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04/18/2008, 01:55 PM
(This post was last modified: 04/18/2008, 01:57 PM by Gottfried.)
I've got some mail, where the author considered
the case from the view of generationfunctions,
which may be interesting for a needed proof of
the method.
Here some exchange:
Code: It seems that e.g.f. of your triangle is:
(exp(y*(exp(y*(exp(x)1))1))1)/y^2
which gives
1,
1,1,1,
1,3,4,3,1,
1,7,13,19,13,6,1,
...
Next mail:
Code: And the e.g.f. connected to your coefficient a_3() is
(exp(y*(exp(y*(exp(y*(exp(x)1))1))1))1)/y^3
which gives triangle
1,
1,1,1,1,
1,3,4,6,4,3,1,
1,7,13,26,31,31,25,13,6,1,
...
And so on.
Then I did not understand, how the generationfunction was
related to the bivariate array:
Code: ´
> However, I did not understand how you arrived
> via the generationfunction process at the
> actual Kmatrices. Would you mind to explain
> this to me?
>
> Gottfried
Final mail:
Code: An example:
Substituting t = exp(y) in your U_t(x,2) we get
exp(y*(exp(y*(exp(x)1))1))1 =
y^2*x+(1/2*y^2+1/2*y^3+1/2*y^4)*x^2+(1/6*y^2+1/2*y^3+2/3*y^4+1/2*y^5+1/6*y^6)*x^3+(1/24*y^2+7/24*y^3+13/24*y^4+19/24*y^5+13/24*y^6+1/4*y^7+1/24*y^8)*x^4+...
.
Gottfried Helms, Kassel
Posts: 1,389
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Joined: Aug 2007
Gottfried Wrote:The symbolic eigensystemdecomposition is enormous resourcesconsuming and also not yet well documented and verified in a general manner. Here I propose a completely elementary (while still matrixbased) way to arrive at the same result as by eigensystemanalysis.

As you might recall from my "continuous iteration" article I've found a matrix representation for the coefficients of the powerseries for the Utetration
, where .
Gottfried, as long as you deal with functions with fixed point at 0, for example or there is no need to apply the Diagonalization method in its full generality.
Note, that I would like to change the name from matrix operator method to diagonalization method, as this is more specific (for example Walker uses the name "matrix method" for his natural Abel method).
For the case of a fixed point at 0, we can use the ordinary formulas of regular iteration. You know there are two cases, first (which Daniel Geisler calls parabolic) and or (which Daniel Geisler calls hyperbolic, however I am not really happy with those names, because where is the elliptic case? Is this in opposite to the which we then would call "properly hyperbolic". Also the hyperbolic and elliptic cases (in the just mentioned sense) exchange by using and hence can be treated quite similarly. This is not the case for hyperbolas and ellipses.)
The formula for the parabolic iteration was already mentioned on this forum under the name double binomial formula.
The formula for hyperbolic iteration (or elliptic iteration) can be derived from the fact that
and by the regularity condition:
For abbreviation let , then the first equation can be written as
where means the th power, by the subscript index we denote the th coefficients of the indexed powerseries.
We can rearrange the above formula to get a recurrence relation for the coefficients of :
the only undetermined coefficient is now but we know already that we set . The coefficient is the entry at the mth row and the nth column of the Carleman matrix of . We see that in the above formula also the term occurs though we dont know all the coefficients of yet, however we have the power formula:
and see that in the biggest index taken at that can occur is . However , the exponent of , can be bigger than 0 only in the case (second line below the sum) and all other . This however can only happen for (first line below the sum). This case is excluded in because and so we have indeed a recurrence formula which gives a polynomial in for .
You see, we not even have to solve a linear equation system for using the diagonalization method on a fixed point at 0.
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04/26/2008, 06:09 PM
(This post was last modified: 04/26/2008, 06:40 PM by Gottfried.)
Hi Henryk 
bo198214 Wrote:Gottfried, as long as you deal with functions with fixed point at 0, for example or there is no need to apply the Diagonalization method in its full generality.
Note, that I would like to change the name from matrix operator method to diagonalization method, as this is more specific (for example Walker uses the name "matrix method" for his natural Abel method).
 yes, I've no problem with it. My motive to retain the name was not to indicate something as new, but to not mix methods, which are possibly different. I hate the cases, where names are borrowed for something other than their original meaning. You said it several times, that my approach is just the regular tetration; but since I've dealt with something more (most prominently infinite series of tetration/its associated matrices), for which I've not seen a reference before although such generalizations are smehow obvious, I could not be without doubt, that I was introducing something going away.
If you identify it fully with the name "diagonalization"method, I've no problem to adapt this in the future, because it contains the name of the general idea behind (which is also widely acknowledged), and that is always a good manner of names... So I'll try to get used to it hoping to not provoke protests and demonstrations...
What I'll keep anyway will be the term "operator" to indicate the class of matrices, whose second column are defined as the coefficients of a function (represented by a polynomial or powerseries) and the other columns are configured such that they provide the consecutive powers of that function, so that the matrixmultiplication of a vandermondevector (consecutive powers of one parameter) will again give a vandermondevector, and is thus (at least finitely often) iterable. (The Bell and Carlemanmatrices are mutatis mutandis instances of such matrix"operators").

(for the following formulae  I've to go through them in detail to see, whether they are of help to compute the coefficients. I think with the help of Aldrovandis Bell/Carleman examples I'll be able to relate the both ways of views soon)

Quote:You see, we not even have to solve a linear equation system for using the diagonalization method on a fixed point at 0.
Yepp, I already seem to have arrived at such a view. My latest Eigensystemsolver for this class of triangular matrices is already very simple, it's just a ~tenliner in Pari/GP
Code: \\ Ut is the lower triangular matrixoperator for the function
\\ Ut := x > (t^x  1)
\\ for t=exp(1) Ut is the factorially scaled matrix of Stirlingnumbers 2'nd kind
\\ Ut may even contain the log(t)parameter u in *symbolic* form
\\ use dim<=32 in this case to prevent excessive memory and timeconsumtion.
\\ Also use exact arithmetic then; provide numfmt=1 instead of =1.0
{ APT_Init2EW(Ut,dim=9999,numfmt=1.0)=local(tu=Ut[2,2],UEW,UEWi,tt=exp(tu),tuv) ;
dim=min(rows(Ut),dim);
tuv=vectorv(dim,r,tu^(r1)); \\ the eigenvalues
UEW=numfmt*matid(dim); \\ the first eigenmatrix UEW
for(c=2,dim1,
for(r=c+1,dim,
UEW[r,c] = sum(k=c,r1,Ut[r,k]*UEW[k,c])/(tuv[c]tuv[r])
));
UEWi=numfmt*matid(dim); \\ the second eigenmatrix UEWi = UEW^1
for(r=3,dim,
forstep(c=r1,2,1,
UEWi[r,c]=sum(k=0,r1c,Ut[rk,c]*UEWi[r,rk]) /(tuv[r] tuv[c])
));
return([[tu,tt,exp(tu/tt)],UEW,tuv,UEWi]);
\\ Ut = UEW * matdiagonal(tuv) * UEWi
}
This is a recursive approach, very simple and I could imagine it implements just the g and f recursions in your post.
Gottfried Helms, Kassel
Posts: 1,389
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Joined: Aug 2007
04/26/2008, 06:47 PM
(This post was last modified: 04/26/2008, 06:48 PM by bo198214.)
Gottfried Wrote:You said it several times, that my approach is just the regular tetration; but since I've dealt with something more (most prominently infinite series of tetration/its associated matrices), for which I've not seen a reference before although such generalizations are smehow obvious, I could not be without doubt, that I was introducing something going away.
Nono, I didnt say that it is just regular tetration, but it is regular tetration if you do it at a fixed point. Strangely we may have different opinions about the interestingness of the application types of this method. I am fascinated to apply this method to nonfixed points, e.g. for the case while at fixed points the case is already explored by the regular iteration and yields no new results. While you, (if I understand a discussion right, that we had some time ago on this forum), whom introduced the method to this forum, rather are fascinated with finding patterns in the matrices, which rather occur at fixed points.
Quote:This is a recursive approach, very simple and I could imagine it implements just the g and f recursions in your post.
Wow, yes this may indeed be, however I am currently too lazy to check this in detail. But at least regarding our comparison here accutely supports that they are equal.
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Just read the book "Advanced combinatorics" of Louis Comtet (pg 14314
about his method of fractional iteration for powerseries. This is just
a binomialexpansion using the Bellmatrix
where
However, with one example, with the matrix for dxp_2(x) (base= 2)
the results of all three methods (Binomialexpansion, Matrix
logarithm, Diagonalization) converge to the same result.
For matrixlog and binomialexpansion I need infinitely
many terms to arrive at exact results (because for the general
case the diagonal of the matrix is not the unitdiagonal and so
the terms of the expansions are not nilpotent) while the diagonali
zationmethod needs only as many terms as the truncationsize 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 Bellmatrix for the function
Then I determined the coefficients for halfiteration Ut°0.5(x) by
Ut^0.5 using all three methods.
Result by Diagonalization / Binomial / Matrixlog
(differences are vanishing when using more terms for the seriesexpansion
of Binomial / matrixlogarithm; I used 200 terms here)
Differences:
Diagonalization  binomial (200 terms)
Binomial  matrixlog (200 terms)
Diagonalization  matrixlog (200 terms)
Gottfried Helms, Kassel
Posts: 763
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08/08/2008, 01:12 PM
(This post was last modified: 08/09/2008, 06:39 PM by Gottfried.)
I've collected basic details of the diagonalizationmethod as I use it for fractional Utetration, including the Pari/GProutine and some more sample coefficients in
[update 9.8.2008]
http://go.helmsnet.de/math/tetdocs/APT.htm
[/update]
Comments welcome 
Gottfried
Gottfried Helms, Kassel
Posts: 763
Threads: 118
Joined: Aug 2007
09/24/2008, 08:22 PM
(This post was last modified: 09/25/2008, 07:38 AM by Gottfried.)
Just derived a method to compute exact entries for powers of the (square) matrixoperator for Ttetration.
It is applicable to positive integer powers only, but for any base.
The restriction to positive integer powers lets look such solutions useless, since integer iterationheight can easily computed just using the scalar values. But I'll use this for further analysis of powerseries, series of powertowers and hopefully one time for the fractional iteration...
Let's use the following notational conventions:
Code: ´
b^^h  the powertower of height h using base b
V(x)  the vandermondecolumnvector containing consecutive powers of its parameter x:
V(x) = column(1,x,x^2,x^3,...)
dV(x)  used as diagonalmatrix
T  the matrix which performs Ttetration to base b (in our forum:"exp_b°t(x)" ):
V(x)~ * T = V(b^x)~
U  the matrix which performs Utetration to base b (in our forum:"dxp_b°t(x)" ):
V(x)~ * U = V(b^x  1) ~
Note, that U is lower triangular.
The triangularity allows to compute exact entries for the integer matrixpowers.
Then the entries for positive integer powers of T can be finitely computed and are "exact", as far as we assume scalar logarithms and exponentials as exact:
Code: ´
T^2 = U*dV(b^^0) * T*dV(b^^1)
T^3 = U*dV(b^^0) * U*dV(b^^1) * T*dV(b^^2)
...
T^h = prod_{k=0}^{h2} (U * dV(b^^k))
* (T * dV(b^^(h1)))
This finding is interesting, because in my matrixmethod I had to use fixpointshift to get exact entries even for integer powers, and since the fixpoints for Ttetration are real only for a small range of bases we had to deal with complexvalued Umatrices when considering the general case. Here we do not need a fixpointshift.
I did not check how this computation is related to Ioannis Galidakis' method for exact entries yet, but I think, this is interesting too.
Here is the top left of the symbolic T^2, where lambda=log(b). Each row has to be multiplied by the entry in the most left column and each column must also be multiplied by the entry in the first row.
Here is the top left of the symbolic T^3. (b^^2 means b^b). Legend as before
The difference between the symbolic computation and the simple matrixpower is interesting. I used dim=64x64, base b=sqrt(2), which provides a good approximation when the simple matrixpower is computed. Here are two (zoomed) images: very good aproximation in the leading 12 columns (abs differences to the exact values <1e20 ), but in the columns 52 to 63 the differences grow up to absolute values greater than 1e10. Surely I "knew" that differences should occur, but I hadn't guessed, that they are so large  I just didn't investigate this in detail.
The leading first twelve columns of the matrix of differences:
The twelve rightmost columns:
The large errors are actually still relatively small for that base b=sqrt(2). A measure for the quality of approximation is, whether the resulting vector of V(x)~*T^3 = Y~ is actually vandermonde and thus Y = V(y) .
This means, that the ratios of logarithms of its entries : log(Y[k])/log(Y[1]), k=0..63, should give the exact sequence [0,1,2,3,...], because this means, that Y contains indeed the consecutive powers of Y[1].
Here is a table of that ratios. ( Remember: we check the colsums of the third power of T, using x=1)
Code: ´
column symbolic "naive" difference

0 1.13831366798E19 0.E201 1.13831366798E19
1 1.00000000000 1.00000000000 0.E202
2 2.00000000000 2.00000000000 1.11189479752E19
3 3.00000000000 3.00000000000 2.20208219209E19
4 4.00000000000 4.00000000000 3.27448139227E19
....
42 42.0000000000 42.0000000000 1.19448154596E11
43 43.0000000000 43.0000000000 3.09972987348E11
44 44.0000000000 43.9999999999 7.77154497620E11
45 45.0000000000 44.9999999998 1.88541576938E10
46 46.0000000000 45.9999999996 0.000000000443257385765
47 47.0000000000 46.9999999990 0.00000000101122118773
48 48.0000000000 47.9999999978 0.00000000224146913405
49 49.0000000000 48.9999999952 0.00000000483322217091
50 50.0000000000 49.9999999899 0.0000000101495726916
51 51.0000000000 50.9999999792 0.0000000207790885254
52 52.0000000000 51.9999999585 0.0000000415151935602
53 53.0000000000 52.9999999190 0.0000000810212593226
54 54.0000000000 53.9999998454 0.000000154592714129
55 55.0000000000 54.9999997114 0.000000288630924056
56 56.0000000000 55.9999994723 0.000000527723771045
57 57.0000000000 56.9999990544 0.000000945602091679
58 58.0000000000 57.9999983383 0.00000166172556358
59 59.0000000000 58.9999971341 0.00000286585838319
60 60.0000000000 59.9999951463 0.00000485372880576
61 61.0000000000 60.9999919223 0.00000807772027087
62 62.0000000000 61.9999867825 0.0000132174922318
63 63.0000000000 62.9999787236 0.0000212764324046
For base b=2 this looks already catastrophic for the "naive"computation:
Code: ´
column symbolic "naive" difference

0 4.36636233681E20 0.E202 4.36636233681E20
1 1.00000000000 1.00000000000 0.E202
2 2.00000000000 2.00000000000 2.46662212171E14
3 3.00000000000 2.99999999998 2.31650095614E11
4 4.00000000000 3.99999999669 0.00000000331064284896
5 5.00000000000 4.99999985867 0.000000141325875791
6 6.00000000000 5.99999733964 0.00000266036059669
....
50 50.0000000000 37.9413247398 12.0586752602
51 51.0000000000 38.3796009369 12.6203990631
52 52.0000000000 38.8098393554 13.1901606446
53 53.0000000000 39.2323121795 13.7676878205
54 54.0000000000 39.6472796473 14.3527203527
55 55.0000000000 40.0549905348 14.9450094652
56 56.0000000000 40.4556826502 15.5443173498
57 57.0000000000 40.8495833279 16.1504166721
58 58.0000000000 41.2369099170 16.7630900830
59 59.0000000000 41.6178702587 17.3821297413
60 60.0000000000 41.9926631496 18.0073368504
61 61.0000000000 42.3614787893 18.6385212107
62 62.0000000000 42.7244992089 19.2755007911
63 63.0000000000 43.0818986820 19.9181013180
It is obvious, that we should use the "exact" (symbolic) description, if we ever explicitely consider powers of the tetrationmatrix T.
Gottfried Helms, Kassel
Posts: 1,389
Threads: 90
Joined: Aug 2007
Gottfried Wrote:Just derived a method to compute exact entries for powers of the (square) matrixoperator for Ttetration.
It is applicable to positive integer powers only, but for any base.
...
Code: T^2 = U*dV(b^^0) * T*dV(b^^1)
T^3 = U*dV(b^^0) * U*dV(b^^1) * T*dV(b^^2)
...
T^h = prod_{k=0}^{h2} (U * dV(b^^k))
* (T * dV(b^^(h1)))
So the finding is that though the matrix multiplication of the T's is infinite, the result is expressible in finite terms (polynomials) of ?
Quote:This finding is interesting, because in my matrixmethod I had to use fixpointshift to get exact entries even for integer powers, and since the fixpoints for Ttetration are real only for a small range of bases we had to deal with complexvalued Umatrices when considering the general case. Here we do not need a fixpointshift.
I did not check how this computation is related to Ioannis Galidakis' method for exact entries yet, but I think, this is interesting too.
That would be useful to compare. Can you derive a recurrence from your matrix formula?
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09/26/2008, 09:56 AM
(This post was last modified: 09/26/2008, 11:28 AM by Gottfried.)
bo198214 Wrote:So the finding is that though the matrix multiplication of the T's is infinite, the result is expressible in finite terms (polynomials) of ?
Exactly. The occuring infinite sums are decomposable into sums of simple exponentialseries for which then exact values are defined (and computable to arbitrary precision). The compositions of the evaluated exponentials contain then only finitely many terms. See the display of the matrix T^2. I found these by studying the dotproducts of the rows and second column of T*T which was then easily generalizable to the other columns of the matrixproduct.
Quote:That would be useful to compare. Can you derive a recurrence from your matrix formula?
Yes. For this I see another streamlining of the formula first:
Note, that T = U * P~ , which can also be seen by the operation
Code: ´
V(x)~ * T = V(b^x)~
V(x)~ * U = V(b^x 1)~
also: V(b^x 1)~ * P~ = V(b^x)~
thus: V(x)~ * (U * P~ ) = V(b^x)~
==> U * P~ = T
A further adaption can be made. We have, that the final term in T^2 = U*dV(b^^0) * T*dV(b^^1) is
Code: ´
T*dV(b^^1) = U *P~ * dV(b^^1)
We can rewrite this in terms of a power of P ( use notation "PPow()" ) by expanding with the trivial product dV(b^^1)*dV(1/b^^1)= I
Code: ´
T*dV(b^^1) = U * dV(b^^1)*dV(1/b^^1) * P~ * dV(b^^1)
= U * dV(b^^1)* (dV(1/b^^1) * P~ * dV(b^^1))
= U * dV(b^^1)* PPow(b^^1) ~
The general productformula changes then to
Code: ´
T^h = prod_{k=0}^{h1} (U * dV(b^^k))
* PPow(b^^(h1)) ~
and a recursion is then
Code: ´
T^(h+1) = T^h * PPow(b^^(h1))~ * U * dV(b^^h) * PPow(b^^h)~
or, even more simple
T^(h+1) = T^h * PPow(b^^(h1))~ * T * dV(b^^h)
where the first part of the product , T^h * PPow(b^^(h1))~ , gives a triangular matrix.
The recursion may also be seen "in action", when evaluated for a parameter x. We need an ascii notation for iterated exponentiation, I use x.b^^h for exp_b°h(x) here, or if x=1, simply b^^h .
We have by definition, that
Code: '
V(x)~ * T^(h+1) = V(x.b^^(h+1)) ~
Using the recursionformula we get
Code: '
V(x)~ * T^(h+1) = V(x) ~*T^h * PPow(b^^(h1))~ * T * dV(b^^h)
= V(x.b^^h) ~ * PPow(b^^(h1))~ * T * dV(b^^h)
= V(x.b^^h  b^^(h1))~ * T * dV(b^^h)
= V(b^(x.b^^h  b^^(h1)))~ * dV(b^^h)
= V(b^(x.b^^h)/b^(b^^(h1)))~ * dV(b^^h)
= V(x.b^^(h+1)/b^^h)~ * dV(b^^h)
= V(x.b^^(h+1))~ * dV(1/b^^h) * dV(b^^h)
= V(x.b^^(h+1))~ * I
= V(x.b^^(h+1))~
which is the same result.
Gottfried Helms, Kassel
