Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Matrix-method: compare use of different fixpoints
#1
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
Gottfried Helms, Kassel
Reply


Messages In This Thread
Matrix-method: compare use of different fixpoints - by Gottfried - 11/04/2007, 12:38 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  The Promised Matrix Add On; Abel_M.gp JmsNxn 2 413 08/21/2021, 03:18 AM
Last Post: JmsNxn
  Revisting my accelerated slog solution using Abel matrix inversion jaydfox 22 29,819 05/16/2021, 11:51 AM
Last Post: Gottfried
  Which method is currently "the best"? MorgothV8 2 7,549 11/15/2013, 03:42 PM
Last Post: MorgothV8
  "Kneser"/Riemann mapping method code for *complex* bases mike3 2 9,867 08/15/2011, 03:14 PM
Last Post: Gottfried
  An incremental method to compute (Abel) matrix inverses bo198214 3 12,891 07/20/2010, 12:13 PM
Last Post: Gottfried
  SAGE code for computing flow matrix for exp(z)-1 jaydfox 4 13,054 08/21/2009, 05:32 PM
Last Post: jaydfox
  regular sexp:different fixpoints Gottfried 6 17,966 08/11/2009, 06:47 PM
Last Post: jaydfox
  Convergence of matrix solution for base e jaydfox 6 14,081 12/18/2007, 12:14 AM
Last Post: jaydfox



Users browsing this thread: 1 Guest(s)