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

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)

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