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Matrix Operator Method
#35
Just read the book "Advanced combinatorics" of Louis Comtet (pg 143-14Cool
about his method of fractional iteration for powerseries. This is just
a binomial-expansion using the Bell-matrix



where



However, with one example, with the matrix for dxp_2(x) (base= 2)
the results of all three methods (Binomial-expansion, Matrix-
logarithm, Diagonalization) converge to the same result.

For matrix-log and binomial-expansion I need infinitely
many terms to arrive at exact results (because for the general
case the diagonal of the matrix is not the unit-diagonal and so
the terms of the expansions are not nilpotent) while the diagonali-
zation-method needs only as many terms as the truncation-size of
the matrix determines, and is then constant for increasing sizes.
(Well, I'm talking of triangular matrices here and without thoroughly
testing...)



So Ut is the Bell-matrix for the function

Then I determined the coefficients for half-iteration Ut°0.5(x) by
Ut^0.5 using all three methods.


Result by Diagonalization / Binomial / Matrix-log
(differences are vanishing when using more terms for the series-expansion
of Binomial / matrix-logarithm; I used 200 terms here)




Differences:

Diagonalization - binomial (200 terms)


Binomial - matrixlog (200 terms)


Diagonalization - matrixlog (200 terms)


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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