Calculating the fractional iterates of \(f(z) = e^z-1\) to reasonable accuracy
#7
Oh yes, absolutely Gottfried.

This program has quite a few faults, runtime being the most obvious, lol. It is descriptive of much of my programming. I was primarily a C programmer, and I specialized in recursion. I have written some unbelievably complicated recursion programs for C. The thing is, Pari-GP does not have the run time benefits that C has when you write things recursively. I think a large part of that is that I try to preserve Taylor data.

For example, the fact I have built in F(1+s,z) as a valid command, which will produce the taylor series in s and the asymptotic taylor series in z--is super helpful if we think of this as an object. It is not helpful pointwise, when you run F(1.5,-0.5). This will run so slow, because much of the code for F(1+s,z) takes priority. I am aware of this, but I still like simple code above all, especially as I am using it on my computer. I am definitely not winning any efficiency awards though, lol Smile

I absolutely know your code, or Sheldon's code, is far superior in runtime. But again, they are not recursion based. Everything I do is recursion based (my brain kinda just works like that, lol).

Sincere Regards, James





Here are some iterates \(f^{\circ s}(z)\):

   

These take an exceptionally long time to compile. And I know this. I am again, just trying to stress that my mathematical construction which is slow to program produces the same results as all of Helms' Trappman's work. We can do all of this with integrals. Obviously, codewise, this is not as good--but mathematically my work works, lol.

This goes really far into the metaphor I always use. We can do Schrodinger mechanics, or we can do Heisenberg mechanics. We can talk about integral transforms, and actions on hilbert spaces--or we can talk about infinite matrices and eigen values. We are doing the same thing in either language, but we must choose a language. I'm just trying to add some integrals, and Schrodinger mechanics to the problem. And I'm doing so using the Mellin transform (which is just a modified fourier transform when you break it down). I'm trying to be Schrodinger to your Heisenberg... I hope this metaphor makes sense. Everyone here is so fixated on infinite matrix equations, and reducing matrix equations, and approximations through \(N\times N\) scenarios--and this is exactly Heisenberg's approach of quantum physics. Schrodinger's approach was to look at the solution functions globally and create differential/integral equations between the functions globally. Then Von Neumann, using Hilbert's work, proved that Heisenberg and Schrodinger were saying the same thing. Heisenberg just used matrices, and Schrodinger used integrals (to be simplistic).

Again, I'm just trying to add integrals to the discussion, and show how they give the exact same thing you guys already have. Just like Schrodinger...

Just understand that, Gottfried. My code is not something groundbreaking. It's just proof that my math is working Smile The integrals are converging pretty good Tongue

Here's a, very slow to produce (took about 24 hrs of cpu time), graph of \(f^{\circ 0.5}(z)\) for \(1.25>\Im(z) > 0\) and \(-2 < \Re(z) < 0.5\)

   

The schrodinger/integral/mellin/fourier view works. It's not as fast or as good as the Heisenberg, matrix solution method, we see everywhere. But mathematically, it's pretty fucking clean. Cool
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RE: Calculating the fractional iterates of \(f(z) = e^z-1\) to reasonable accuracy - by JmsNxn - 11/19/2022, 04:26 AM

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