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 inBollettino della Unione matematica italiana (2008) Vol. 16; no. 1; pp. 81 - 102
Main Authors Giscard, Pierre-Louis, Pozza, Stefano
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.03.2023
Springer Nature B.V
Subjects
Algorithms
Control theory
Differential equations
Exponential functions
Mathematics
Mathematics and Statistics
Model reduction
Subspaces
System dynamics
Tridiagonal matrices
Matrix differential equations
Ordinary differential equations
Lanczos algorithm
Matching moments
Time-ordered exponential
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Summary: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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ISSN:1972-6724
2198-2759
DOI:10.1007/s40574-022-00328-6