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 Matrix Operator Method Gottfried Ultimate Fellow Posts: 758 Threads: 117 Joined: Aug 2007 07/08/2008, 06:46 AM Just read the book "Advanced combinatorics" of Louis Comtet (pg 143-14 about his method of fractional iteration for powerseries. This is just a binomial-expansion using the Bell-matrix $\hspace{24}B^t = \sum_{k=0}^{\infty} ({t \\k})*B1^k$ where $\hspace{24} B1 = B - ID$ 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...) $\hspace{24} U_t = {}^dV(\log(2))* fS2F$ So Ut is the Bell-matrix for the function $\hspace{24} Ut(x) = 2^x - 1$ 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) $\hspace{24} \begin{matrix} {rrrrrrr} 1.00000000000 & . & . & . & . & . & . & . \\ 0 & 0.832554611158 & . & . & . & . & . & . \\ 0 & 0.157453119779 & 0.693147180560 & . & . & . & . & . \\ 0 & 0.0100902384840 & 0.262176641827 & 0.577082881386 & . & . & . & . \\ 0 & -0.000178584914170 & 0.0415928340834 & 0.327414558137 & 0.480453013918 \\ 0 & 0.0000878420556305 & 0.00288011566971 & 0.0829028563527 & 0.363454000182 \\ 0 & -0.00000218182495620 & 0.000191842025839 & 0.0114684142796 & 0.126396502873 \\ 0 & -0.00000702051219082 & 0.0000204251058104 & 0.00104695045599 & 0.0258020272404 \end{matrix}$ Differences: Diagonalization - binomial (200 terms) $\hspace{24} \begin{matrix} {rrrrrrr} 0.E-414 & . & . & . & . & . & . & . \\ 0 & -1.06443358765E-107 & . & . & . & . & . & . \\ 0 & 1.60236818692E-61 & -1.41872090005E-61 & . & . & . & . & . \\ 0 & -1.75734170509E-39 & 2.93458536041E-39 & -1.29912638612E-39 & . & . & . & . \\ 0 & 8.68085535070E-27 & -2.05384773365E-26 & 1.74805552817E-26 & -5.15903675680E-27 & . & . \\ 0 & -7.73773729412E-19 & 2.30948495672E-18 & -2.83611081486E-18 & 1.62496041665E-18 \\ 0 & 1.30102806362E-13 & -4.60061066251E-13 & 7.25093796854E-13 & -6.05100795132E-13 \\ 0 & -0.000000000461000932910 & 0.00000000185781313787 & -0.00000000352662825863 & 0.00000000381278773133 \end{matrix}$ Binomial - matrixlog (200 terms) $\hspace{24} \begin{matrix} {rrrrrrr} 0.E-414 & . & . & . & . & . & . & . \\ 0 & 1.55109064503E-66 & . & . & . & . & . & . \\ 0 & -5.98902754444E-56 & 5.30262558750E-56 & . & . & . & . & . \\ 0 & -1.75734170498E-39 & 2.93458536023E-39 & -1.29912638604E-39 & . & . & . & . \\ 0 & 8.68085535070E-27 & -2.05384773365E-26 & 1.74805552817E-26 & -5.15903675680E-27 & . \\ 0 & -7.73773729412E-19 & 2.30948495672E-18 & -2.83611081486E-18 & 1.62496041665E-18 \\ 0 & 1.30102806362E-13 & -4.60061066251E-13 & 7.25093796854E-13 & -6.05100795132E-13 \\ 0 & -0.000000000461000932910 & 0.00000000185781313787 & -0.00000000352662825863 \end{matrix}$ Diagonalization - matrixlog (200 terms) $\hspace{24} \begin{matrix} {rrrrrrr} 0.E-414 & . & . & . & . & . & . & . \\ 0 & -1.55109064503E-66 & . & . & . & . & . & . \\ 0 & 5.98904356812E-56 & -5.30263977471E-56 & . & . & . & . & . \\ 0 & -1.03921639872E-49 & 1.73538878994E-49 & -7.68248541586E-50 & . & . & . & . \\ 0 & 3.03477814704E-45 & -7.18038111513E-45 & 6.11155244587E-45 & -1.80377946480E-45 & . \\ 0 & -9.50474725632E-42 & 2.83773329816E-41 & -3.48603689927E-41 & 1.99810508742E-41 \\ 0 & 7.26187711067E-39 & -2.57084380097E-38 & 4.05741175331E-38 & -3.39109179435E-38 \\ 0 & -1.42479127409E-29 & 5.74050200445E-29 & -1.08939157931E-28 & 1.17741370976E-28 \end{matrix}$ Gottfried Helms, Kassel « Next Oldest | Next Newest »

 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 Diagonalization for dxp/basic facts/Pari-routine - by Gottfried - 08/08/2008, 01:12 PM Exact entries for T-tetration Bell-matrix - by Gottfried - 09/24/2008, 08:22 PM RE: Exact entries for T-tetration Bell-matrix - by bo198214 - 09/26/2008, 07:30 AM RE: Exact entries for T-tetration Bell-matrix - by Gottfried - 09/26/2008, 09:56 AM

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