[AIS] (alternating) Iteration series: Half-iterate using the AIS?

[sheldonison]

Hmm, perhaps I have made some basic error now, or something is around which I did not understand correctly from the beginning.
It is clear, that the \( \operatorname{asum}(x) \) is 2-periodic, that means \( \operatorname{asum}(x_{-1})=\operatorname{asum}(x)=\operatorname{asum}(x_1)=... \)

The same seems obvious to me for the asum-derivatives in that periods. ....
Gottfried

In general, asum(z+x) <> -asum(f(z+x)) <> asum(f(f(z))), for arbitrary values of x. This is sort of obvious since the difference of adjacent iterations goes to zero as you approach the fixed point. This explains what Gottfried is seeing, which matches expectations. The derivatives are not the same for the asum of the different points in the sequence.

asum(f^z) is 2-periodic, where f^z is a superfunction of b^z-1 developed from either fixed point, or even developed from the asum itself.
\( \text{asum}(f^{[z]}(m)) = \sum_{n=1,3,5,7...} a_n\exp(n\pi i z) + \overline{a_n}\exp(-n\pi i z) \)

If f^z is developed from the asum itself than asum(f^z) simplifies to following, where the amplitude is |a1|. This is exactly a scale/shift factor of the cosine function, since that is the definition of the asum super function.
\( \text{asum}(f_a^{[z]}(m)) = a_1\exp(\pi i z) + \overline{a_1}\exp(-\pi i z) \)

For example, take b=2. f(z)=2^z-1. This has two fixed points, zero and one. Take m=0.5, halfway in between. From the upper fixed point, z=1, generate f^z(0.5), which is entire. I choose to use the upper fixed point instead of the lower fixed point of zero, because the upper fixed point f^z is entire. Then asum(f^z(0.5)) is a 2-periodic function. Approximations for the first 16 a_n values for this function are listed below; this is the 2-periodic function I posted graphs for earlier in this thread. Carried to an infinite number of terms, this series converges if |imag(z)|<=
8.9236674*I ...because the singularity closest to the real axis is approximately 0.882829631453880483797 - 8.92366740700108685788*I. See my previous post.

Code:

a1=   3.78754977991134100255081 E-12 - 6.66245137733325455039148 E-12*I
a2=   0
a3=  -8.56153837665403679362997 E-37 + 8.20652440803445391242505 E-37*I
a4=   0
a5=  -6.17868689127540317916996 E-62 - 1.40879460535098258045236 E-61*I
a6=   0
a7=   2.69912106767933014466137 E-86 - 2.15007688226231959664247 E-86*I
a8=   0
a9=   1.05225771708086615143776 E-110 + 7.87768609052959699900088 E-111*I
a10=  0
a11= -2.36626909058726397741512 E-135 + 5.04117442927247251360066 E-135*I
a12=  0
a13= -2.24772598940997534805500 E-159 - 5.71838296679997560382250 E-160*I
a14=  0
a15=  5.57292972598194663451264 E-185 - 9.35998235598449485787905 E-184*I
a16=  0
a17=  3.64641604760555650221415 E-208 - 5.29800340799999004645756 E-209*I
a18=  0
a19=  5.14869237875480583568436 E-233 + 1.30845860927041313071075 E-232*I
a20=  0
a21= -4.16086823237678155003564 E-257 + 3.11723329796534944165436 E-257*I
a22=  0
a23= -1.52379617020143136387884 E-281 - 1.06783455902859766268479 E-281*I
a24=  0
a25=  1.48535905453654011759961 E-306 - 6.29485803391119512951806 E-306*I
a26=  0
a27=  2.15687832412641855258872 E-330 - 5.04239044035898262834287 E-331*I
a28=  0
a29=  5.48790370118910315845613 E-355 + 5.47038230073900010514912 E-355*I
a30=  0
a31= -4.38623706870332597707645 E-380 + 2.88842212562768901721380 E-379*I
... terms up to a65 required, for results accurate to 32 decimal digits for |imag(z)|<8.57

Now the next step in my algorithm is to generate the asum superfunction itself. Again, starting from the upper fixed point entire function, f^z(0.5), I calculate a 1-cyclic theta(z) function, which also has very small coefficients. But like any periodic function, the terms in theta(z) grow exponentially as imag(z) increases, as an*exp(2n pi).
\( \theta(z)= b_0 + \sum_{n=1}^{\infty} b_n\exp(2n\pi i z) + \overline{b_n}\exp(-2n\pi i z) \)
\( f_a^z=f^{z+\theta(z)} \)
\( \text{asum}(f_a^{[z]}) = \text{amplitude}\times\cos(\pi i z) \)
\( \text{amplitude}\approx 1.53275949438437638108869\times10^{-11} \)

Theta(z) has a radius of convergence that is slightly smaller, |imag(z)|<8.7793i. The nearest singularity occurs when the asum(f^z) has a derivative of zero, where the acos(asum(z)/amplitude) has a singularity, but has a defined value. Here, where f'(z)=0, the acos(f^z/amplitude)= 0.445180168159228295119296 - 8.77930110300507236428937*I. This is a simple branch singularity, but the function is starting to behave pretty poorly at this point, and there are many other singularities nearby, where the acos has singularities, or where the asum(f^z) has singularities, so it seems impossible to extend the function in any meaningful way much beyond this point in the complex plane.

The pari-gp code I have works for b in the range of 1.1 to around 2.5. With no attempt at optimizations, it takes about 8 seconds to generate the asum and theta fourier series results accurate to 32 decimal digits, for b=2, accurate in the range |imag(z)|<8.57. For 64 digits accuracy, triple the time required. I could post the code (after cleaning it up a little).

Here are the theta(z) coefficients, normalized so that fa(z)=amplitude*cos(z). This is the best way to accurately represent the asum superfunction over fairly wide range of values, approaching within 0.3i of the nearest singularity to the real axis. At the real axis, theta(z) has very small values, and the two superfunctions are nearly identical. So theta(z) provides one clear way to document the small differences between the functions, in a way that also shoes how those small differences grow as imag(z) increases.
- Sheldon

Code:

b0=   0.335456219829438730039
b1=   3.33669195272218669860 E-26 + 3.62344900310350155726 E-26*I
b2=  -1.80769101620114018113 E-50 - 7.89110277430440224394 E-51*I
b3=   1.20097489213760696637 E-74 + 2.51814953821047901586 E-76*I
b4=  -8.06184317720671848678 E-99 + 3.02174062523520392543 E-99*I
b5=   5.01787673640943299717 E-123 - 4.50349412257750817497 E-123*I
b6=  -2.54119020978779808029 E-147 + 4.97980255860701751204 E-147*I
b7=   5.24368727163436325210 E-172 - 4.79825514607678170351 E-171*I
b8=   1.06388349485199970614 E-195 + 4.16155040777703883391 E-195*I
b9=  -2.22978881631121368592 E-219 - 3.21665525346589892794 E-219*I
b10=  2.97621100822729857968 E-243 + 2.08692670740436833975 E-243*I
b11= -3.31361772171087796881 E-267 - 8.84006742271380167537 E-268*I
b12=  3.26611209557096457375 E-291 - 2.88873515071575647588 E-292*I
b13= -2.87456434306604547765 E-315 + 1.33777405738055441555 E-315*I
b14=  2.19755467009001549963 E-339 - 2.18111402505261136382 E-339*I
b15= -1.31035397720908540692 E-363 + 2.75379610980160491952 E-363*I
b16=  3.02024403194138164472 E-388 - 3.01156935122381145850 E-387*I
... 50 terms required for 32 decimal digits accuracy for |imag(z)|<8.57

- Sheldon