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Optimal residuals and the Dahlquist test problem

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Abstract

We show how to compute the optimal relative backward error for the numerical solution of the Dahlquist test problem by one-step methods. This is an example of a general approach that uses results from optimal control theory to compute optimal residuals, but elementary methods can also be used here because the problem is so simple. This analysis produces some new insights into the numerical solution of stiff problems.

The optimal backward error for the Dahlquist test problem when the midpoint method is used. Computed via a Padé approximant.

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Acknowledgements

Thanks are also due to the Rotman Institute of Philosophy, the Fields Institute for Research in the Mathematical Sciences, the Ontario Research Center for Computer Algebra and the University of South Australia which supported a visit of the first author to the second. We are grateful for detailed critical remarks by the referees, which helped us to improve the paper, and for their pointing out the relevant works of Higham, Calvo, Montijano and Rández. Special thanks are owed to John C. Butcher for his hosting the beautiful ANODE conferences over the years, and his long-term support of research in the Runge-Kutta world.

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Part of this work was supported by the Natural Sciences and Engineering Research Council of Canada.

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Correspondence to Robert M. Corless.

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Corless, R.M., Kaya, C.Y. & Moir, R.H.C. Optimal residuals and the Dahlquist test problem. Numer Algor 81, 1253–1274 (2019). https://doi.org/10.1007/s11075-018-0624-x

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  • DOI: https://doi.org/10.1007/s11075-018-0624-x

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