Convergence of matrix solution for base e

[jaydfox]

I could extend this logic up to, but not including, the radius of convergence, at which point, the total error for any finite truncation would in theory be infinite, making comparison...difficult

After further reflection, I think the infinity issue is only a problem for the finite natural solutions. The accelerated solutions will not suffer from this problem.

I have two reasons for coming to this conclusion.

The first is conceptual: the infinity arises from the fact that there is a singularity in the natural solution. However, after deriving the power series for the "residue" as I call it, by subtracting the logarithms from the accelerated solution, we are left with a removable singularity. I.e., the value of the "residue" at a primary fixed point is the difference of two infinities, so it is technically a singularity, but it has a distinct finite value in the limit, much like \( \frac{sin(z)}{z} \) evaluated at 0.

As such, all values of the residue along the radius of convergence are finite.

The second reason for me to come to this conclusion is numerical: testing with my 1200-term accelerated solution indicates that, even at the radius of convergence, the magnitude of the terms of the power series decreases as we go further and further into the series. The decrease is not quite exponential, but it at least appears that the error is bounded asymptotically on the radius of convergence. Outside the radius, I would of course expect divergence, but it would take at least tens of thousands of terms to notice it appreciably for points very close to the radius, and I only have 1200.

Anyway, with all this said, I think I can calculate the cumulative absolute error at the radius of convergence, and assuming I see a similar linear plot on the log-log graph, we should be able to come up with a very conservative upper bound for the error on the entire domain of the power series. But only for accelerated solutions. So all this presupposes that the accelerated and natural solutions converge on the same infinite solution.