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Preprints, Working Papers, ... Year : 2024

Rounding Error Analysis of an Orbital Collision Probability Evaluation Algorithm

Florent Bréhard
Mioara Joldes
Marc Mezzarobba

Abstract

We present an error analysis of an algorithm due to Serra et al. (Journal of Guidance Control and Dynamics, 2016) for computing the orbital collision probability in the short term encounter model. The algorithm reduces the numerical computation of the collision probability to that of the sum of a series whose coecients are produced by a linear recurrence relation, and is specically designed to avoid cancellation issues in the evaluation of the sum. The authors derived a bound on the method error arising from the truncation of the series, and observed experimentally that the computation of the terms of the sum is numerically stable, but did not study the evaluation error. Here we give a rigorous bound on the accumulated rounding error when Serra et al.'s algorithm is implemented in oating-point arithmetic. For a unit roundo u and a truncation order N , the bound is of the form (N + A)u + o(u) where A is an explicit constant depending on the problem parameters and o(u) stands for explicitly bounded small terms compared to u. Our analysis is based on the observation that the generating series of the errors aecting each individual term is solution to a perturbed form of a dierential equation satised by the Laplace transform of a function related to the collision probability.
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Dates and versions

hal-04466875 , version 1 (19-02-2024)

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  • HAL Id : hal-04466875 , version 1

Cite

Denis Arzelier, Florent Bréhard, Mioara Joldes, Marc Mezzarobba. Rounding Error Analysis of an Orbital Collision Probability Evaluation Algorithm. 2024. ⟨hal-04466875⟩
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