07/28/2026, 05:36 AM
As promised I have done some error analysis.
So I have made a spreadsheet where it shows the absolute error(\(|approximation - Kneser tetration|\)) compared to fatou.gp(made by sheldonison on the tetration forum). The google sheet link is https://docs.google.com/spreadsheets/d/e...ingle=true.
Anyways the google sheet link shows the absolute error between my approximation and fatou.gp for \(\Im(x) \in [-1,-0.9,-0.8,-0.7,-0.6,-0.5,-0.4,-0.3,-0.2,-0.1,0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1]\) and \(\Re(x) \in [-1,-0.9,-0.8,-0.7,-0.6,-0.5,-0.4,-0.3,-0.2,-0.1,0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1]\).
The maximum absolute error in the spreadsheet is at \(x = -1 - 0.8i\) with a value of \(1.20310869470002500544715591410430055784 \cdot 10^{-11}\) which is quite good considering that my approximation was only designed for the interval \([0,1]\)
Adding all the absolute errors up and dividing them by 441(the amount of absolute errors there are) i get an average error of \(3.10353284479205457020122170976028317270 \cdot 10^{-13}\) and this is over 441 sample values as a grid over the complex plane, my approximation was designed for the interval \([0,1]\), and so \(3.10353284479205457020122170976028317270 \cdot 10^{-13}\) is quite good.
So I have made a spreadsheet where it shows the absolute error(\(|approximation - Kneser tetration|\)) compared to fatou.gp(made by sheldonison on the tetration forum). The google sheet link is https://docs.google.com/spreadsheets/d/e...ingle=true.
Anyways the google sheet link shows the absolute error between my approximation and fatou.gp for \(\Im(x) \in [-1,-0.9,-0.8,-0.7,-0.6,-0.5,-0.4,-0.3,-0.2,-0.1,0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1]\) and \(\Re(x) \in [-1,-0.9,-0.8,-0.7,-0.6,-0.5,-0.4,-0.3,-0.2,-0.1,0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1]\).
The maximum absolute error in the spreadsheet is at \(x = -1 - 0.8i\) with a value of \(1.20310869470002500544715591410430055784 \cdot 10^{-11}\) which is quite good considering that my approximation was only designed for the interval \([0,1]\)
Adding all the absolute errors up and dividing them by 441(the amount of absolute errors there are) i get an average error of \(3.10353284479205457020122170976028317270 \cdot 10^{-13}\) and this is over 441 sample values as a grid over the complex plane, my approximation was designed for the interval \([0,1]\), and so \(3.10353284479205457020122170976028317270 \cdot 10^{-13}\) is quite good.
