Hmm, I think we crossed wires there. I was doing the Abel matrix for \( e^x \), centered at x=0, which works fine.
I then recentered the Abel matrix of \( e^x \) to x=1, which is not a fixed point and which is within the radius of convergence. It was at x=3 that things broke down, as this is well outside the radius of convergence.
For comparison's sake, I then then inverted the Bell matrix of \( e^x-1 \), because the inverse has a logarithmic singularity, like the solution to the Abel matrix. I then tried recentering to x=2 and inverting. This is on the radius of convergence, which would imply potential divergence, which is what I was observing.
However, we should be able to solve at points other than x=1, or so I thought. It took a little tinkering to realize that the problem was that I was both pre- and post-multiplying a lower triangular matrix by uppertriangular matrices, which "mixed up" the results sufficiently that the resulting matrix is only an approximation of the infinite system. Because it's an approximation, the coefficients in the right-most columns of a finite truncation are wildly inaccurate, causing the entire inversion to fail. However, a truncation of a finite truncation dampens the effect of the inaccuracies, which allowed me to see convergence.
Getting back to the Abel matrix for \( e^x \), the simple solution to the divergence problem is either to determine a way to calculate with coefficients of the infinite matrix without resorting to approximations, or to start with a system far larger than the system I intend to solve, and move the center in small steps, truncating along the way. Either way, steps smaller than the radius of convergence must be used.
I then recentered the Abel matrix of \( e^x \) to x=1, which is not a fixed point and which is within the radius of convergence. It was at x=3 that things broke down, as this is well outside the radius of convergence.
For comparison's sake, I then then inverted the Bell matrix of \( e^x-1 \), because the inverse has a logarithmic singularity, like the solution to the Abel matrix. I then tried recentering to x=2 and inverting. This is on the radius of convergence, which would imply potential divergence, which is what I was observing.
However, we should be able to solve at points other than x=1, or so I thought. It took a little tinkering to realize that the problem was that I was both pre- and post-multiplying a lower triangular matrix by uppertriangular matrices, which "mixed up" the results sufficiently that the resulting matrix is only an approximation of the infinite system. Because it's an approximation, the coefficients in the right-most columns of a finite truncation are wildly inaccurate, causing the entire inversion to fail. However, a truncation of a finite truncation dampens the effect of the inaccuracies, which allowed me to see convergence.
Getting back to the Abel matrix for \( e^x \), the simple solution to the divergence problem is either to determine a way to calculate with coefficients of the infinite matrix without resorting to approximations, or to start with a system far larger than the system I intend to solve, and move the center in small steps, truncating along the way. Either way, steps smaller than the radius of convergence must be used.
~ Jay Daniel Fox
