Matrix Operator Method

[Gottfried]

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.

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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)

\( \hspace{24}
\begin{matrix} {rrrrrrr}
1.00000000000 & . & . & . & . & . & . & . \\
0 & 0.832554611158 & . & . & . & . & . & . \\
0 & 0.157453119779 & 0.693147180560 & . & . & . & . & . \\
0 & 0.0100902384840 & 0.262176641827 & 0.577082881386 & . & . & . & . \\
0 & -0.000178584914170 & 0.0415928340834 & 0.327414558137 & 0.480453013918 \\
0 & 0.0000878420556305 & 0.00288011566971 & 0.0829028563527 & 0.363454000182 \\
0 & -0.00000218182495620 & 0.000191842025839 & 0.0114684142796 & 0.126396502873 \\
0 & -0.00000702051219082 & 0.0000204251058104 & 0.00104695045599 & 0.0258020272404
\end{matrix}
\)

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Differences:

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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)
\( \hspace{24}
\begin{matrix} {rrrrrrr}
0.E-414 & . & . & . & . & . & . & . \\
0 & -1.55109064503E-66 & . & . & . & . & . & . \\
0 & 5.98904356812E-56 & -5.30263977471E-56 & . & . & . & . & . \\
0 & -1.03921639872E-49 & 1.73538878994E-49 & -7.68248541586E-50 & . & . & . & . \\
0 & 3.03477814704E-45 & -7.18038111513E-45 & 6.11155244587E-45 & -1.80377946480E-45 & . \\
0 & -9.50474725632E-42 & 2.83773329816E-41 & -3.48603689927E-41 & 1.99810508742E-41 \\
0 & 7.26187711067E-39 & -2.57084380097E-38 & 4.05741175331E-38 & -3.39109179435E-38 \\
0 & -1.42479127409E-29 & 5.74050200445E-29 & -1.08939157931E-28 & 1.17741370976E-28
\end{matrix}
\)

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