Tetration

Gottfried Helms - Univ Kassel 07' 2007 - 2008

 

 

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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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, x
2, x3,...)
dV(x) - used as diagonal-matrix

T - the matrix which performs T-tetration to base b (in our forum:"exp
_bt(x)" ):
such that V(x)~ * T = V ( b
x )~

U - the matrix which performs U-tetration to base b (in our forum:"dxp_bt(x)" ):
such that V(x)~ * U = V( b
x 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:

 

 

 

 

 

 

T 1 =

T * dV(b^^0) = T * dV(1) = T

 

 

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 4 =

U * dV(b^^0) * U * dV(b^^1) * U * dV(b^^2) * T * dV(b^^3)

 

 

...

 

 

 

T h =

(kh=-02 U * dV(b^^k) ) * T * dV(b^^h-1)

 

 

Using the decomposition T = U * P~ we can systematize even more:

 

 

Th=

(kh=-01 U * dV(b^^k) ) * P^^h-1~

 

 

 

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 * b
b^x 1
= b
b^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

 

 

col 1

col 1, common powers of lambda distributed

 

1*

 

b

b*1

 

x/1!*

 

bλ(λ)

b*( 1λ )

 

x/2!*

 

bλ(λ+λ)

b*( 1λ + 1λ4 )

 

x/3!*

 

bλ(1λ+3λ+1λ)

b*( 1λ4 + 3λ5 + 1λ6 )

 

x4/4!*

 

bλ4(1λ+7λ+6λ+1λ4)

b*( 1λ5 + 7λ6 + 6λ7 + 1λ8 )

 

 

and then, if x=1 (which mimics the common tetration) collecting like powers of λ :

bb = b(1+1λ+1/2 λ+ 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 = 1λ+1/2 λ+ 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
------------------------------------------------------------
  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
------------------------------------------------------------
  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