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Matrix Operator Method
#19
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 of individual entries 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
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Messages In This Thread
Matrix Operator Method - by Gottfried - 08/12/2007, 08:08 PM
RE: Matrix Operator Method - by bo198214 - 08/13/2007, 04:15 AM
RE: Matrix Operator Method - by jaydfox - 08/13/2007, 05:40 AM
RE: Matrix Operator Method - by Gottfried - 08/13/2007, 09:22 AM
RE: Matrix Operator Method - by bo198214 - 08/14/2007, 03:43 PM
RE: Matrix Operator Method - by Gottfried - 08/14/2007, 04:15 PM
RE: Matrix Operator Method - by bo198214 - 08/26/2007, 12:18 AM
RE: Matrix Operator Method - by Gottfried - 08/26/2007, 11:24 AM
RE: Matrix Operator Method - by bo198214 - 08/26/2007, 11:39 AM
RE: Matrix Operator Method - by Gottfried - 08/26/2007, 04:22 PM
RE: Matrix Operator Method - by Gottfried - 08/26/2007, 10:54 PM
RE: Matrix Operator Method - by bo198214 - 08/27/2007, 08:29 AM
RE: Matrix Operator Method - by Gottfried - 08/27/2007, 11:04 AM
RE: Matrix Operator Method - by bo198214 - 08/27/2007, 11:35 AM
RE: Matrix Operator Method - by Gottfried - 08/27/2007, 11:58 AM
RE: Matrix Operator Method - by bo198214 - 08/27/2007, 12:13 PM
RE: Matrix Operator Method - by Gottfried - 08/27/2007, 01:19 PM
RE: Matrix Operator Method - by Gottfried - 08/27/2007, 02:29 PM
RE: Matrix Operator Method - by bo198214 - 08/27/2007, 02:36 PM
RE: Matrix Operator Method - by Gottfried - 08/27/2007, 03:09 PM
RE: Matrix Operator Method - by bo198214 - 08/27/2007, 07:15 PM
RE: Matrix Operator Method - by Gottfried - 08/27/2007, 08:15 PM
RE: Matrix Operator Method - by bo198214 - 08/29/2007, 05:28 PM
RE: Matrix Operator Method - by Gottfried - 08/27/2007, 12:43 PM
RE: Matrix Operator Method - by Gottfried - 10/08/2007, 12:11 PM
RE: Matrix Operator Method - by Gottfried - 10/14/2007, 09:32 PM
RE: Matrix Operator Method - by Gottfried - 04/04/2008, 09:41 AM
RE: Matrix Operator Method - by Gottfried - 04/17/2008, 09:21 PM
RE: Matrix Operator Method - by bo198214 - 04/25/2008, 03:39 PM
RE: Matrix Operator Method - by Gottfried - 04/26/2008, 06:09 PM
RE: Matrix Operator Method - by bo198214 - 04/26/2008, 06:47 PM
RE: Matrix Operator Method - by Gottfried - 04/18/2008, 01:55 PM
RE: Matrix Operator Method - by Gottfried - 07/08/2008, 06:46 AM

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