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
The computed matrix POLY begins as follows
so we have to compute the column-vector A_h of coefficents for Ut°h(x)
Check:
For h=1 we get the beginning of A_1
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':
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:
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
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
... ....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)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
...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
+ ...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