Hey, everyone!
So I successfully implemented a Taylor Series Method with this tetration. This code is definitely suboptimal; but it'll get the job done on the real-line. The previous protocol for tetration is desired for \( |\Im(z)| > 1 \); this method is intended for a domain near the real-line. I'm still working on making everything perfect, but it seems to be working. I'm able to get about 20 digit precision with the Taylor Series Method. But it works perfectly up to that. I had to modify the code significantly. So I'll attach here everything you need for the Taylor series approach at calculating \( \text{tet}_\beta \).
But first, here's a graph of \( \text{tet}_\beta(s-x_0) \) for \( |\Re(z)| \le 0.5 \) and \( |\Im(z)| \le 0.5 \). (The value \( x_0 \approx 1 \).) There is absolutely no short circuiting; as was happening before; and it agrees perfectly with the old protocol on the real line (upto 20 digits, I mean).
I'm going to work on compiling all my code into a nice bundle. But for the moment, I'll post how I'm grabbing the Taylor series.
Here's a print out of the first 50 taylor coefficients with 200 digit accuracy, which is precise to about 20 digits, when \( A=1 \) and \( |z| \le 1/2 \). And we're summing,
\(
\text{tet}_\beta(z-x_0) = \sum_{j=0}^\infty a_j (z-1)^j\\
\)
After defining a recursive function. Here's the graph of \( \text{tet}_\beta(z-x_0) \) for \( 1 \le \Re(z) \le 6 \) and \( |\Im(z)| \le 0.7 \). Again \( x_0 \approx 1 \).
I think we're well on the way; that this tetration is holomorphic. At this point; Is it Kneser's tetration?
So I successfully implemented a Taylor Series Method with this tetration. This code is definitely suboptimal; but it'll get the job done on the real-line. The previous protocol for tetration is desired for \( |\Im(z)| > 1 \); this method is intended for a domain near the real-line. I'm still working on making everything perfect, but it seems to be working. I'm able to get about 20 digit precision with the Taylor Series Method. But it works perfectly up to that. I had to modify the code significantly. So I'll attach here everything you need for the Taylor series approach at calculating \( \text{tet}_\beta \).
But first, here's a graph of \( \text{tet}_\beta(s-x_0) \) for \( |\Re(z)| \le 0.5 \) and \( |\Im(z)| \le 0.5 \). (The value \( x_0 \approx 1 \).) There is absolutely no short circuiting; as was happening before; and it agrees perfectly with the old protocol on the real line (upto 20 digits, I mean).
I'm going to work on compiling all my code into a nice bundle. But for the moment, I'll post how I'm grabbing the Taylor series.
Code:
\ps 50 //set series precision to 50
\p 200 //set precision to 200; this makes the Taylor series work better
//This code estimates the depth of recursion needed for the iteration to work about A.
//The value 0.0001 can be made smaller, but it risks an overflow for other A's.
//This, ideally, is meant for A a real number; and expanding a taylor series about a real number.
//If you plug in complex values it suffers the same fate as Tet(z,n) for z near the real line.
Tet_GRAB_k(A,n) ={
my(k=0);
while( 1/real(beta(A+k,n)) >= 0.0001, k++);
return(k);
}
//This code will run the code of Tet; but the value k fixes the depth of recursion. The goal is to guess the value of k; which the previous function does.
Tet_taylor(z,n,k) = {
my(val = beta(z+k,n));
for(i=1,k,val = log(val));
return(val);
}
//This code will create an array of size 50, of the Taylor coefficients. 50 seems good enough for 20-digit precision.
TAYLOR_SERIES(A,n) = {
my(ser = vector(50,i,0));
for(i=1,50, ser[i] = polcoeff(Tet_taylor(A+z,n,Tet_GRAB_k(A,n)),i-1,z));
return(ser);
}Here's a print out of the first 50 taylor coefficients with 200 digit accuracy, which is precise to about 20 digits, when \( A=1 \) and \( |z| \le 1/2 \). And we're summing,
\(
\text{tet}_\beta(z-x_0) = \sum_{j=0}^\infty a_j (z-1)^j\\
\)
Code:
0.03259650901600602497933792676384984393747893444466645403696826747470297040854769936473446193681691248
1.064566018894822439459121680191873880002289031679825196946198206070023707777321010554487553032188483
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0.0023049646597304620702001627477045893100607516041809170910352492486428007634527I think we're well on the way; that this tetration is holomorphic. At this point; Is it Kneser's tetration?