A Lanczos-like method for non-autonomous linear ordinary differential equations
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 exponenti...
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| Published in | Bollettino della Unione matematica italiana (2008) Vol. 16; no. 1; pp. 81 - 102 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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Springer International Publishing
01.03.2023
Springer Nature B.V |
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| ISSN | 1972-6724 2198-2759 |
| DOI | 10.1007/s40574-022-00328-6 |
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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. |
|---|---|
| AbstractList | 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. 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. |
| Author | Giscard, Pierre-Louis Pozza, Stefano |
| Author_xml | – sequence: 1 givenname: Pierre-Louis surname: Giscard fullname: Giscard, Pierre-Louis organization: Univ. Littoral Côte d’Opale, UR 2597, LMPA, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville – sequence: 2 givenname: Stefano orcidid: 0000-0003-1529-8420 surname: Pozza fullname: Pozza, Stefano email: pozza@karlin.mff.cuni.cz organization: Faculty of Mathematics and Physics, Charles University |
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| Keywords | Tridiagonal matrices Matrix differential equations Ordinary differential equations Lanczos algorithm Matching moments Time-ordered exponential |
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| SubjectTerms | Algorithms Control theory Differential equations Exponential functions Mathematics Mathematics and Statistics Model reduction Subspaces System dynamics |
| Title | A Lanczos-like method for non-autonomous linear ordinary differential equations |
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