bo198214 Wrote:I thought b=exp(1) was anyway outside the range of convergence, thatswhy I wanted to be sure what you mean by it. Also your next statement is mysteries assuming that range of convergence.Well, take it with a grain of salt... Surely you're right, "eigenvalues" of [1,1,1,..,] or [1,-1,1,-1,...] should not verbally included into a statement about convergence...
Quote:Quote:For the case of b outside this range I found that always a part of the eigenvalues(truncated matrices) converge to that logarithms, but another part vary wildly;How can a part of the eigenvalues converge to that logarithms, if there are no real fixed points for \( b>e^{1/e} \) and hence no logarithms of that fixed points?
Yes, there we have the two different approaches.
In the analytical view (assuming infinite matrices), there should "no part of the set" with special behaviour occur; I meant this statement in the context of sets of eigenvalues of truncated matrices, and series of such sets, when the size of matrices increases. What I observed was just that: if I ordered the empirical eigenvalues, then parts of them could be identified which stabilized to certain values, while others were wildly varying. Again: eigenvalues computed on the base of finite matrices, with a canned eigensystem-solver. Maybe, that were not the logarithms - have it not in mind currently. I'll look at it later this evening.
[update] I just see, that I was partly in error here. The case described was for bases 1/e^e<b<1. Here we have one part of eigenvalues of each set converging to the expected values. For bases b outside [1/e^e..e^(1/e)] I didn't find documents - I think, they were truely random [/update]
Perhaps you try this with mathematica/maple since I heard, that they are much faster than pari/gp, with which it was nearly impossible/extremely time-expensive to get beyond the size=24x24
Gottfried Helms, Kassel