Matrix-method: compare use of different fixpoints

[Gottfried]

I thought, I'll check my matrix-method and compare, what happens, if I used different fixpoints for initialization.
I used s=sqrt(2) as base-parameter, because of simpliness of the two fixpoints t1 =2 and t2=4

This is the reference matrix CHK = Bs , based on s=sqrt(2). It is constructed the simple method by
Bs = dV(log(s)) * B
CHK = Bs
and the terms of the second column, taken as coefficients for powers of x give the series for
s^x = sum {r=0,inf} log(s)^r / r! * x^r
or, iterated
s^s^x = sum {r=0,inf} log(s)^r / r! * (s^x)^r
------------------------------------------------------------------------

Code:

chk =
  1                     1                   1                  1
  0        0.346573590280      0.693147180560      1.03972077084
  0       0.0600566267398      0.240226506959     0.540509640658
  0      0.00693801358310     0.0555041086648     0.187326366744
  0     0.000601133069227    0.00961812910763    0.0486917786074
  0    0.0000416673692076    0.00133335581464    0.0101251707174
  0   0.00000240680162397   0.000154035303934   0.00175455838387
  0  0.000000119161982844  0.0000152527338041  0.000260607256480

------------------------------------------------------------------------

This matrix should also result by my analytic matrix-composition whatever initial fixpoint I use.
My parameters for composition are
s : the base, here s = sqrt(2)
t=h(s) : one of its fixpoints, such that s = t^(1/t), here I use 2 and 4 as examples
tl = log(t), principal branch

I compose Bs analytically by its eigensystem
Bs = W * D * W^-1
where D is diagonal diag(tl^0,tl^1,tl^2,...)
and W may have complex entries, if the fixpoint is complex.

Furthermore, I construct W and W^-1 by the composition
W = dV(1/t) *(P^-1)~ * X
W^-1 = X^-1 * P~ * dV(t)
where P is the pascal-matrix and X is triangular.
and compute X^-1 by my analytical description using the parameters t and tl, and then X by (X^-1) numerical inversion.

Note, that in
W = dV(1/t) *(P^-1)~ * X
we may have divergent summation, which is numerically problematic, although I use Euler-summation, but only the same order for all implicite vectorial products (I'll have to improve that...).
But it seems, that my procedure overcame this problem, anyway, the errors seem to cancel out nicely.
Only few rows and columns are displayed in the following. I used matrix-dimension 24 and 32, float-precision 200 digits

A) Construction using Fixpoint 4 =============================================
(t=4, tl=log(4), s=t^(1/t) = sqrt(2)
APT_Init1(tl,t,s,24)

Code:

X=
  1               0               0                 0
  0               1               0                 0
  0  -1.79434972478   1.79434972478  3.236949254E-194
  0   3.39342592088  -6.43938186964     3.04595594876
  0  -6.53979952045   17.9552372772    -16.3965306551
  0   12.7228628576  -45.3209402902     60.4297878795
  0  -24.8909724939   108.433343624    -188.637901738
  0   48.8779300721  -250.895176969     535.783894633

D = diag(
              1
  1.38629436112
  1.92181205567
  2.66419721591
  3.69336157733
  5.12008632823
  7.09794680527
  9.83984363168
)
X^-1 =
  1  .               .               .
  0  1               .               .
  0  1  0.557304959111               .
  0  1   1.17818494749  0.328304156994
  0  1   1.86938951398   1.08069640221
  0  1   2.63817091372   2.37347688706
  0  1   3.49231192153   4.34693229384
  0  1   4.44015854200   7.16930292361

The eigenmatrices:

Code:

W = (questionable due to divergent summation in computation of entries)
  1.00000000000  -705146.375907   10061636.1490  -68537708.6133
              0   3489389.51207  -50276479.5673   345593823.396
              0  -8286390.34681   120465028.795  -835106222.017
              0   12559030.2520  -184103274.475   1286494011.00
              0  -13630974.9241   201383129.397  -1417931872.62
              0   11269155.8714  -167724501.261   1189493061.90
              0  -7370028.22071   110466238.936  -788853148.479
              0   3908997.17173  -58986047.0304   424035285.025
              0  -1710236.53780   25974624.1942  -187925954.460
              0   624480.079318  -9543783.34494   69478898.2726
              0  -191804.578207   2949043.40534  -21598718.7055
              0   49793.5937275  -770081.778535   5673170.37787
              0  -10950.8376873   170326.013172  -1261963.18919
              0   2040.06459378  -31906.8402260   237720.549789
              0  -321.147009511   5050.00587604  -37830.1150461
              0   42.5064378702  -671.951323793   5060.53629729

W^-1 = (see the reflection of the fixpoint in first row)
  1  4             16             64            256           1024
  0  4             32            192           1024           5120
  0  4  40.9168793458  299.002552149  1880.02041719  10826.8027813
  0  4  50.8509591599  439.222975966  3169.87553611  20546.4484299
  0  4  61.9102322236  620.087356425  5053.40975295  36356.7666541
  0  4  74.2107346195  850.431336206  7738.67988445  61208.7538347
  0  4  87.8769907445  1140.72755574  11494.5329580  99241.7940545
  0  4  103.042536672  1503.34582718  16666.0898919  156199.665184
  0  4  119.850522634  1952.85063934  23693.7469193  239976.475820
  0  4  138.454392309  2506.34113631  33136.4265599  361329.444672
  0  4  159.018634839  3183.83930575  45699.9515027  534805.127091
  0  4  181.719604981  4008.73276953  62271.5816187  779937.761743
  0  4  206.746407269  5008.27930299  83961.9507878  1122793.40038
  0  4  234.301841121  6214.18103467  112155.871364  1597952.06203
  0  4  264.603405272  7663.23720718  148573.745782  2251043.13061
  0  4  297.884361368  9398.08542413  195345.643872  3141977.57828

and the completely composed matrix Bs:

Code:

BS = W*D*W^-1
  1.00000000000         1.00000000000       1.00000000000      1.00000000000
              0        0.346573590280      0.693147180560      1.03972077084
              0       0.0600566267398      0.240226506959     0.540509640658
              0      0.00693801358310     0.0555041086648     0.187326366743
              0     0.000601133069227    0.00961812910763    0.0486917786092
              0    0.0000416673692076    0.00133335581464    0.0101251707171
              0   0.00000240680162397   0.000154035303932   0.00175455838251
              0  0.000000119161982842  0.0000152527338064  0.000260607257382

Compare with chk:

Code:

chk =
  1                     1                   1                  1
  0        0.346573590280      0.693147180560      1.03972077084
  0       0.0600566267398      0.240226506959     0.540509640658
  0      0.00693801358310     0.0555041086648     0.187326366744
  0     0.000601133069227    0.00961812910763    0.0486917786074
  0    0.0000416673692076    0.00133335581464    0.0101251707174
  0   0.00000240680162397   0.000154035303934   0.00175455838387
  0  0.000000119161982844  0.0000152527338041  0.000260607256480

================================================================================




B) Construction using Fixpoint 2 =============================================
(t=2, tl=log(t), s=t^(1/t) = sqrt(2)
APT_Init1(tl,t,s,24)

Code:

X=
  1              0               0                  0
  0              1               0                  0
  0  1.12944567664  -1.12944567664  -1.473607464E-196
  0  1.19858472554  -2.55129507294      1.35271034740
  0  1.24745912391  -4.14824726765      4.58343856085
  0  1.28563007597  -5.87581793333      10.0407587481
  0  1.31707187412  -7.70930267379      17.9985821965
  0  1.34390534108  -9.63260543481      28.6872598916

D=diag(
                1
   0.693147180560
   0.480453013918
   0.333024651989
   0.230835098583
   0.160002697757
   0.110905418832
  0.0768737783725
)

X^-1
  1  .                .               .
  0  1                .               .
  0  1  -0.885390081778               .
  0  1   -1.66990025441  0.739256561405
  0  1   -2.36595087773   2.01369042283
  0  1   -2.98470320802   3.66529728561
  0  1   -3.53611631924   5.57317309542
  0  1   -4.02901260446   7.64597311973

The Eigenmatrices

Code:

W=  (the implicite summation due to vectorproducts is convergent here)
  1.00000000000         -0.455962801825        -0.234814101634      -0.128231007102
              0         0.0983187801477         0.101265436309      0.0829509522811
              0         0.0304091151843        0.0204026096323     0.00776935638663
              0        0.00825961118386       0.00175355081901    -0.00281012813971
              0        0.00216911546942     -0.000644677201669    -0.00169340744299
              0       0.000562477735004     -0.000469774810429   -0.000550860865246
              0       0.000144695356452     -0.000201904229153   -0.000130846641810
              0      0.0000370240728932    -0.0000732383771608  -0.0000128640201235
              0     0.00000943358845693    -0.0000238649227543  -0.0000129727419446
              0     0.00000241175018837   -0.00000861154790223   0.0000422504430010
              0    0.000000581709580799  -0.000000122411796083  -0.0000825732852465
              0    0.000000207755871710   -0.00000489124349097    0.000159345857287
              0  -0.0000000528566347196    0.00000687141685920   -0.000260121747386
              0    0.000000148437431700    -0.0000104743725876    0.000375063362779
              0   -0.000000181557806492     0.0000134996273957   -0.000475016406556
              0    0.000000216378004102    -0.0000154711877332    0.000530629958372

W^-1 =(see again the reflection of the used fixpoint in the first row)
  1  2               4               8              16              32
  0  2               8              24              64             160
  0  2   4.45843967289   2.75063803733  -20.9974478507  -123.324826169
  0  2   1.32039898237  -10.1635536145  -48.9980044932  -137.805981761
  0  2  -1.46380351094  -16.6732976830  -43.7639053514  -47.8114270701
  0  2  -3.93881283208  -18.3104987076  -21.5532162896   56.6576374180
  0  2  -6.14446527695  -16.2814068984   6.80218701173   132.051592175
  0  2  -8.11605041784  -11.5285175492   34.3368653819   162.669701386
  0  2  -9.88460902634  -4.78251972225   56.7947041603   148.542098335
  0  2  -11.4772621111   3.39451450481   71.7814561860   97.7532147986
  0  2  -12.9175616969   12.5735922704   78.1526668904   21.7500514052
  0  2  -14.2258516379   22.4298721260   75.5782437643  -67.3871580389
  0  2  -15.4196265145   32.7193707710   64.2377859836  -158.417173279
  0  2  -16.5138780006   43.2604036860   44.6113289218  -241.630000643
  0  2  -17.5214203670   53.9189447244   17.3386826132  -309.075022032
  0  2  -18.4531894090   64.5971838286  -16.8726754476  -354.528979793

The composed matrix

Code:

Bs= W * D * W^-1
  1.00000000000         1.00000000000       1.00000000000      1.00000000000
              0        0.346573590280      0.693147180560      1.03972077084
              0       0.0600566267398      0.240226506959     0.540509640658
              0      0.00693801358310     0.0555041086648     0.187326366744
              0     0.000601133069225    0.00961812910761    0.0486917786076
              0    0.0000416673692068    0.00133335581466    0.0101251707175
              0   0.00000240680162443   0.000154035303952   0.00175455838364
              0  0.000000119161983188  0.0000152527338009  0.000260607256418

Compare

Code:

chk =
  1                     1                   1                  1
  0        0.346573590280      0.693147180560      1.03972077084
  0       0.0600566267398      0.240226506959     0.540509640658
  0      0.00693801358310     0.0555041086648     0.187326366744
  0     0.000601133069227    0.00961812910763    0.0486917786074
  0    0.0000416673692076    0.00133335581464    0.0101251707174
  0   0.00000240680162397   0.000154035303934   0.00175455838387
  0  0.000000119161982844  0.0000152527338041  0.000260607256480

================================================================================

Different eigensystems compose the same tetration-matrix Bs; hmm.
The different fixpoints are reflected in the first row of W^-1 . This is necessarily so, since by the eigen-composition
Bs = W * D * W^-1
we have

W^-1 * Bs = D * W^-1

and for the first row of W^-1, associated with the first eigenvalue e0=1 , this means that it remains unchanged by right-multiplication with Bs, which is just the property of the (powerseries of the) fixpoint.


Gottfried

[bo198214]

I dont get it. Isnt the matrix decomposition independent of the fixed point?

[Gottfried]

I dont get it. Isnt the matrix decomposition independent of the fixed point?

It means, we can have different eigenmatrices / sets of eigenvectors resulting in the same composition for the infinite matrix Bs. This agrees with the fact, that we have multiple fixpoints for a base s.
For e^-e < s < e^(1/e) we have even two real fixpoints; for other s the fixpoints are complex.

So we may write ( I assume the continuation to more than two solutions)

Bs = W0 * D0 * W0^-1
= W1 * D1 * W1^-1
= W2 * D2 * W2^-1

where W0 and D0 are different to W1 and D1 (and assumed to W2 and D2,...) according to the different initialization with different fixpoints

[bo198214]

It means, we can have different eigenmatrices / sets of eigenvectors resulting in the same composition for the infinite matrix Bs.

Oh you mean for the infinite matrices! But the truncated matrices have a unique decomposition. And you use truncated matrices to approximate the solution. So how does this fit into your considerations?

[Gottfried]

It means, we can have different eigenmatrices / sets of eigenvectors resulting in the same composition for the infinite matrix Bs.

Oh you mean for the infinite matrices! But the truncated matrices have a unique decomposition. And you use truncated matrices to approximate the solution. So how does this fit into your considerations?

... approximately... ;-) You may have different series, which asymptotically compute gamma(something), truncated give good approximations as well.
The terms of X*D*X^-1 are finite polynomials in t and tl, and - a guess for answering - they may be expressed differently by the different roots of these polynomials... Then they are also summed by weights of binomial-coefficients and finally weighted by powers of 1/t to produce the terms for Bs.

[update]
A situation, where two different infinite series occur, is the case of
1/3 = lim 1/2 - 1/4 + 1/8 - 1/16 ... = 1/2 / (1 + 1/2)
1/3 = lim 1 -2 +4 - 8 +16 ... // analytic continuation for 1 / (1 + 2)
Possibly this analogy is better, since in the example we have an alternating convergent series for t=2 and an alternating divergent series for t=4 for the final construction of terms.
[/update]

[bo198214]

I simply dont see how you get fixed points involved.
You have Carleman matrix \( B_b \) of \( b^x \) and you uniquely decompose the finite truncations \( {B_b}_{|n} \) into
\( {B_b}_{|n}=W_{|n} D_{|n} {W_{|n}}^{-1} \) and then you define

\( {B_b}^t = \lim_{n\to\infty} W_{|n} {D_{|n}}^t {W_{|n}}^{-1} \).

Where are the fixed points used?

[Gottfried]

I simply dont see how you get fixed points involved.
You have Carleman matrix \( B_b \) of \( b^x \) and you uniquely decompose the finite truncations \( {B_b}_{|n} \) into
\( {B_b}_{|n}=W_{|n} D_{|n} {W_{|n}}^{-1} \) and then you define

\( {B_b}^t = \lim_{n\to\infty} W_{|n} {D_{|n}}^t {W_{|n}}^{-1} \).

Where are the fixed points used?

Assume the eigensystem-decomposition Bs = W0 * D0 * W0^-1
where the small s indicates the baseparameter s=b=base

Now this means also
W0^-1 * Bs = D0 * W0^-1

Now look at the first row of W0^-1, call this vector Y0. Then, since I assume the first eigenvalue d0_0 =1 we have
Y0 * Bs = 1 * Y0 = Y0
Now I observed, that Y0 has the form of a powerseries of the scalar parameter y0, so I may note,

V(y0)~ * Bs = V(y0)~

But on the other hand I know by construction of Bs that
V(x)~ * Bs = V(s^x)~

so the previous means
V(y0)~ * Bs = V(y0)~ = V(s^y0) ~
thus
y0 = s^y0

and y0 is obviously a fixpoint.

The observation actually was: the first row in W^-1 is the powerseries in the first fixpoint y0.

Now consider this backways. Using another fixpoint y1 this relation holds again,
y1 = s^y1
V(y1)~ * Bs = V(s^y1)~ = V(y1)~
First row of W1^-1 is V(y1)~
and W0^-1 <> W1^-1 in their first rows and supposedly the same for the whole matrices W0 <> W1 and

W1^-1 * Bs = 1 * W1^-1

is another solution, provided that the form of the first row in W1^-1 is still that of a powerseries.

And in fact; if I introduce the coefficients u1 = alpha + beta*I and t1=exp(u1) in my analytical eigensystem-constructor as parameters, I get valid eigen-matrices W1 and W1^-1 based on these other fixpoints for the cases I've checked where it holds that
Bs = W1*D1*W1^-1 (but see note 1)

These had in fact the powerseries of the second fixpoint in the first row in W1^-1 - and I suppose, this will be the same with all other fixpoints (however the numerical computations become too difficult to do a general conjecture based on and backed by reliable approximations)

Gottfried
---------------

(1) This is directly related to your introductory posting in the "Bummer"-thread, and we must solve the discrepancy between these two statements!

[bo198214]

Now consider this backways. Using another fixpoint y1 this relation holds again,
y1 = s^y1
V(y1)~ * Bs = V(s^y1)~ = V(y1)~
First row of W1^-1 is V(y1)~
and W0^-1 <> W1^-1 in their first rows and supposedly the same for the whole matrices W0 <> W1 and

W1^-1 * Bs = D1 * W1^-1

is another solution, provided that the form of the first row in W1^-1 is still that of a powerseries.

Sorry, the Eigensystem decomposition for Bs truncated to n is unqiue (up to permutations of eigenvalues), how can there be two solutions???

[Gottfried]

Sorry, the Eigensystem decomposition for Bs truncated to n is unqiue (up to permutations of eigenvalues), how can there be two solutions???

Hmm, we are talking about truncated matrices only as approximations, no?
When I compute the eigensystem not by numerical eigensystem-solver for finite matrices (as implemented in a software) but based on an analytical description of each entry, then I actually work with finite truncations of an assumed infinite matrix, which may provide multiple solutions for the same composed theoretical result matrix.

1/2 - 1/4 + 1/8 - ... = 1 - 2 + 4 -8 ... independently of the fact, that all finite truncations of lhs and rhs are different (and even do not converge to the same value)

Don't we agree in this?

[bo198214]

Hmm, we are talking about truncated matrices only as approximations, no?
When I compute the eigensystem not by numerical eigensystem-solver for finite matrices (as implemented in a software) but based on an analytical description of each entry, then I actually work with finite truncations of an assumed infinite matrix, which may provide multiple solutions for the same composed theoretical result matrix.

But this is than another method. The matrix operator method is to use truncated Matrices with unique Eigensystem decomposition and hence with a unique limit. That was the absolute good thing about this method as we dont have to take fixed points (with the attached question of which to choose) into consideration.

For the infinite matrices there is not even only one solution for each fixed point but also all the other possible solutions come into play, i.e. the non-regular Schroeder functions. So how should this help? If we know there are some solutions out there which we also want to have and than modify our method (that was designed to be unique) also to include the other solutions???