The time-ordered exponential is defined as the function that solves any system of coupled first o... more The time-ordered exponential is defined as the function that solves any system of coupled first order linear differential equations with constant or non-constant coefficients. In spite of it being at the heart of much system dynamics, control theory and model reduction problems, the time-ordered exponential function remains elusively difficult to evaluate. Here we present a Lanczos-like algorithm capable of evaluating it by producing a tridiagonalization of the original differential system. The algorithm is presented in a theoretical setting. Nevertheless, a natural strategy for its numerical implementation is also outlined and will form the basis of a future work.
Under what conditions can one tridiagonalize a system of coupled linear differential equations wi... more Under what conditions can one tridiagonalize a system of coupled linear differential equations with non-constant coefficients by appealing only to 'well-behaved' distributions? In this work we show that if all the non-constant coefficients are smooth functions then tridiagonalisation is always possible using only piecewise smooth functions and isolated Dirac delta distributions. As a corollary, we formally establish the convergence and good behavior of the recently published Lanczos-like algorithm for solving arbitrary linear differential systems with smooth coefficients via tridiagonalization. This is a key piece in evaluating the hitherto elusively difficult ordered exponential function, both formally and numerically.
The time-ordered exponential is defined as the function that solves any system of coupled first o... more The time-ordered exponential is defined as the function that solves any system of coupled first order linear differential equations with constant or non-constant coefficients. In spite of it being at the heart of much system dynamics, control theory and model reduction problems, the time-ordered exponential function remains elusively difficult to evaluate. Here we present a Lanczos-like algorithm capable of evaluating it by producing a tridiagonalization of the original differential system. The algorithm is presented in a theoretical setting. Nevertheless, a natural strategy for its numerical implementation is also outlined and will form the basis of a future work.
Under what conditions can one tridiagonalize a system of coupled linear differential equations wi... more Under what conditions can one tridiagonalize a system of coupled linear differential equations with non-constant coefficients by appealing only to 'well-behaved' distributions? In this work we show that if all the non-constant coefficients are smooth functions then tridiagonalisation is always possible using only piecewise smooth functions and isolated Dirac delta distributions. As a corollary, we formally establish the convergence and good behavior of the recently published Lanczos-like algorithm for solving arbitrary linear differential systems with smooth coefficients via tridiagonalization. This is a key piece in evaluating the hitherto elusively difficult ordered exponential function, both formally and numerically.
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