11/30/2007, 06:42 PM
I tried with various modifications of the series for Sl to eliminate possible effects of divergence or only conditional convergence.
Using b as base , lb(x) = log(1+x)/log(b)
Sla(x) = x - lb(x)/d + lb(lb(x))/d^2 - lb(lb(lb(x)))/d^3 + ... - ...
Slb(x) = x/0! - lb(x)/1! + lb(lb(x))/2! - lb(lb(lb(x)))/3! + ... - ...
where I used 2 <= d <=32 for Sla(x)
The observed differences between serial and matrix-based summation are still occuring with no apparent diminuation, so I think, the effect is based already in very early terms of the matrix-based coefficients for the powerseries.
Note, that the matrix-based results are all consistent between each other - using the matrix for iterated logarithm, for iterated exponentiation, the sums, the reciprocity of the sum-matrices, the summing of Sl(x)+Su(x) = x , even the sum Su(x) via matrix- and serial summation.
Only the Sl-summation between serial and matrix-summations differs, and the difference is small, sometimes zero and of roughly sinusoidal dilated periodicity.
So, one needs really a new idea
Using b as base , lb(x) = log(1+x)/log(b)
Sla(x) = x - lb(x)/d + lb(lb(x))/d^2 - lb(lb(lb(x)))/d^3 + ... - ...
Slb(x) = x/0! - lb(x)/1! + lb(lb(x))/2! - lb(lb(lb(x)))/3! + ... - ...
where I used 2 <= d <=32 for Sla(x)
The observed differences between serial and matrix-based summation are still occuring with no apparent diminuation, so I think, the effect is based already in very early terms of the matrix-based coefficients for the powerseries.
Note, that the matrix-based results are all consistent between each other - using the matrix for iterated logarithm, for iterated exponentiation, the sums, the reciprocity of the sum-matrices, the summing of Sl(x)+Su(x) = x , even the sum Su(x) via matrix- and serial summation.
Only the Sl-summation between serial and matrix-summations differs, and the difference is small, sometimes zero and of roughly sinusoidal dilated periodicity.
So, one needs really a new idea
Gottfried Helms, Kassel