I'll add two links from my work which might be specifically helpful.
One essay on Andy's slog, matrix and matrix-inverse taken by LR-decomposition makes better informed decision possible, how the coefficients of the solution-power series converge when matrixsize is increased
https://math.eretrandre.org/tetrationfor...hp?tid=251
One essay on the slog as comparision with the concept of indefinite summation, where I find an example, where the indefinite summation works to give a valid powerseries, and is shown that the matrix-method is closely related (if not identical!) to the Andy's /Walkers matrixmethod
https://math.eretrandre.org/tetrationfor...hp?tid=709
Two more links:
https://math.eretrandre.org/tetrationfor...74#pid4874
problem of reproducing the identy by (Bb-I)*SLOG =?= I0 this might indicate a principal problem of accuracy of matrix-method, even when dimension is increased/powerseries solution is prolonged
https://math.eretrandre.org/tetrationfor...hp?tid=179 "Exact entries" for the matrix-method by Andrew 13.3.2009. I discussed his found formula, but could not fix the relation with the "regular" method via Schröder function. Moreover, maybe the introduction of the integrals is in fact a *modification* of the Walker/Robbins-method, and not only an explication.
At the moment I feel unable to come over with a quick-shot to put this together; maybe I'll find time and concentration next days. For an introduction into that matrix-thinking I could offer a live-session using zoom where I show my matrix-analyses in action, so anyone who is interested should be able to analyze the above linked articles/essays/postings.
Gottfried
