Numerical Analysis of Ordinary Differential Equations in Isabelle/HOL

Abstract. Ordinary Differential Equations (ODEs) are ubiquitous in physical applications of mathematics. The Picard-Lindelöf theorem is the first fundamental theorem in the theory of ODEs. It allows one to solve differential equations numerically. We provide a constructive development of the Picard-Lindelöf theorem which includes a program together with sufficient conditions for its correctness. The proof/program is written in the Coq proof assistant and uses the implementation of efficient real numbers from the CoRN library and the MathClasses library. Our proof makes heavy use of operators and functionals, functions on spaces of functions. This is faithful to the usual mathematical description, but a novel level of abstraction for certified exact real computation.

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“…We mention related work by Immler and Hölzl in Isabelle [11]. They go further in that they also implement the Euler method.…”

Section: Resultsmentioning

confidence: 99%

The Picard Algorithm for Ordinary Differential Equations in Coq

Makarov

Spitters

2013

Interactive Theorem Proving

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“…We are working towards a verified implementation of a simple ODE-solver in Coq. We mention related work by Immler and Hölzl in Isabelle [11]. They go further in that they also implement the Euler method.…”

Section: Resultsmentioning

confidence: 99%

Interactive Theorem Proving

2013

Lecture Notes in Computer Science

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Ordinary Differential Equations (ODEs) are ubiquitous in physical applications of mathematics. The Picard-Lindelöf theorem is the first fundamental theorem in the theory of ODEs. It allows one to solve differential equations numerically. We provide a constructive development of the Picard-Lindelöf theorem which includes a program together with sufficient conditions for its correctness. The proof/program is written in the Coq proof assistant and uses the implementation of efficient real numbers from the CoRN library and the MathClasses library. Our proof makes heavy use of operators and functionals, functions on spaces of functions. This is faithful to the usual mathematical description, but a novel level of abstraction for certified exact real computation.

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“…Despite these methods produce rigorous results, in case of numerical integration methods based on Taylor series [5,32] or based on Runge-Kutta methods [10,2], the distinction between round-off errors and method errors is not considered, and so no information on the propagation of round-off errors is provided nor considered. Note that an implementation of the approach of [10] has been formally proved in [25] but without giving much attention on the propagation on round-off errors as they rely on multi-precision. A special case is the work [35] in which a more precise study on the propagation of roundoff errors in the numerical integration method is considered in order to reduce the pessimism of interval computations.…”

Section: Related Workmentioning

confidence: 99%

Round-Off Error and Exceptional Behavior Analysis of Explicit Runge-Kutta Methods

Boldo

Faissole

Chapoutot

2020

IEEE Trans. Comput.

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Numerical integration schemes are mandatory to understand complex behaviors of dynamical systems described by ordinary differential equations. Implementation of these numerical methods involve floating-point computations and propagation of round-off errors. This paper presents a new fine-grained analysis of round-off errors in explicit Runge-Kutta integration methods, taking into account exceptional behaviors, such as underflow and overflow. Linear stability properties play a central role in the proposed approach. For a large class of Runge-Kutta methods applied on linear problems, a tight bound of the round-off errors is provided. A simple test is defined and ensures the absence of underflow and a tighter round-off error bound. The absence of overflow is guaranteed as linear stability properties imply that (computed) solutions are non-increasing.

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Numerical Analysis of Ordinary Differential Equations in Isabelle/HOL

Cited by 38 publications

References 12 publications

The Picard Algorithm for Ordinary Differential Equations in Coq

The Picard Algorithm for Ordinary Differential Equations in Coq

Interactive Theorem Proving

Round-Off Error and Exceptional Behavior Analysis of Explicit Runge-Kutta Methods

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