I've tried doing something similar. I used Faa di Bruno's formula, which, if I understand correctly, is similar to using a Bell polynomial (I can even see the reference in the article you linked to).
However, the system is of non-linear equations (as you point out, and Andrew points this out in his paper). The sytem can be solved iteratively by plugging approximations back into the system, and it converges exponentially for the first 11 terms. When I tried to solve the 12th term (with 11 unknowns, the first term being known), the system was unstable. I was able to manually pick the 11th unknown and solve the 10 unknown system, but this required two sets of iterations.
Solving beyond this would appear to require manually tweaking all the terms beyond the 10th and then iterating, and this becomes infeasible beyond about three or four additional terms, due to the non-linearity. If someone has a spiffier method that come overcome the non-linearity, it might be doable, but I don't see it being possible to solve for, say, a 100x100 system. To be honest, I was blown away that the slog function was a simple linear system, given how chaotic the sexp function is. Hats off to Andrew for his approach.
However, the system is of non-linear equations (as you point out, and Andrew points this out in his paper). The sytem can be solved iteratively by plugging approximations back into the system, and it converges exponentially for the first 11 terms. When I tried to solve the 12th term (with 11 unknowns, the first term being known), the system was unstable. I was able to manually pick the 11th unknown and solve the 10 unknown system, but this required two sets of iterations.
Solving beyond this would appear to require manually tweaking all the terms beyond the 10th and then iterating, and this becomes infeasible beyond about three or four additional terms, due to the non-linearity. If someone has a spiffier method that come overcome the non-linearity, it might be doable, but I don't see it being possible to solve for, say, a 100x100 system. To be honest, I was blown away that the slog function was a simple linear system, given how chaotic the sexp function is. Hats off to Andrew for his approach.
~ Jay Daniel Fox
