Looking back at the article on the alternating iteration-series of exponential there was some confirmation for the matrix-based method missing. While I could use the serial summation (Abel- or Eulersummation of the explicite iterates) for the crosscheck of the matrix-method for the bases, where the powertower of infinite height converges, I could not do that for the other bases due to too fast growth of terms/iterated exponentials.
But well, if I take the (complex) fixpoint t as initial value, then the alternating series \( spa_b(t)= t-b^t+ b^{b^t}-...+... \) reduces to \( spa_b(t)= t-t+ t-...+... =t/2 \text{(Euler sum)} \) , which should be meaningful for each base, whether its exponential fixpoint is real or not.
With this I have now (at least) one check-value by serial summation for the comparision with the matrix-method.
The matrix-method, dimension 32x32, for instance for base e, which has a divergent iteration-series, comes out near the expected result \( spa_e(t)=t/2 \approx 0.159066 + 0.668618*I \) to three/four digits and the same was true for the conjugate of t. If the convergence could be accelerated, then this gives another confirmation for the applicability of this method for the iteration-series.
But well, if I take the (complex) fixpoint t as initial value, then the alternating series \( spa_b(t)= t-b^t+ b^{b^t}-...+... \) reduces to \( spa_b(t)= t-t+ t-...+... =t/2 \text{(Euler sum)} \) , which should be meaningful for each base, whether its exponential fixpoint is real or not.
With this I have now (at least) one check-value by serial summation for the comparision with the matrix-method.
The matrix-method, dimension 32x32, for instance for base e, which has a divergent iteration-series, comes out near the expected result \( spa_e(t)=t/2 \approx 0.159066 + 0.668618*I \) to three/four digits and the same was true for the conjugate of t. If the convergence could be accelerated, then this gives another confirmation for the applicability of this method for the iteration-series.
Gottfried Helms, Kassel