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Lanczos-like algorithm for the time-ordered exponential: The * -inverse problem

Pierre-Louis Giscard;  Stefano Pozza — 2020

Applications of Mathematics

The time-ordered exponential of a time-dependent matrix 𝖠 ( t ) is defined as the function of 𝖠 ( t ) that solves the first-order system of coupled linear differential equations with non-constant coefficients encoded in 𝖠 ( t ) . The authors have recently proposed the first Lanczos-like algorithm capable of evaluating this function. This algorithm relies on inverses of time-dependent functions with respect to a non-commutative convolution-like product, denoted by * . Yet, the existence of such inverses, crucial to...

The ☆ -product approach for linear ODEs: A numerical study of the scalar case

Pozza, Stefano;  Van Buggenhout, Niel — 2023

Programs and Algorithms of Numerical Mathematics

Solving systems of non-autonomous ordinary differential equations (ODE) is a crucial and often challenging problem. Recently a new approach was introduced based on a generalization of the Volterra composition. In this work, we explain the main ideas at the core of this approach in the simpler setting of a scalar ODE. Understanding the scalar case is fundamental since the method can be straightforwardly extended to the more challenging problem of systems of ODEs. Numerical examples illustrate the...

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