$L$-derivative of an approximate solution to $\Phi ’=\mathbf {A}(t)\Phi$: series and product formulae for left corrections

Given an initial approximation ${\Phi _0}$ to the fundamental matrix of solutions for $\Phi ’ = \textrm {A}\left ( t \right )\Phi$, it is shown that a left correction, $\Gamma {\Phi _0}$, is locally more accurate than a right correction, ${\Phi _0}\Gamma$. For each relative error function considered, there is a left correction $\Gamma$ and the associated differential equation. The common feature is the same integrable part whose forcing function is the difference between L-derivatives of the exact and the initial solution.