Schroeder Summations
Experimental Results

Limitations: Only tested on Mathematica 4.1. Uses the Iterate.m file. Assumes that there is a non-superattracting fixed point [Graphics:Images/index_gr_1.gif] - [Graphics:Images/index_gr_2.gif]!=0; . Assumes that the function is [Graphics:Images/index_gr_3.gif]. Software written with intent to extend it to handle matrix functions, but currently assume that z is complex to avoid non-communatve multiplication.

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Validation

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The following routine validates the Schroeder summations by computing the error between the Schroeder summations and the derivatives of iterated functions where k is the range of derivatives tested and n is the number of iterations.

[Graphics:Images/index_gr_28.gif] ... [Graphics:Images/index_gr_29.gif]
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... ...
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0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0

Labled Hierarchies

This adds up the coefficients of the nth Schroeder summation giving the hierarchies of n just as adding up the coefficients of the nth Bell summation gives the nth Bell number.

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The following is an uses the pointing operator to generate the instances of labeled hierarchies. Only the first six values of the hierarchies structure are computed due to time. Even with the algorithmic complexity of computing Schroder summations, they can be used to calculate the values of  the hierarchies structure faster than using the pointing operator because the Schroder summations are actually generating the unlabeled heirachies combinatoric structure which grows much for sloly than the labeled version.

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Unlabled Hierarchies

This adds up the number of terms of the nth Schroeder summation giving the unlabeled hierarchies of n just as adding up the number of terms of the nth Bell summation gives the nth partition number. Both unlabeled structures serve as indexes to their label versions. The initial value is wrong due to the way the program is implemented, it should be one, not zero.

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Hierarchies of Height n

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Iterated Quadratic

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Converted by Mathematica      February 23, 2005