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A Lanczos-like method for non-autonomous linear ordinary differential equations

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Abstract

The time-ordered exponential is defined as the function that solves a system of coupled first-order linear differential equations with generally non-constant coefficients. In spite of being at the heart of much system dynamics, control theory, and model reduction problems, the time-ordered exponential function remains elusively difficult to evaluate. The \(*\)-Lanczos algorithm is a (symbolic) algorithm capable of evaluating it by producing a tridiagonalization of the original differential system. In this paper, we explain how the \(*\)-Lanczos algorithm is built from a generalization of Krylov subspaces, and we prove crucial properties, such as the matching moment property. A strategy for its numerical implementation is also outlined and will be subject of future investigation.

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Notes

  1. The \(\#\)P class is the class of problems equivalent to counting the number of accepting paths of a polynomial-time non-deterministic Turing machine. Problems in \(\#\)P-complete are at least as hard as NP-complete problems.

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Acknowledgements

We thank Francesca Arrigo, Des Higham, Jennifer Pestana, and Francesco Tudisco for their invitation to the University of Strathclyde, without which this work would not have come to fruition. The first author was supported in part by 2019 Alcohol project ANR-19-CE40-0006 and 2020 Magica project ANR-20-CE29-0007. The second author was supported by Charles University Research programs No. PRIMUS/21/SCI/009 and UNCE/SCI/023.

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  1. These authors contributed equally to this work.

Authors and Affiliations

  1. Univ. Littoral Côte d’Opale, UR 2597, LMPA, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville, F-62100, Calais, France

    Pierre-Louis Giscard

  2. Faculty of Mathematics and Physics, Charles University, Prague 8, Czech Republic

    Stefano Pozza

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  1. Pierre-Louis Giscard
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  2. Stefano Pozza
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Correspondence to Stefano Pozza.

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Giscard, PL., Pozza, S. A Lanczos-like method for non-autonomous linear ordinary differential equations. Boll Unione Mat Ital 16, 81–102 (2023). https://doi.org/10.1007/s40574-022-00328-6

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