There is one question open for me with this computation, maybe it is dealt elsewhere.
(In my matrix-notation) the coefficients for the slog-function to some base b are taken from the SLOGb-vector according to the idea
(I - Bb)*SLOGb = [0,1,0,0,...]
where I is the identity operator and Bb the operator which performs x->b^x
Because the matrix (I - Bb) is not invertible, Andrew's proposal is to remove the empty first column - and the last(!) row to make it invertible, let's call this "corrected(I - Bb)"
Then the coefficients for the slog-powerseries are taken from SLOGb, from a certain sufficiently approximate finite-dimension solution for
SLOGb = (corrected(I - Bb))^-1*[0,1,0,0,...]
and because the coefficients stabilize for higher dimension, the finite SLOGb is taken as a meaningful and also valid approximate to the true (infinite dimensional) SLOGb .
Btw, this approach resembles the problem of the iteration series for powertowers in a nice way: (I - Bb)^-1 would be a representation for I+Bb + BB^2 + BB^3 + ... which could then be used for the iteration-series of h^^b whith increasing heights h. Obviously such a discussion needed some more consideration because we deal with a nasty divergent series here, so let's leave this detail here.
The detail I want to point out is the following.
Consider the coefficients in the SLOGb vector. If we use a "nice" base, say b=sqrt(2), then for dimension=n the coefficients at k=0..n-1 decrease when k approaches n-2, but finally, at k=n-1, one relatively big coefficient follows, which supplies then the needed value for a good approximation of the order-n-polynomial for the slog - a suspicious effect!
This can also be seen with the partial sums; for the slog_b(b)-slog_b(1) we should get partial sums which approach 1. Here I document the deviation of the partial sums from the final value 1 at the last three terms of the n'th-order slog_b-polynomial
(For crosscheck see Pari/GP excerpt at end of msg)
Examples, always the ("partial sums" - 1) up to terms at k=n-3, k=n-2,k=n-1 are given, for some dimension n
While we see generally nice convergence with increasing dimension, there is a "step"-effect at the last partial sum (which also reflects an unusual relatively big last term)
Looking at some more of the last coefficients with dim n=64 we see the following
where we nearly get convergence to an error-result (of about 3e-10), which stabilizes for many terms and is only corrected by a jump due to the very last coefficient.
What does this mean if dimension n->infinity: then, somehow, the correction term "is never reached" ?
Well, the deviation of the partial sums from 1 decreases too, so in a rigorous view we may find out, that this effect can indeed be neglected.
But I'd say, that this makes also a qualitative difference for the finite-dimension-based approximations for the superlog/iteration-height by the other known methods for tetration and its inverse.
What do you think?
Gottfried
(In my matrix-notation) the coefficients for the slog-function to some base b are taken from the SLOGb-vector according to the idea
(I - Bb)*SLOGb = [0,1,0,0,...]
where I is the identity operator and Bb the operator which performs x->b^x
Because the matrix (I - Bb) is not invertible, Andrew's proposal is to remove the empty first column - and the last(!) row to make it invertible, let's call this "corrected(I - Bb)"
Then the coefficients for the slog-powerseries are taken from SLOGb, from a certain sufficiently approximate finite-dimension solution for
SLOGb = (corrected(I - Bb))^-1*[0,1,0,0,...]
and because the coefficients stabilize for higher dimension, the finite SLOGb is taken as a meaningful and also valid approximate to the true (infinite dimensional) SLOGb .
Btw, this approach resembles the problem of the iteration series for powertowers in a nice way: (I - Bb)^-1 would be a representation for I+Bb + BB^2 + BB^3 + ... which could then be used for the iteration-series of h^^b whith increasing heights h. Obviously such a discussion needed some more consideration because we deal with a nasty divergent series here, so let's leave this detail here.
The detail I want to point out is the following.
Consider the coefficients in the SLOGb vector. If we use a "nice" base, say b=sqrt(2), then for dimension=n the coefficients at k=0..n-1 decrease when k approaches n-2, but finally, at k=n-1, one relatively big coefficient follows, which supplies then the needed value for a good approximation of the order-n-polynomial for the slog - a suspicious effect!
This can also be seen with the partial sums; for the slog_b(b)-slog_b(1) we should get partial sums which approach 1. Here I document the deviation of the partial sums from the final value 1 at the last three terms of the n'th-order slog_b-polynomial
(For crosscheck see Pari/GP excerpt at end of msg)
Examples, always the ("partial sums" - 1) up to terms at k=n-3, k=n-2,k=n-1 are given, for some dimension n
Code:
dim n=4
...
-0.762957567623
-0.558724904310
-0.150078240781
dim n=8
...
-0.153309829172
-0.120439792559
-0.00882912480664
dim n=16
...
-0.00696424577339
-0.00629653092984
-0.0000322687018600
dim n=32
...
-0.0000228720888610
-0.0000223192966457
-0.000000000473007074189
dim n=64
...
-0.000000000331231525320
-0.000000000330433387110
-0.000000000000000000108While we see generally nice convergence with increasing dimension, there is a "step"-effect at the last partial sum (which also reflects an unusual relatively big last term)
Looking at some more of the last coefficients with dim n=64 we see the following
Code:
...
-0.000000000626200198250
-0.000000000492336933075
-0.000000000417440371765
-0.000000000376261655863
-0.000000000354008626669
-0.000000000342186109314
-0.000000000336009690814
-0.000000000332835946403
-0.000000000331231525320
-0.000000000330433387110
- - - - - - - - - - - - - - - -
-0.000000000000000000108 (= -1.08608090167E-19)What does this mean if dimension n->infinity: then, somehow, the correction term "is never reached" ?
Well, the deviation of the partial sums from 1 decreases too, so in a rigorous view we may find out, that this effect can indeed be neglected.
But I'd say, that this makes also a qualitative difference for the finite-dimension-based approximations for the superlog/iteration-height by the other known methods for tetration and its inverse.
What do you think?
Gottfried
Code:
b = sqrt(2)
N=64
\\ (...) computation of Bb
tmp = Bb-dV(1.0) ;
corrected = VE(tmp,N-1,1-N); \\ keep first n-1 rows and last n-1 columns
\\ computation for some dimension n<=N
n=64;
tmp=VE(corrected,n-1)^-1; \\ inverse of the dim-top/left segment of "corrected"
SLOGb = vectorv(n,r,if(r==1,-1,tmp[r-1,1])) \\ shift resulting coefficients; also set "-1" in SLOGb[1]
partsums = VE(DR,dim)*dV(b,dim)*SLOGb \\ DR provides partial summing
disp = partsums - V(1,dim) \\ partial sums - 1 : but document the last three entries for msg
Gottfried Helms, Kassel