non-neglectable Infinitesimals in the Bell-matrix?

[Gottfried]

The discrepancy between the serial summation to the infinite alternating series of powertowers and the matrix-based summation is going to make me crazy...

In my other discussion I considered the tetration x->b^x and there are some caveats, like impossibility of inversion of the infinite Bell-matrix.

But the discrepancy occurs also with the x->exp(x)-1 tetration, resp. its inverse x-> log(1+x). This is even more curious, since we have a row-finite operator, for which an exact reciprocal (and its unchanged truncation) can be computed and used.

But I ran out of ideas, where the source of the discrepancy may be.

Here I give a list of results of the serial summation and matrix-based summation of
b=1.3 and b=eta=e^(1/e) for

ub(x) = b^x - 1
lb(x) = log(1+x)/log(b)

which are the inverse operations and are computed by

V(x)~ * S2b = V(ub(x))~
V(x)~ * S1b = V(lb(x))~

where S2b = dV(log(b))*S2 and S1b = S1*dV(1/log(b)) , parametrized versions of the Stirlingmatrices (see appendix) where the results come out correctly and satisfy the definitions of the Stirling-numbers of the two kinds.

--------

The alternating series

Su = x - ub(x) + ub(ub(x)) - ... +...
Sl = x - lb(x) + lb(lb(x)) - ... + ...

are implemented by the matrixoperators

Mu = I - S2b + S2b^2 - S2b^3 + ... = (I + S2b)^-1
Ml = I - S1b + S1b^2 - S1b^3 + ... = (I + S1b)^-1

and the matrix-computation for Su(x) and Sl(x) is

V(x)~ * Mu = V(Su(x)) ~
V(x)~ * Ml = V(Sl(x)) ~

This comes out to be correct for Su(x), when crosschecked by serial summation, but differ for Sl(x)



Moreover, by matrixtheory it is

Mu + Ml = I

so the sum of the two results are

V(x)~ * Mu + V(x)~* Ml = V(x)~ *( Mu + Ml) = V(x)~ * I = V(x)~

which again comes out correctly to given real-precision, when numerically checked.



But the serial computation of Su(x) + Sl(x) =/= x, and the reason is in the occuring discrepancy for Sl(x) only.

Also the differences for the Sl(x)-versions are relatively small and also roughly periodic with varying freqency, so I assume a systematic small error in the definition of S1 , and then also in S2.
Or - for any other reason the geometric-series-based computation of Ml is somehow incompatible with the serial summation of the lb()-towers

I have currently no idea, where to look into...

Gottfried

---- Appendix ----------

Here are two plots:

   

   

Code:

base=1.3, Sl(x)
       x               matrix-based       serial            difference
    1.00000000000    0.235301957128     0.245204215222  -0.00990225809450
   0.975000000000    0.228701914313     0.239151131819   -0.0104492175061
   0.950000000000    0.222141129686     0.233094325597   -0.0109531959119
   0.925000000000    0.215619404337     0.227027895542   -0.0114084912053
   0.900000000000    0.209136539762     0.220945608398   -0.0118090686363
   0.875000000000    0.202692337870     0.214840897906   -0.0121485600364
   0.850000000000    0.196286600996     0.208706869038   -0.0124202680427
   0.825000000000    0.189919131913     0.202536308811   -0.0126171768976
   0.800000000000    0.183589733845     0.196321705675   -0.0127319718303
   0.775000000000    0.177298210472     0.190055280039   -0.0127570695670
   0.750000000000    0.171044365948     0.183729029165   -0.0126846632164
   0.725000000000    0.164828004906     0.177334790567   -0.0125067856610
   0.700000000000    0.158648932469     0.170864329198   -0.0122153967297
   0.675000000000    0.152506954259     0.164309455153   -0.0118025008943
   0.650000000000    0.146401876408     0.157662180530   -0.0112603041216
   0.625000000000    0.140333505569     0.150914926535   -0.0105814209661
   0.600000000000    0.134301648917     0.144060795071  -0.00975914615380
   0.575000000000    0.128306114167     0.137093923193  -0.00878780902604
   0.550000000000    0.122346709574     0.130009944135  -0.00766323456122
   0.525000000000    0.116423243947     0.122806585589  -0.00638334164115
   0.500000000000    0.110535526655     0.115484444901  -0.00494891824556
   0.475000000000    0.104683367633     0.108047992534  -0.00336462490093
   0.450000000000   0.0988665773892     0.100506870001  -0.00164029261148
   0.425000000000   0.0930849670147    0.0928775672741  0.000207399740560
   0.400000000000   0.0873383481879    0.0851855875555   0.00215276063243
   0.375000000000   0.0816265331820    0.0774682336608   0.00415829952120
   0.350000000000   0.0759493348711    0.0697781768756   0.00617115799554
   0.325000000000   0.0703065667364    0.0621879866737   0.00811858006277
   0.300000000000   0.0646980428720    0.0547957847775   0.00990225809450
   0.275000000000   0.0591235779909    0.0477320835127    0.0113914944782
   0.250000000000   0.0535829874301    0.0411675435700    0.0124154438601
   0.225000000000   0.0480760871563    0.0353205246358    0.0127555625205
   0.200000000000   0.0426026937709    0.0304611391266    0.0121415546443
   0.175000000000   0.0371626245147    0.0269031449478    0.0102594795668
   0.150000000000   0.0317556972729    0.0249614784615   0.00679421881140
   0.125000000000   0.0263817305798    0.0248181823565   0.00156354822330
   0.100000000000   0.0210405436227    0.0261459357564  -0.00510539213364
  0.0750000000000   0.0157319562467    0.0270897957341   -0.0113578394875
  0.0500000000000   0.0104557889581    0.0217178308705   -0.0112620419124
  0.0250000000000  0.00521186292883  -0.00152823441650   0.00674009734534

Code:

S2
  1      .     .    .  .  .
  0      1     .    .  .  .
  0    1/2     1    .  .  .
  0    1/6     1    1  .  .
  0   1/24  7/12  3/2  1  .
  0  1/120   1/4  5/4  2  1
[\code]

[code] S1
  1     .      .     .   .  .
  0     1      .     .   .  .
  0  -1/2      1     .   .  .
  0   1/3     -1     1   .  .
  0  -1/4  11/12  -3/2   1  .
  0   1/5   -5/6   7/4  -2  1