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 the inconsistency depending on fixpoint-selection Gottfried Ultimate Fellow Posts: 786 Threads: 121 Joined: Aug 2007 03/06/2008, 04:06 PM (This post was last modified: 03/06/2008, 05:15 PM by Gottfried.) Hi Henryk - I'm rereading the "bummer"-posts and I just tested your example-function $\hspace{24} f(x) = x^2 + x - 1/16$ with my matrix-analysis. However, it is not clear to me, * how the different fixpoints are involved here (update:but maybe I've got it now) * what the plot in your first msg describes exactly. What I've got so far is a good result for the half-iterate, which seems to be exact to more than 20 digits. I've also used a shifting of x->x' to get a triangular matrix with exact terms, where I denote $\hspace{24} \begin{matrix} x' &=& x/(1/4) - 1 \\ x'' &=& (x+1)*1/4 \\ g(z)&=& 1/4 z^2 + 3/2 z \end{matrix} $ and with the above then $\hspace{24} f(x) = g(x')''$ The matrix-operator G for g(x) is then triangular. Results: $\hspace{24} \begin{matrix} {llll} & f^{(h)}(x) & g^{(h)}(x) \\ h=0 & 1.0000000000000000000 & 3 \\ h=0.5 & 1.3427402879772577347 & 4.3709611519090309387 \\ h=1 & 1.9375000000000000000 & 6.7500000000000000000 \end{matrix}$ [update] If I understood the fix-point-problem correctly, then the other transformations x', x", and the second function g2(x) are $\hspace{24} \begin{matrix} x' &=& x/(-1/4) + 1 \\ x'' &=& (x-1)*(-1/4) \\ g2(z)&=& -1/4 z^2 + 3/2 z \end{matrix} $ Then I get identical results for the back-shifted solutions (checked up to 20 digits): g2(x) = $\hspace{24} \begin{matrix} {r} 1.0000000000000000000 & -3 \\ 1.3427402879772577347 & -4.3709611519090309387 \\ 1.9375000000000000000 & -6.7500000000000000000 \end{matrix}$ ---------------------------------------------------------------------- Documents: Matrix-operator F for f(x) Code:´   1  -1/16  1/256  -1/4096  1/65536  -1/1048576   0      1   -1/8    3/256  -1/1024     5/65536   0      1    7/8  -45/256  23/1024  -155/65536   0      0      2      5/8   -13/64     35/1024   0      0      1    45/16   35/128   -405/2048   0      0      0        3     13/4     -17/128Because this operator is not rowfinite, I use the substitution x->x' and f(x)=g(x')" , replacing the original matrix-multiplication $\hspace{24} V(x)\sim * F = V(f(x))\sim$ by $\hspace{24} V(x')\sim * G = V(g(x'))\sim$ and finally apply re-substitution. Matrix-operator G for g(x) Code:´   1    .     .      .      .       .   0  3/2     .      .      .       .   0  1/4   9/4      .      .       .   0    0   3/4   27/8      .       .   0    0  1/16  27/16  81/16       .   0    0     0   9/32   27/8  243/32 The fractional powers for g(x) are expressed by fractional powers of G, so we need the eigensystem-decomposition G = W * D * W^-1 W = Code:´   1         .        .      .     .  .   0         1        .      .     .  .   0      -1/3        1      .     .  .   0      2/15     -2/3      1     .  .   0   -49/855    17/45     -1     1  .   0  158/6175  -58/285  11/15  -4/3  1 D = diag() (which is a divergent sequence of eigenvalues!) Code:´   1  3/2  9/4  27/8  81/16  243/32 W^-1 = Code:´   1            .         .    .    .  .   0            1         .    .    .  .   0          1/3         1    .    .  .   0         4/45       2/3    1    .  .   0      52/2565     13/45    1    1  .   0  2048/500175  256/2565  3/5  4/3  1 The half-power G^0.5 = Code:´   1.0000000               .              .              .           .          .           0       1.2247449              .              .           .          .           0     0.091751710      1.5000000              .           .          .           0   -0.0067347010     0.22474487      1.8371173           .          .           0    0.0010135508  -0.0080782047     0.41288269   2.2500000          .           0  -0.00019936640   0.0012468417  0.00062493491  0.67423461  2.7556760 And using the second column of G^0.5 with x'=3 according to $\hspace{24} V(x')\sim * G^{0.5}$ we get the diverging sequence of terms of the powerseries (first 64 terms) Code:´                0        3.6742346       0.82576539      -0.18183693      0.082097618     -0.048446035      0.032977002     -0.024500419      0.019276236 ...       -1.1067517        2.6939736       -5.1977227        9.0095267       -14.654213        22.823197       -34.410169        50.545198       -72.619728        102.28920       -141.43120        192.02334       -255.88447        334.19212       ... which seems to be Euler-summable to $\hspace{24} V(3)\sim * G^{0.5} = V(4.3709611519090309387...)\sim$ so $\hspace{24} g^{(0.5)}(3)=4.3709611519090309387...$ The next iteration is even more divergent, but again seems to be Euler-summable to get $\hspace{24} V(4.3709611519090309387...)\sim * G^{0.5} = V(6.750)\sim$ The backtransformation of these values gives $\hspace{24} \begin{matrix} 3'' &=& 1 \\ 4.37096115...'' &=& 1.3427... \\ 6.750'' &=& 1.9375 \end{matrix}$ Appendix: [update] The half-power G2^0.5 (developed using the second fixpoint) is Code:´   1.0000000               .              .              .            .          .           0       1.2247449              .              .            .          .           0    -0.091751710      1.5000000              .            .          .           0   -0.0067347010    -0.22474487      1.8371173            .          .           0   -0.0010135508  -0.0080782047    -0.41288269    2.2500000          .           0  -0.00019936640  -0.0012468417  0.00062493491  -0.67423461  2.7556760 where apparently only the signs have changed, but the numerical digits are the same up to the 20'th digit. ======================================================================== Now, at some place I need the idea, how to introduce the other fixpoint correctly (if I didn't get it this way), and then, what computation your plot shows. Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread the inconsistency depending on fixpoint-selection - by Gottfried - 02/07/2008, 03:06 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 02/07/2008, 05:20 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/04/2008, 10:14 AM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/04/2008, 11:17 AM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/04/2008, 11:25 AM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/04/2008, 11:32 AM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/04/2008, 11:43 AM RE: the inconsistency depending on fixpoint-selection - by GFR - 02/07/2008, 09:02 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 02/07/2008, 09:49 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/03/2008, 11:20 AM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/06/2008, 04:06 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/06/2008, 04:58 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/06/2008, 05:25 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/07/2008, 03:16 AM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/07/2008, 10:56 AM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/07/2008, 07:17 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/07/2008, 08:30 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/07/2008, 09:38 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/08/2008, 01:22 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/08/2008, 01:52 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/08/2008, 04:21 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/09/2008, 09:30 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/14/2008, 02:10 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/14/2008, 04:41 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/14/2008, 07:19 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 04/29/2008, 08:15 AM

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