Gottfried Wrote:If the sequence of corresponding entries of the powers of (A-I)^n/n do not converge well with finite dimension, but at least alternate in sign, I try to approximate the sums \( s^{^{(\infty)}}_{i,k} \) of individual entries \( b^{^{(n)}}_{i,k} \) of the n'th powers of the matrix B=(A-I) by Euler-summation via partial sums, which is sometimes an option.
Well, I should add a remark concerning the appropriateness of the Euler-summation here.
Ideed I observed some cases, where Euler-summation did not satisfyingly well with the series, which occur in our cases. So to say, the "exotic" cases occured in fact. That was for instance at some difficult-cases of the (s^x-1)-iteration
Thus I tried other variants similar to the Euler-summation and possibly found a tool, which is more appropriate. While Euler-summation employs binomial-coefficients to "limit" a sequence of oscillating divergent partial sums, I tried to apply the Stirling-numbers 2'nd kind themselves, since the whole computation involves powers of a matrix containing just that numbers.
That seemed to be promising; I plotted a comparision of the two summation methods, where the theoretical limit was known.
To come near that limit I needed Euler-summation of very high order; very high means here of about the same order as the dimension of my matrices are. But that means: I have only dim=32 or dim=64 terms and Euler-summation of order 27 or 60 to get the partial sums to converge to a limit - but what after that number of terms/size of dimension? I learned, not to trust Euler-summation if it does not converge in, say the last 8 partial sums with orders of 3...7 when only 32 terms of a series are available. So I didn't rely my computations on such results at nasty parameters s (near the critical bounds).
If I preconditioned the occuring series with a transformation using Stirling-numbers of 2'nd kind, a much more clear convergence occured with an appended Euler-transform of low order like 3 or 4.
An example graph, with which I studied a special case with Euler-orders of about to 10, can be seen here:
Summation-Comparision. The situation is not so difficult for our usual cases, where the base-parameter s is in a "friendly" region, like 1+eps < s < e^(1/e)-eps and the like.
So this is another path of exploration: to find optimal methods for approximation of the occuring series in our tetration-context; that's where I am involved currently at about 50% .
Gottfried
Gottfried Helms, Kassel