fractional iteration with complex bases/a bit of progress Gottfried Ultimate Fellow     Posts: 889 Threads: 130 Joined: Aug 2007 07/21/2008, 05:26 PM (This post was last modified: 07/21/2008, 09:36 PM by Gottfried.) An intermediate result for fractional iterates of T-tetration of non-convergent bases (complex fixpoints) Using base b=2, then the fixpoint-shift is performed by t = 0.824678546142 + 1.56743212385*I u = 0.571623609127 + 1.08646115737*I = log(t) such that b = t^(1/t) = exp(u/t) . Using the fixpointshift x' = x/t-1 x"=(x+1)*t and the according triangular Bell-matrix Ut I could now compute some fractional iterates, which was not possible before. There might be some mistake in my diagonalization-procedure due to fractional powers if the argument is complex. Because I could not locate the error yet I employed my new exponential polynomial interpolation. *In principle* this seems a start, the error seems to not to occur here. (For the method see : http://go.helms-net.de/math/tetdocs/Expo...lation.pdf ) To check: Ut begins as follows Code:1                           .                        .                        .                        .                      .   0      0.57162361+1.0864612*I                        .                        .                        .                      .   0    -0.42682215+0.62104685*I  -0.85364430+1.2420937*I                        .                        .                      .   0   -0.30624163-0.036240215*I  -1.8374498-0.21744129*I  -1.8374498-0.21744129*I                        .                      .   0  -0.033920340-0.088358850*I  -0.47488476-1.2370239*I   -1.2211322-3.1809186*I  -0.81408817-2.1206124*I                      .   0   0.015321758-0.017472227*I  0.45965275-0.52416682*I    2.2982637-2.6208341*I    3.6772220-4.1933346*I  1.8386110-2.0966673*I   ...     ....The computed matrix POLY begins as follows Code:0                           .                          .                          .                            .                           .   0                   1.0000000                          .                          .                            .                           .   0    -0.34295950+0.39829087*I    0.34295950-0.39829087*I                          .                            .                           .   0    0.011293278-0.29398501*I   0.082028794+0.54639055*I  -0.093322072-0.25240554*I                            .                           .   0    0.086289977+0.13933724*I   -0.34931427-0.28800518*I    0.39760954+0.14818664*I  -0.13458524+0.00048129196*I                           .   0  -0.080900876-0.038017321*I  0.33173844+0.0088923457*I   -0.42115742+0.18129664*I      0.18386237-0.21507654*I  -0.013542508+0.062904872*I   ...      .... so we have to compute the column-vector A_h of coefficents for Ut°h(x) Code:´ A_h         = POLY * V(u^h) V(x)~ * A_h = Ut°h(x)Check: For h=1 we get the beginning of A_1 Code:´ A_1:                            0       0.57162361+1.0864612*I     -0.42682215+0.62104685*I    -0.30624163-0.036240215*I   -0.033920340-0.088358850*I    0.015321758-0.017472227*I    ...which is of course the same as the 2nd column in Ut itself. The vector A_0.5 for Ut°0.5(x') begins with first 64 coefficients as follows and seems to have a slowly divergent sequence of entries. Here is Ut°0.5(x') given as powerseries in x': Code:´                                        0 +           (0.94849474+0.57272914*I)*x'   +       (0.075363236+0.32629361*I)*x'^2 +      (-0.017446691+0.022563745*I)*x'^3   +   (-0.0034925347-0.0020992659*I)*x'^4 +  (-0.000015974734+0.0036674771*I)*x'^5   +    (0.0045440026-0.0055371510*I)*x'^6 +      (-0.010881477+0.012802630*I)*x'^7   +     (0.0094010926-0.027893929*I)*x'^8 +      (0.0068992435+0.039389807*I)*x'^9   +    (-0.029755850-0.034623204*I)*x'^10 +      (0.043933293+0.013888987*I)*x'^11   +    (-0.041102377+0.011322125*I)*x'^12 +      (0.024005637-0.028919052*I)*x'^13   +   (-0.0011784702+0.033375740*I)*x'^14 +     (-0.019605161-0.025397896*I)*x'^15   +    (0.033453991+0.0081087366*I)*x'^16 +     (-0.036878144+0.014840833*I)*x'^17   +     (0.027006852-0.038148976*I)*x'^18 +    (-0.0036781248+0.053912245*I)*x'^19   +    (-0.027708275-0.053842125*I)*x'^20 +      (0.056195979+0.033961262*I)*x'^21   +   (-0.069317259+0.0017624491*I)*x'^22 +      (0.058909002-0.041734262*I)*x'^23   +    (-0.025253063+0.070928405*I)*x'^24 +     (-0.022366540-0.076595337*I)*x'^25   +     (0.068379966+0.053024259*I)*x'^26 +    (-0.095592452-0.0041201605*I)*x'^27   +     (0.090634618-0.056569702*I)*x'^28 +      (-0.049205371+0.10866605*I)*x'^29   +     (-0.020385356-0.13073050*I)*x'^30 +       (0.097340351+0.10806611*I)*x'^31   +     (-0.15348021-0.040025855*I)*x'^32 +       (0.16266104-0.056519720*I)*x'^33   +      (-0.11153240+0.15003727*I)*x'^34 +      (0.0074207583-0.20364656*I)*x'^35   +       (0.12034367+0.18834832*I)*x'^36 +      (-0.22729198-0.095748995*I)*x'^37   +      (0.26774097-0.054555400*I)*x'^38 +       (-0.21197038+0.21665590*I)*x'^39   +      (0.060973606-0.33029389*I)*x'^40 +        (0.14726472+0.34088251*I)*x'^41   +      (-0.34428288-0.22181063*I)*x'^42 +      (0.45090197-0.0090434807*I)*x'^43   +      (-0.40560503+0.28635743*I)*x'^44 +        (0.19241344-0.51222367*I)*x'^45   +       (0.14274919+0.58702754*I)*x'^46 +       (-0.49615266-0.44787066*I)*x'^47   +       (0.73375762+0.10126897*I)*x'^48 +       (-0.73663932+0.36383272*I)*x'^49   +       (0.45073248-0.79217029*I)*x'^50 +        (0.077418484+1.0077057*I)*x'^51   +      (-0.69596341-0.87761013*I)*x'^52 +         (1.1830357+0.37401427*I)*x'^53   +       (-1.3174698+0.39105319*I)*x'^54 +         (0.96551733-1.1773648*I)*x'^55   +       (-0.15336955+1.6810426*I)*x'^56 +        (-0.90616777-1.6389445*I)*x'^57   +        (1.8539135+0.94127137*I)*x'^58 +        (-2.2912467+0.28977904*I)*x'^59   +         (1.9273758-1.6940103*I)*x'^60 +        (-0.71865274+2.7591658*I)*x'^61   +        (-1.0565265-2.9866144*I)*x'^62 +          (2.8316963+2.0905013*I)*x'^63   + ...The series converges very slowly for x=1 (x'=1/t -1), if at all, and is very poorly Euler-summable. But using more appropriate example-values for x, such that x'=1,x'=1.5 or the like we get some meaningful values, well approximatable (needs only small orders for Euler-summation). Some results: Code:x'            =  1   y=Ut°0.5(x')  =  1.0035209 + 0.92089313*I  abs(y) = 1.3620200   z=Ut°0.5(y)   = -0.17532145 + 1.5674321*I  abs(z) = 1.5772067   check: t^x'-1 = -0.17532145 + 1.5674321*I   x  =          =  1.6493571  + 3.1348642*I  abs(x) = 3.5422807   y" =Tb°0.5(x) =  0.20882326 + 3.8998239*I  abs(y")= 3.9054108   z" =Tb°0.5(y")= -1.7767488  + 2.5852553*I  abs(z")= 3.1369382      check:2^x     = -1.7767488  + 2.5852553*I -----------------------------------------------------------------   x'            =  1.2500000   y=Ut°0.5(x')  =  1.2626569  + 1.2681218*I  abs(y)= 1.7895349   z=Ut°0.5(y)   = -0.56863584 + 1.9971715*I  abs(z)= 2.0765454   check: t^x'-1 = -0.56863584 + 1.9971715*I   x  =          =  1.8555267  + 3.5267223*I  abs(x )= 3.9850658   y" =Tb°0.5(x) = -0.12173015 + 4.5923540*I  abs(y")= 4.5939670   z" =Tb°0.5(y")= -2.7746940  + 2.3231585*I  abs(z")= 3.6188385   check:2^x     =  -2.7746940 + 2.3231585*I -----------------------------------------------------------------   x'            =   1.5000000   y=Ut°0.5(x')  =   1.5206787  + 1.6658045*I  abs(y) = 2.2555195   z=Ut°0.5(y)   =  -1.1387426  + 2.3530211*I  abs(z) = 2.6140855   check: t^x'-1 =  -1.1387426  + 2.3530211*I   x  =          =   2.0616964  + 3.9185803*I  abs(x )= 4.4278509   y" =Tb°0.5(x) =  -0.53228577 + 5.3247459*I  abs(y")= 5.3512847   z" =Tb°0.5(y")=  -3.8026189  + 1.7230164*I  abs(z")= 4.1747690   check:2^x     =  -3.8026189  + 1.7230164*I ----------------------------------------------------------------- Anyway - things are still not yet fully satisfying. The range for parameters is very much limited, and there are still incompatibilities. The ugliest problem is the divergence with complex numbers involved - the common summation-methods are simply not taylored for complex series. Sigh... Gottfried Gottfried Helms, Kassel Gottfried Ultimate Fellow     Posts: 889 Threads: 130 Joined: Aug 2007 07/21/2008, 10:58 PM Gottfried Wrote:Some results: Code:x'            =  1   y=Ut°0.5(x')  =  1.0035209 + 0.92089313*I  abs(y) = 1.3620200   z=Ut°0.5(y)   = -0.17532145 + 1.5674321*I  abs(z) = 1.5772067   check: t^x'-1 = -0.17532145 + 1.5674321*IHere is a plot for fractional heights from h=-3 to h=3 by 1/10-steps using x'=1. Gottfried Helms, Kassel « Next Oldest | Next Newest »

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