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Analytic entries for T-tetration-operator
Symbolic description for entries of the integer-powers of the T-tetration Bellmatrix
Vers 2.0
This is a reformatted (and slightly extended) excerpt
of the thread "Matrix Operator
Method" (printable version)
in "tetrationforum"
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Matrix Operator Method
- Printable Version
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Exact entries for T-tetration Bell-matrix - Gottfried – 24.09.08 21:22
Just derived a method to compute exact entries for powers of the (square) matrix-operator for T-tetration.
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 iteration-height 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:
b^^h - the powertower of
height h using base b
V(x) - the vandermonde-column-vector containing consecutive powers of
its parameter x:
V(x) = column(1,
x, x2, x3,...)
dV(x) - used as diagonal-matrix
T - the matrix which
performs T-tetration to base b (in
our forum:"exp_b°t(x)" ):
such that V(x)~
* T = V ( bx )~
U - the matrix which
performs U-tetration to base b (in
our forum:"dxp_b°t(x)" ):
such that V(x)~
* U = V( bx – 1) ~
Note, that U
is lower triangular.
The
triangularity allows to compute exact entries for the integer matrix-powers.
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:
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T 1 = |
T * dV(b^^0) = T * dV(1) = T |
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T 2 = |
U * dV(b^^0) * T * dV(b^^1) |
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T 3 = |
U * dV(b^^0) * U * dV(b^^1) * T * dV(b^^2) |
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T 4 = |
U * dV(b^^0) * U * dV(b^^1) * U * dV(b^^2) * T * dV(b^^3) |
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... |
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T h = |
(∏kh=-02 U * dV(b^^k) ) * T * dV(b^^h-1) |
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Using the decomposition T = U * P~ we can systematize even more:
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Th= |
(∏kh=-01 U * dV(b^^k) ) * P^^h-1~ |
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This finding is interesting, because in my matrix-method I had to use fixpoint-shift to get exact entries even for integer powers, and since the fixpoints for T-tetration are real only for a small range of bases we had to deal with complex-valued U-matrices when considering the general case. Here we do not need a fixpoint-shift.
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 T2, 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.
Additionally, the rows are related to the powers of x in a powerseries. So from the second column (headed by b^^1 = b) we have the powerseries for b^b^x:
b^b^x = b^^1*[ 1/0!*x0*(1 )
+ λ /1!*x1*
(1*λ)
+
λ2 /2!*x2* (1*λ+1* λ2)
+
λ3 /3!*x3* (1*λ+3* λ2+1* λ3)
+
... ]
= b* [
1 +
+ λ1 x1/1! *
(1λ)
+ λ2 x2/2! *
(1λ + 1λ2)
+ λ3 x3/3! *
(1λ + 3λ2
+ 1λ3)
+ λ4 x4/4! *
(1λ + 7λ2 + 6λ3 +
1λ4)
+
...]
If we change order of computation, and look at the columns in the triangle of the rhs, we recognize, that these are just the columns of the matrix of stirling-numbers 2nd kind, where each column has a power of log(b) as cofactor. So we can compute each of the columns first (using the common factors at the head of each row giving simple powers of exponentialseries-minus-1 according to the known transformation-property of the stirling-matrix) to get
= b*[ 1+
λ*(exp(λx)-1) + λ2*(exp(λx)-1)2/2! +
λ3*(exp(λx)-1)3/3! +... ]
= b*[
1+ λ*( bx -1
) + λ2*( bx
– 1 )2/2! + λ3*( bx – 1 )3/3! +... ]
Now the bracketed expression is just another exponentialseries; this is for exp(log(b)*(bx – 1)) so we have
= b *
exp(log(b) * (bx –1))
= b * bb^x –1
= bb^x
which is the result, which we expect if we write
V(x)~ T2 = V(bb^x)~
The next table shows analoguously the top left of the symbolic T3. (b^^2 is just b^b). Legend as before
What is now the powerseries for bb (without the variable x) in terms of log(b)?
One interpretation would be to use the powerseries of bx developed for powers of x evaluated at x=b. We get then simply
bx = exp(log(b)*x) =x=b 1 + b*log(b)/1! + b2*log(b)2/2! + ...
Considering things the way which we discussed here, this means we have the formula for bb^x developed for powers of x (which we already got in the above tables), evaluated at x=1 and then like powers of log(b) collected:
Reordering in col 1 of the table for T2 for like powers of λ gives first
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col 1 |
col 1, common powers of lambda distributed |
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1* |
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b |
b*1 |
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x/1!* |
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bλ(λ) |
b*( 1λ² ) |
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x²/2!* |
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bλ²(λ+λ²) |
b*( 1λ³ + 1λ4 ) |
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x³/3!* |
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bλ³(1λ+3λ²+1λ³) |
b*( 1λ4 + 3λ5 + 1λ6 ) |
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x4/4!* |
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bλ4(1λ+7λ²+6λ³+1λ4) |
b*( 1λ5 + 7λ6 + 6λ7 + 1λ8 ) |
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and then, if x=1 (which mimics the common tetration) collecting like powers of λ :
bb = b(1+1λ²+1/2 λ³+ 2/3 λ4 + 13/24 λ5 + 7/15 λ6 + ...
Note, that here no higher powers of b occur, different from the first example.
To rewrite this as powerseries depending only on λ and having zero constant, we rearrange the b and the 1 to the left:
bb-1 – 1 = 1λ²+1/2 λ³+ 2/3 λ4 + 13/24 λ5 + 7/15 λ6 + ...
Using Pari/GP the latter series can be crosschecked (using "u" for "λ"):
Pari/GP: exp(‘u*(exp(‘u)-1))-1
%208 = u^2 + 1/2*u^3 + 2/3*u^4 + 13/24*u^5 + 7/15*u^6 + 271/720*u^7 +
739/2520*u^8 + 1781/8064*u^9 + 1465/9072*u^10 + 417091/3628800*u^11 +
1593301/19958400*u^12 + 3710011/68428800*u^13 + 56148167/1556755200*u^14 +
2051959183/87178291200*u^15 + 9865284983/653837184000*u^16 +
28429908007/2988969984000*u^17 + O(u^18)
The numerical difference between the symbolic computation, call it Tksym , and the "naive" matrixpower of the empirical (truncated) matrix T is interesting.
I used dim=64x64, base b=sqrt(2), and compare the 3rd powers which provides still a good approximation for this base when the "naive" matrix-power is computed (just compute the product of the truncated T: T3naive =T64x64*T64x64*T64x64).
Below are two (zoomed) images for the difference T3sym – T3naive: very good approximation in the leading 12 columns (abs differences to the exact values <1e-20 ), 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)~* T3naive = Y~
is actually approximately vandermonde and thus satisfies the expectation of the analytic formula such that Y =approx 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 col-sums of the "naive" third power of T, using x=1)
column symbolic "naive"
difference
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0
-1.1383136E-19 0.E-201
-1.13831366798E-19
1
1.00000000000 1.00000000000 0.E-202
2
2.00000000000 2.00000000000 1.11189479752E-19
3
3.00000000000 3.00000000000 2.20208219209E-19
4
4.00000000000 4.00000000000 3.27448139227E-19
....
42 42.0000000000 42.0000000000 1.19448154596E-11
43 43.0000000000 43.0000000000 3.09972987348E-11
44 44.0000000000 43.9999999999 7.77154497620E-11
45 45.0000000000 44.9999999998 1.88541576938E-10
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
However, for base b=2 this looks already catastrophic for the "naive"-computation:
column symbolic "naive"
difference
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0 -4.36636233681E-20 0.E-202 -4.36636233681E-20
1
1.00000000000 1.00000000000
0.E-202
2
2.00000000000 2.00000000000
2.46662212171E-14
3
3.00000000000 2.99999999998
2.31650095614E-11
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 tetration-matrix T.
Gottfried Helms, 27.9.2008