Lanczos-like algorithm for the time-ordered exponential: The $\ast$-inverse problem

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. Here we present a Lanczoslike algorithm capable of evaluating it by producing a tridiagonalization of the original differential system. The algorithm is presented in a theoretical setting. Nevertheless, a strategy for its numerical implementation is also outlined and will be subject of future investigation.

Computations of general Heun functions from their integral series representations

“…This product makes D into a non-commutative algebra [10,11] with unit element 1 * ≡ δ(z ′ −z). Now consider the ensemble Sm Θ D of distributions of D for which all smooth coefficients fi (z ′ , z) = 0, i.e.…”

Tridiagonalization of systems of coupled linear differential equations with variable coefficients by a Lanczos-like method

“…Only three formal methods have been devised to calculate ordered exponentials analytically, only one of which is guaranteed to produce an exact answer in a finite number of steps. These are: the Floquet approach, applicable when A(t ′ ) is periodic and which produces an infinite perturbative expansion of the solution 1 usually too complicated to be evaluated beyond its first or second terms [5]; the Magnus series expansion [17], which presents the solution as the matrix exponential of an increasingly intricate infinite series of nested commutators plagued by incurable divergence issues 2 ; and the path-sum approach, which expresses the solution exactly as a finite fraction [8,7] but requires solving an NP-hard problem [10,12].…”

“…Recently P.-L. G. and S. P. proposed a constructive method to tridiagonalize systems of linear differential equations with non-constant coefficients [10,9], from which one can easily evaluate w H U(t ′ , t)v for any two vectors w, v with w H v = 1. Here w H denotes the Hermitian transpose of w. Under the assumptions that the tridiagonalized system coefficients are "well-behaved" distributions (to be made precise below) and that the method does not breakdown, this approach -a Lanczos-like algorithmis able to produce the tridiagonalization.…”

“…The * -product extends directly to distributions for which the smooth coefficients are matrices of smooth coefficients by using the ordinary matrix product between the integrands in (1.2) (see [10] for more details). It is also well defined for distributions of D whose smooth coefficients depend on less than two variables.…”

“…the series on the right-hand side converging when A elements are bounded. Now we can recall results in [10]: baring breakdowns, the * -Lanczos algorithm reproduced here in Table 1 produces a sequence of tridiagonal matrices T n , 1 ≤ n ≤ N , of the form…”