andydude Wrote:I'm still confused as to why these are different. It seems to me that "dsexp" and "rsexp" should be identical everywhere,
No, absolutely not. Though you can regard rsexp as diagonalization method, the difference is *where* you apply it.
dsexp is applied to 0, while rsexp is applied to the lower fixed point.
It is not clear at all that the result of the diagonalization method does not depend on the development point. In the same way as it is not clear that the natural Abel method does not depend on the development point.
If you apply the diagonalization method to the fixed point you get nice triangular matrices with the powers of \( f'(a) \) on the diagonal (the eigenvalues). However if you apply it to 0, you neither have triangular matrices nor are the eigenvalues powers of something (edit: though I should check the latter before asserting).
You can always apply the diagonalization method to analytic functions, also if they have no real fixed point (e.g. \( e^x \)).
Also I dont know of a way yet how to compute the result of the diagonalization method (at a non-fixed point) without powerseries coefficients, i.e. with an iterative formula like there is for the regular iteration.