01/23/2019, 09:44 PM
(This post was last modified: 01/23/2019, 10:25 PM by sheldonison.)
This is kind of a cool error plot where I calculated Jay's slog to 120 terms and computed an 8 term theta mapping:
\( \theta_{Rk}\mid\;\text{JaySlog}(z)\approx\text{KneserSlog}(z)+\theta_{Rk}(\text{KneserSlog}(z)) \)
\( \text{errplot}=\frac{10^{-29}}{\text{JaySlog}(z)-\text{KneserSlog}(z)-\theta_{Rk}(\text{KneserSlog})} \)
The Jay's slog Taylor series is centered at zero! The region of "good convergence" extends all the way out to nearly real(2)! The plot goes from real(-1) to real(2). On the sides, the error is dominated by approximately the 100th term of the JaySlog series, and in the vertical, it is limited by both Jay's slog and the corresponding 8 term theta mapping. For the most part, these two functions are consistent to 10^-32 or so. Without the theta mapping, the error is much larger; around 10^-17 or so.
If I optimize the theta mapping closer to the real axis with the same 120 term JaySlog solution, its possible to do even better, with consistencies better than 10^-64, but over a smaller region. This error plot uses a 22 term theta mapping, with 10^-62 in the numerator of the error plot.
I think the next step is to figure out how to compute the reverse theta mapping \( \theta_{Rj} \) starting from only the Schroder complex valued Abel function and without starting with Kneser's slog, and then using \( \theta_{Rj} \) to approximate Kneser's slog from Jay's slog. More later...
\( f(z)=z+\theta_{Rk}(z);\;\;\;f^{-1}(z)=z+\theta_{Rj}(z) \)
\( \text{KneserSlog}(z)\approx\text{JaySlog}(z)+\theta_{Rj}(\text{JaySlog}(z)) \)
\( \theta_{Rk}\mid\;\text{JaySlog}(z)\approx\text{KneserSlog}(z)+\theta_{Rk}(\text{KneserSlog}(z)) \)
\( \text{errplot}=\frac{10^{-29}}{\text{JaySlog}(z)-\text{KneserSlog}(z)-\theta_{Rk}(\text{KneserSlog})} \)
The Jay's slog Taylor series is centered at zero! The region of "good convergence" extends all the way out to nearly real(2)! The plot goes from real(-1) to real(2). On the sides, the error is dominated by approximately the 100th term of the JaySlog series, and in the vertical, it is limited by both Jay's slog and the corresponding 8 term theta mapping. For the most part, these two functions are consistent to 10^-32 or so. Without the theta mapping, the error is much larger; around 10^-17 or so.
If I optimize the theta mapping closer to the real axis with the same 120 term JaySlog solution, its possible to do even better, with consistencies better than 10^-64, but over a smaller region. This error plot uses a 22 term theta mapping, with 10^-62 in the numerator of the error plot.
I think the next step is to figure out how to compute the reverse theta mapping \( \theta_{Rj} \) starting from only the Schroder complex valued Abel function and without starting with Kneser's slog, and then using \( \theta_{Rj} \) to approximate Kneser's slog from Jay's slog. More later...
\( f(z)=z+\theta_{Rk}(z);\;\;\;f^{-1}(z)=z+\theta_{Rj}(z) \)
\( \text{KneserSlog}(z)\approx\text{JaySlog}(z)+\theta_{Rj}(\text{JaySlog}(z)) \)
- Sheldon