Abstract
We analyze the convergence of quasi-Newton methods in exact and finite precision arithmetic. In particular, we derive an upper bound for the stagnation level and we show that any sufficiently exact quasi-Newton method will converge quadratically until stagnation. In the absence of sufficient accuracy, we are likely to retain rapid linear convergence. We confirm our analysis by computing square roots and solving bond constraint equations in the context of molecular dynamics. We briefly discuss implications for parallel solvers.
P. García-Risueño—Independent scholar.
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Acknowledgments
Prof. I. Argyros commented on an early draft of this paper and provided the reference to the work of I. P. Mysovskii. The first author is supported by eSSENCE, a collaborative e-Science programme funded by the Swedish Research Council within the framework of the strategic research areas designated by the Swedish Government. This work has been partially supported by the Spanish Ministry of Science and Innovation (contract PID2019-107255GB-C21/AEI/10.13039/501100011033), by the Generalitat de Catalunya (contract 2017-SGR-1328), and by Lenovo-BSC Contract-Framework Contract (2020).
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Kjelgaard Mikkelsen, C.C., López-Villellas, L., García-Risueño, P. (2023). How Accurate Does Newton Have to Be?. In: Wyrzykowski, R., Dongarra, J., Deelman, E., Karczewski, K. (eds) Parallel Processing and Applied Mathematics. PPAM 2022. Lecture Notes in Computer Science, vol 13826. Springer, Cham. https://doi.org/10.1007/978-3-031-30442-2_1
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