With some more analytical matrix-operations and numerical checks I now tend to the conclusion, that Asimov's proposal may be proven to be false.
I checked the occuring matrices in more depth and found very convincing numerical results, that -at least for the case b=sqrt(2), t0=2 and t1=4- the results are equal. For integer heights this is obvious from the scalar expression only (sqrt(2)^2 = 2, sqrt(2)^4=4 and so on), the problem occurs with fractional heights. I checked this now for h=1/2, using the powerseries, which I get using eingensystem-decomposition/diagonalization.
I used a better routine to compute the eigen-matrices up to dimension 160x160 and some transformations get those different eigenmatrices summable to the same values. Although this is only in certain numerical approximation the involved transformations are simple binomial-transformations, which then may be analytically derived as well. So my challenge is now to put some effort to perform those analytical derivations since the possibility of success is backed better.
Let b=sqrt(2), t0 = 2, t1 = 4, h the height using h=1/2
Then, in my usual notation we should get
V(x)~ * Bb^0.5 =V(y)~
where y = T_b°0.5(x)
With the fixpoints, using the h()-function with indexes
t0 = h_0(b)=2 t1 = h_1(b)=4
this is equivalent to the two different fixpoint-based matrix expressions (using PInv for P^-1)
so we request:
Rearrange dV(1/t0):
Now set t1/t0=a and the part dV(1/a)*PInv ~ in the rhs can be expanded according to
and we get, with a= 2, also writing U2 for U_t0 and U4 for U_t1
and this gives then, arranging P^-2 and P^2 to the left:
We see, that -unfortunately- we still have infinite sums in the lhs, namely all rows of P~ are infinite as well as the columns of U2^0.5. So we have either to determine these sums analytically (for what I've no solution currently) or employ accelerating methods, like Euler-transform/summation. It occurs, that these row/column-products are not well Euler-accelerable; the Euler-sum does not converge well. The second multiplication, U2^0.5 * PInv~, however provides a simple result: since only the second column of the result is interesting, only the top left 2x2-segment of PInv~ is of relevance, and this simply subtracts the first [1,0,0,0,...] column of U2^0.5 from its second column - which is happily trivial.
Applying Euler-summation anyway we get for the first few items of the lhs and rhs identity within a certain range of accuracy, see end of msg.
Some more tests gave even better accuracy with an additional powerseries in x, which makes -for abs(x)<1- the resulting powerseries convergent using the first 160 terms:
where the lhs can be rewritten using the binomial-theorem and makes
where the implicte infinite series in the matrix-product P~*U2^0.5 are now removed.
Tests with various x, which make the matrix-multiplication convergent, should then give the same results for the lhs and rhs. However, using various different x does not prove the identity, but makes it more likely.
(see last example, where x was set x=-1/2)
The matrix U2^0.5 was constructed from the analytic eigen-decomposition (160x160):
Let u0=log(t0)=log(2), u1 = log(t1)=log(4)
where W2 is the matrix of eigenvectors and D2 the matrix of eigenvalues of the matrix U2=dV(u0)*S2. Since the matrix U2 is triangular, their eigenvalues can be taken from the diagonal (and are thus identical to the entries in dV(u0)) and their eigenvector-matrices are assumed to be triangular, too. Using my analytical description for the eigenmatrices we get exact terms for any dimension. The same method was applied to U4 (based on the second fixpoint):
=======================================================================
Documents:
-------------------------------------------------------------
Always: rows 0..10 , 149..159, columns 0..3; col 1 is of interest in U2^0.5 and U4^0.5
Comparision of first 8 terms, produced by the different fixpoints-matrices.
With Euler-summation of different orders for the non-converging vector-products in P~ * (U2^0.5*P^-1 ~) I get for the first eight terms
Here are partial sums if the matrices are used as coefficients for a powerseries in x
Here is x=-1/2 for the two versions from U2 and U4. The approximations are very good and both results seem to be equal
(only the 2'nd columns are relevant, but also the other columns provide equal results).
The partial sums are sequentially rowwise, according to the increasing number of involved terms.
=======================================================================
Conclusion:
Although essential approximations are poor I'm now much confident, that either with a better tool for convergence-acceleration or with an analytical approach based on the formal description of terms using my solution for the eigen-system, a chance to get a result becomes more realistic and should now be worth a more serious effort.
Gottfried
I checked the occuring matrices in more depth and found very convincing numerical results, that -at least for the case b=sqrt(2), t0=2 and t1=4- the results are equal. For integer heights this is obvious from the scalar expression only (sqrt(2)^2 = 2, sqrt(2)^4=4 and so on), the problem occurs with fractional heights. I checked this now for h=1/2, using the powerseries, which I get using eingensystem-decomposition/diagonalization.
I used a better routine to compute the eigen-matrices up to dimension 160x160 and some transformations get those different eigenmatrices summable to the same values. Although this is only in certain numerical approximation the involved transformations are simple binomial-transformations, which then may be analytically derived as well. So my challenge is now to put some effort to perform those analytical derivations since the possibility of success is backed better.
Let b=sqrt(2), t0 = 2, t1 = 4, h the height using h=1/2
Then, in my usual notation we should get
V(x)~ * Bb^0.5 =V(y)~
where y = T_b°0.5(x)
With the fixpoints, using the h()-function with indexes
t0 = h_0(b)=2 t1 = h_1(b)=4
this is equivalent to the two different fixpoint-based matrix expressions (using PInv for P^-1)
Code:
´
V(x)~ * Bb^0.5 = V(x)~ * dV(1/t0)*PInv ~ * U_t0^0.5 * P~ * dV(t0)
V(x)~ * Bb^0.5 = V(x)~ * dV(1/t1)*PInv ~ * U_t1^0.5 * P~ * dV(t1)Code:
´
dV(1/t0)*PInv ~ * U_t0^0.5 * P~ * dV(t0)
= dV(1/t1)*PInv ~ * U_t1^0.5 * P~ * dV(t1)Rearrange dV(1/t0):
Code:
´
PInv~ * U_t0^0.5 * P~
= dV(t0/t1)*PInv ~ * U_t1^0.5 * P~ * dV(t1/t0)Now set t1/t0=a and the part dV(1/a)*PInv ~ in the rhs can be expanded according to
Code:
´
dV(1/a)*P^-1 ~ = (dV(1/a) P^-1~ dV(a))*dV(1/a)
= (dV(a) P^-1 dV(1/a)) ~ * dV(1/a)
= P^-a ~ * dV(1/a)and we get, with a= 2, also writing U2 for U_t0 and U4 for U_t1
Code:
´
P^-1 ~ * U2^0.5 * P~
= P^-2 ~ * dV(1/2) * U4^0.5 * dV(2)* P^2 ~and this gives then, arranging P^-2 and P^2 to the left:
Code:
´
P ~ * U2^0.5 * PInv ~
= dV(1/2) * U4^0.5 * dV(2)We see, that -unfortunately- we still have infinite sums in the lhs, namely all rows of P~ are infinite as well as the columns of U2^0.5. So we have either to determine these sums analytically (for what I've no solution currently) or employ accelerating methods, like Euler-transform/summation. It occurs, that these row/column-products are not well Euler-accelerable; the Euler-sum does not converge well. The second multiplication, U2^0.5 * PInv~, however provides a simple result: since only the second column of the result is interesting, only the top left 2x2-segment of PInv~ is of relevance, and this simply subtracts the first [1,0,0,0,...] column of U2^0.5 from its second column - which is happily trivial.
Applying Euler-summation anyway we get for the first few items of the lhs and rhs identity within a certain range of accuracy, see end of msg.
Some more tests gave even better accuracy with an additional powerseries in x, which makes -for abs(x)<1- the resulting powerseries convergent using the first 160 terms:
Code:
´
V(x)~* P ~ * U2^0.5 * PInv ~
=V(x)~* dV(1/2) * U4^0.5 * dV(2)where the lhs can be rewritten using the binomial-theorem and makes
Code:
´
V(x+1)~ * U2^0.5 * PInv ~
=V(x)~* dV(1/2) * U4^0.5 * dV(2)where the implicte infinite series in the matrix-product P~*U2^0.5 are now removed.
Tests with various x, which make the matrix-multiplication convergent, should then give the same results for the lhs and rhs. However, using various different x does not prove the identity, but makes it more likely.
(see last example, where x was set x=-1/2)
The matrix U2^0.5 was constructed from the analytic eigen-decomposition (160x160):
Let u0=log(t0)=log(2), u1 = log(t1)=log(4)
Code:
´
U2 = dV(u0) * S2 // S2 is the factorially similarity-scaled matrix if Stirling-numbers 2'nd kind
= W2 * D2 * W2^-1
U2^0.5 = W2 * D2^0.5 * W2^-1where W2 is the matrix of eigenvectors and D2 the matrix of eigenvalues of the matrix U2=dV(u0)*S2. Since the matrix U2 is triangular, their eigenvalues can be taken from the diagonal (and are thus identical to the entries in dV(u0)) and their eigenvector-matrices are assumed to be triangular, too. Using my analytical description for the eigenmatrices we get exact terms for any dimension. The same method was applied to U4 (based on the second fixpoint):
Code:
´
U4 = dV(u1) * S2
= W4 * D4 * W4^-1
U4^0.5 = W4 * D4^0.5 * W4^-1=======================================================================
Documents:
-------------------------------------------------------------
Always: rows 0..10 , 149..159, columns 0..3; col 1 is of interest in U2^0.5 and U4^0.5
Code:
´
W2:
1.0000000 . . .
0 1.0000000 . .
0 1.1294457 1.0000000 .
0 1.1985847 2.2588914 1.0000000
0 1.2474591 3.6728170 3.3883370
0 1.2856301 5.2023909 7.4226968
0 1.3170719 6.8257401 13.305570
0 1.3439053 8.5286133 21.207245
0 1.3673703 10.300960 31.276282
0 1.3882575 12.135263 43.645475
0 1.4071054 14.025656 58.435556
...
0 1.9669663 450.73244 48243.326
0 1.9685780 454.52192 49020.007
0 1.9701804 458.31761 49803.860
0 1.9717734 462.11948 50594.901
0 1.9733573 465.92749 51393.148
0 1.9749320 469.74163 52198.619
0 1.9764978 473.56185 53011.331
0 1.9780547 477.38813 53831.301
0 1.9796029 481.22044 54658.547
0 1.9811424 485.05875 55493.086
0 1.9826733 488.90302 56334.934
...
W2^-1:
1.0000000 . . .
0 1.0000000 . .
0 -1.1294457 1.0000000 .
0 1.3527103 -2.2588914 1.0000000
0 -1.6826504 3.9810682 -3.3883370
0 2.1512781 -6.4209265 7.8850737
0 -2.8091004 9.9333059 -15.655603
0 3.7304380 -15.029982 28.522828
0 -5.0228111 22.457753 -49.302115
0 6.8411612 -33.311773 82.314637
0 -9.4087785 49.202176 -134.16796
...
0 1.2260250E25 -1.6721839E26 1.5866458E27
0 -1.8513579E25 2.5287005E26 -2.4033047E27
0 2.7957655E25 -3.8240626E26 3.6403727E27
0 -4.2221159E25 5.7831789E26 -5.5143067E27
0 6.3764412E25 -8.7462552E26 8.3530285E27
0 -9.6304175E25 1.3227914E27 -1.2653332E28
0 1.4545551E26 -2.0006638E27 1.9167862E28
0 -2.1970166E26 3.0260093E27 -2.9036895E28
0 3.3185955E26 -4.5769860E27 4.3988012E28
0 -5.0129450E26 6.9231218E27 -6.6638632E28
0 7.5726685E26 -1.0472183E28 1.0095442E29
...
U2^0.5:
1.0000000 . . .
0 0.83255461 . .
0 0.15745312 0.69314718 .
0 0.010090238 0.26217664 0.57708288
0 -0.00017858491 0.041592834 0.32741456
0 0.000087842056 0.0028801157 0.082902856
0 -0.0000021818250 0.00019184203 0.011468414
0 -0.0000070205122 0.000020425106 0.0010469505
0 0.0000016647900 -0.000010572403 0.000090362152
0 0.00000060587940 0.00000048584930 -0.0000059493912
0 -0.00000023525463 0.0000013999264 -0.0000016127504
...
0 2.3828881E-14 5.4299315E-14 8.9934143E-14
0 2.0689227E-14 4.8670874E-14 8.2750369E-14
0 1.7351973E-14 4.2487147E-14 7.4493739E-14
0 1.3883561E-14 3.5884110E-14 6.5367962E-14
0 1.0347935E-14 2.8995152E-14 5.5577029E-14
0 6.8056776E-15 2.1949028E-14 4.5321681E-14
0 3.3132833E-15 1.4868077E-14 3.4796250E-14
0 -7.7428773E-17 7.8667258E-15 2.4185926E-14
0 -3.3197744E-15 1.0502594E-15 1.3664433E-14
0 -6.3725353E-15 -5.4861413E-15 3.3921124E-15
0 -9.2001722E-15 -1.1658107E-14 -6.4855900E-15
Comment: the order of 1E-14 is reached already in the ~ 30'th row and seems to decrease extremely slowly from then.
------------------------------------------------------------
W4:
1.0000000 . . .
0 1.0000000 . .
0 -1.7943497 1.0000000 .
0 3.3934259 -3.5886994 1.0000000
0 -6.5397995 10.006543 -5.3830492
0 12.722863 -25.257585 19.839351
0 -24.890972 60.430440 -61.930607
0 48.877930 -139.82513 175.90008
0 -96.234662 316.19924 -469.95850
0 189.84909 -703.21881 1202.8562
0 -375.10397 1544.2179 -2982.3442
...
0 1.9370531E44 -9.7178394E45 2.5558655E47
0 -3.8712771E44 1.9538433E46 -5.1699354E47
0 7.7369377E44 -3.9282074E46 1.0456809E48
0 -1.5462725E45 7.8973968E46 -2.1148554E48
0 3.0903312E45 -1.5876640E47 4.2769087E48
0 -6.1762668E45 3.1916737E47 -8.6486345E48
0 1.2343806E46 -6.4159924E47 1.7487744E49
0 -2.4670281E46 1.2897183E48 -3.5358114E49
0 4.9306147E46 -2.5924578E48 7.1484829E49
0 -9.8543949E46 5.2109217E48 -1.4451354E50
0 1.9695217E47 -1.0473784E49 2.9212818E50
...
W4^-1:
1.0000000 . . .
0 1.0000000 . .
0 1.7943497 1.0000000 .
0 3.0459559 3.5886994 1.0000000
0 4.9810929 9.3116028 5.3830492
0 7.9195802 20.893206 18.796941
0 12.310657 42.992653 53.513592
0 18.780115 83.386686 134.64033
0 28.192642 154.79615 311.28394
0 41.734039 277.67324 676.14910
0 61.019154 484.41060 1399.2604
...
0 3.2233680E13 7.0896715E18 4.1258282E22
0 3.6756663E13 8.5329610E18 5.1552857E22
0 4.1895929E13 1.0264451E19 6.4375659E22
0 4.7732997E13 1.2340592E19 8.0337875E22
0 5.4359849E13 1.4828683E19 1.0019614E23
0 6.1880239E13 1.7808913E19 1.2488663E23
0 7.0411146E13 2.1376785E19 1.5556698E23
0 8.0084418E13 2.5645989E19 1.9366789E23
0 9.1048598E13 3.0751791E19 2.4095655E23
0 1.0347098E14 3.6855062E19 2.9961454E23
0 1.1753991E14 4.4147026E19 3.7233358E23
...
U4^0.5
1.0000000 . . .
0 1.1774100 . .
0 0.37481156 1.3862944 .
0 0.040296534 0.88261376 1.6322369
0 0.00092111549 0.23537479 1.5587974
0 0.00044447921 0.032376274 0.66380933
0 -0.00023342988 0.0033609687 0.16318455
0 0.00010396861 -0.00014225795 0.027006196
0 -0.000025291882 0.00010651332 0.0026823939
0 -0.000014529123 0.00000038544170 0.00028006948
0 0.000027464012 -0.000045026171 0.000053888787
...
0 1.1073259E11 -1.3152838E11 1.1289422E11
0 -1.6910057E11 2.1323008E11 -1.9169761E11
0 2.3128241E11 -3.2059300E11 3.0724298E11
0 -2.5297584E11 4.2671391E11 -4.5420508E11
0 1.0978170E11 -4.3584528E11 5.8615974E11
0 5.0346351E11 8.7346252E10 -5.4905202E11
0 -2.2791585E12 1.2476824E12 -6.5225955E10
0 6.7065899E12 -4.9809105E12 2.2265937E12
0 -1.6860759E13 1.4124712E13 -8.0902843E12
0 3.8878344E13 -3.4859132E13 2.2220558E13
0 -8.4639528E13 7.9448940E13 -5.3923149E13
...
------------------------------------------------------------------------
dV(1/2)*U4^0.5*dV(2)
1.0000000 . . .
0 1.1774100 . .
0 0.18740578 1.3862944 .
0 0.010074133 0.44130688 1.6322369
0 0.00011513944 0.058843697 0.77939872
0 0.000027779951 0.0040470343 0.16595233
0 -0.0000072946837 0.00021006054 0.020398069
0 0.0000016245095 -0.0000044455608 0.0016878873
0 -0.00000019759283 0.0000016642707 0.000083824810
0 -0.000000056754387 0.0000000030112633 0.0000043760856
0 0.000000053640648 -0.00000017588348 0.00000042100615
...
0 -6.6807008E-16 1.4964761E-15 -2.4746068E-15
0 5.6873578E-16 -1.3041345E-15 2.1989300E-15
0 -4.6685677E-16 1.1041230E-15 -1.9077276E-15
0 3.6391421E-16 -8.9954404E-16 1.6058326E-15
0 -2.6129500E-16 6.9333235E-16 -1.2978634E-15
0 1.6028132E-16 -4.8822996E-16 9.8817980E-16
0 -6.2042494E-17 2.8676590E-16 -6.8084596E-16
0 -3.2370731E-17 -9.1240918E-17 3.7960165E-16
0 1.2202987E-16 -9.6283351E-17 -8.7840394E-17
0 -2.0613040E-16 2.7399096E-16 -1.9140555E-16
0 2.8399169E-16 -4.4031439E-16 4.5547407E-16
...
------------------------------------------------------------------------Comparision of first 8 terms, produced by the different fixpoints-matrices.
With Euler-summation of different orders for the non-converging vector-products in P~ * (U2^0.5*P^-1 ~) I get for the first eight terms
Code:
´
Euler-sum | Compare
P~ * U2^0.5 * P^-1 ~ | dV(1/2)*U4^0.5 *dV(2)
-----------------------+--------------------------
3.1082947E-14 | .
1.1774100 | 1.1774100
0.18740578 | 0.18740578
0.010074195 | 0.010074133
0.00011681686 | 0.00011513944
0.000027854237 | 0.000027779951
-0.0000073207548 | -0.0000072946837
0.0000016098065 | 0.0000016245095
-0.00000016722781 | -0.00000019759283Here are partial sums if the matrices are used as coefficients for a powerseries in x
Here is x=-1/2 for the two versions from U2 and U4. The approximations are very good and both results seem to be equal
(only the 2'nd columns are relevant, but also the other columns provide equal results).
The partial sums are sequentially rowwise, according to the increasing number of involved terms.
Code:
´
partial sums of
dV(-1/2) * (P~ * U2^0.5*PInv~ ) | dV(-1/2) * (dV(1/2)*U4^0.5*dV(2)
= dV(-1/2+1) * U2^0.5*PInv~ |
1.0000000 -1.0000000 1.0000000 -1.0000000 | 1.0000000 . . .
1.0000000 -0.58372269 0.16744539 0.24883192 | 1.0000000 -0.58870501 . .
1.0000000 -0.54435941 0.26200562 -0.15293863 | 1.0000000 -0.54185357 0.34657359 .
1.0000000 -0.54309813 0.29225514 -0.17533567 | 1.0000000 -0.54311283 0.29141023 -0.20402961
1.0000000 -0.54310930 0.29487702 -0.16270440 | 1.0000000 -0.54310564 0.29508796 -0.15531719
1.0000000 -0.54310655 0.29496153 -0.16037546 | 1.0000000 -0.54310651 0.29496149 -0.16050320
1.0000000 -0.54310659 0.29496460 -0.16020536 | 1.0000000 -0.54310662 0.29496477 -0.16018448
1.0000000 -0.54310664 0.29496487 -0.16019783 | 1.0000000 -0.54310663 0.29496481 -0.16019767
1.0000000 -0.54310663 0.29496481 -0.16019733 | 1.0000000 -0.54310663 0.29496481 -0.16019734
1.0000000 -0.54310663 0.29496481 -0.16019734 | 1.0000000 -0.54310663 0.29496481 -0.16019735
1.0000000 -0.54310663 0.29496481 -0.16019735 | 1.0000000 -0.54310663 0.29496481 -0.16019735
1.0000000 -0.54310663 0.29496481 -0.16019735 | 1.0000000 -0.54310663 0.29496481 -0.16019735
1.0000000 -0.54310663 0.29496481 -0.16019735 | 1.0000000 -0.54310663 0.29496481 -0.16019735
...Conclusion:
Although essential approximations are poor I'm now much confident, that either with a better tool for convergence-acceleration or with an analytical approach based on the formal description of terms using my solution for the eigen-system, a chance to get a result becomes more realistic and should now be worth a more serious effort.
Gottfried
Gottfried Helms, Kassel