A Magnus expansion for the equation \(Y^{\prime} = AY - YB\) (Q2708324)

A solution to the initial value problem defined by the linear ordinary differential system \((*)\) \(Y'= AY- YB\), \(Y(0)= Y_0\), with \(t\geq 0\), \(Y_0\) belonging to the set \(M_m\) of \(m\times m\)-matrices, is represented in the form \(Y(t)= e^{\Omega(t)}Y_0\) with \(\Omega\) being a generalization of the classical Magnus expansion [see \textit{W. Magnus}, Commun. Pure Appl. Math. 7, 649-673 (1954; Zbl 0056.34102); for details on the convergence and numerical computation of Magnus series, see \textit{A. Iserles} and \textit{S. P. Norsett}, Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 357, No. 1754, 983-1019 (1999; Zbl 0958.65080)]. This approach toward the solution to \((*)\) investigated by the author leads to a new practical algorithm which is interesting in some applications and other alternative approximation methods; in particular, as an easy means of computing the so-called Baker-Campbell-Hausdorff (bch) formula \(e^{tR} e^{tS}= e^{\text{bch}(t; R,S)}\) with \(R, S\in M_m\), \(|t|\) sufficiently small, and of its symmetric generalization.