Obua, S.: Flyspeck II: The Basic Linear Programs. Ph.D. thesis, München (2008)
O’Connor, R., Spitters, B.: A computer verified, monadic, functional implementation of the integral. Theoretical Computer Science 411(37), 3386–3402 (2010)
O’Connor, R.: Certified Exact Transcendental Real Number Computation in COQ. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 246–261. Springer, Heidelberg (2008)
Reinhardt, H.J.: Numerik gewöhnlicher Differentialgleichungen. de Gruyter (2008)
Spitters, B.: Numerical integration in COQ, Mathematics, Algorithms, and Proofs (MAP 2010) (November 2010), www.unirioja.es/dptos/dmc/MAP2010/Slides/Slides/talkSpittersMAP2010.pdf
Walter, W.: Ordinary Differential Equations, 1st edn. Springer (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Immler, F., Hölzl, J. (2012). Numerical Analysis of Ordinary Differential Equations in Isabelle/HOL. In: Beringer, L., Felty, A. (eds) Interactive Theorem Proving. ITP 2012. Lecture Notes in Computer Science, vol 7406. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32347-8_26
Download citation
DOI: https://doi.org/10.1007/978-3-642-32347-8_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32346-1
Online ISBN: 978-3-642-32347-8
eBook Packages: Computer ScienceComputer Science (R0)