But where does all this analysis even get us? Do we have any hope of unlocking the secrets of this bizarre function?
Regardless of whether Andrew's solution ultimately converges on the correct solution, it's at least very close, at least for base e. Even if it's not correct, it gives me a good starting point for analyzing the nature of the slog function and its many branches.
The first few terms of Andrew's slog indicate that it's converging on a solution that includes singularities at the two primary fixed points. In fact, using some basic matrix math, I was able to extract the two logarithms that I predicted would exist in the slog solution: \( \log_{a_0}\left(z-a_0\right) + \log_{\overline{a_0}}\left(z-\overline{a_0}\right) \)
By extracting the singularities, I was able to greatly increase the rate of convergence. In fact, I can calculate the first 50 terms more accurately with a 50x50 matrix than I can with a 600x600 matrix using Andrew's original solution. There is some prep work, of course, but it can be done separately to produce a single column vector, which is solved in place of the [1, 0, ..., 0] vector in Andrew's solution. It's that simple.
But rather than stop at 50x50, I just went ahead and solved a 640x640 system. I figure it's probably at least as accurate as a 2000x2000 system using Andrew's matrix, and possibly more accurate than a 3000x3000 system. I haven't investigated the rate of convergence of either system to be able to make an accurate prediction, and at any rate, it would take a supercomputer to solve a 3000x3000 system, or a regular desktop several weeks using fast hard disk arrays in place of RAM.
Below I show the first few terms of the power series of this conjugate-logarithm pair, then the slog solution, then the "residue" after removing the singularities. Notice that I intentionally omit the constant term, as I don't feel it's well-defined yet. We can at best say that it's a real number.
\(
\begin{array}{c|ccc} \text{Term} & \log_{a_0}\left(z-a_0\right) + \log_{\overline{a_0}}\left(z-\overline{a_0}\right) & \text{slog}\left(z\right) & \text{slog}\left(z\right)\ -\ {\normalsize \left(\log_{a_0}\left(z-a_0\right) + \log_{\overline{a_0}}\left(z-\overline{a_0}\right)\right)} \\
\hline
1 & 0.945130773 & 0.915946056 & -0.029184717 \\
2 & 0.248253691 & 0.249354599 & 0.001100908 \\
3 & -0.111008639 & -0.11046476 & 0.00054388 \\
4 & -0.093733042 & -0.093936255 & -0.000203213 \\
5 & 0.01000001 & 0.010003233 & 3.22281E-06 \\
6 & 0.035879455 & 0.035897922 & 1.84669E-05 \\
7 & 0.006575953 & 0.006573401 & -2.55239E-06 \\
8 & -0.012304686 & -0.01230686 & -2.17352E-06 \\
9 & -0.006390236 & -0.006389803 & 4.33349E-07 \\
10 & 0.003273231 & 0.00327359 & 3.59009E-07 \\
11 & 0.003769267 & 0.003769203 & -6.43927E-08 \\
12 & -0.000280141 & -0.000280217 & -7.58188E-08 \\
13 & -0.001775114 & -0.001775107 & 7.30188E-09 \\
14 & -0.000427988 & -0.00042797 & 1.83129E-08 \\
15 & 0.000679723 & 0.000679723 & 4.01473E-10 \\
16 & 0.000412797 & 0.000412793 & -4.67886E-09 \\
17 & -0.000186597 & -0.000186598 & -7.82734E-10 \\
18 & -0.00025355 & -0.000253549 & 1.1938E-09 \\
\end{array}
\)
The residue is not due to numerical inaccuracies. Rather, it's due to the fact that the two singularities I removed are only one component of the slog. These two singularities are the primary affectors, when a power series is derived at the origin. I've already predicted singularities at 2k*pi*i offsets of these primary singularities, though those logarithms are so far from the origin that they hardly affect more than the first half dozen terms. The root test of the "residue" seems to indicate another pair of singularities only slightly further from the origin than the first two. The root test for a 640x640 solution climbs as high as 0.67, but studying the progression for smaller solutions indicates that the root test will climb higher, probably to about 0.69, give or take. If I multiply the residue by a few thousand, the root test seems to flatten out, so this seems to indicate that the singularity, whereever it is, is in an extremely narrow "well".
It's also possible that multiple singularities lying on the same line of sight from the origin are adding together and appearing to inflate the root test, such that if I had several thousand coefficients, I'd see the root test start to go back down. I have too little information at this point, so I have to rely on good old fashioned though experiments to try to figure this out.
One thought I've had is that, because the "worlds" on either side of the singularity are completely different from each other, I'm getting interference as the derivatives "refract" around the singularity I thought I'd removed. I may need to evaluate the slog function at other points closer to the primary singularities and perhaps at various rotations around them, to see if I can locate this next closest singularity, identify it, remove it, and extract even more precision in the power series constructed at the origin.
Regardless of whether Andrew's solution ultimately converges on the correct solution, it's at least very close, at least for base e. Even if it's not correct, it gives me a good starting point for analyzing the nature of the slog function and its many branches.
The first few terms of Andrew's slog indicate that it's converging on a solution that includes singularities at the two primary fixed points. In fact, using some basic matrix math, I was able to extract the two logarithms that I predicted would exist in the slog solution: \( \log_{a_0}\left(z-a_0\right) + \log_{\overline{a_0}}\left(z-\overline{a_0}\right) \)
By extracting the singularities, I was able to greatly increase the rate of convergence. In fact, I can calculate the first 50 terms more accurately with a 50x50 matrix than I can with a 600x600 matrix using Andrew's original solution. There is some prep work, of course, but it can be done separately to produce a single column vector, which is solved in place of the [1, 0, ..., 0] vector in Andrew's solution. It's that simple.
But rather than stop at 50x50, I just went ahead and solved a 640x640 system. I figure it's probably at least as accurate as a 2000x2000 system using Andrew's matrix, and possibly more accurate than a 3000x3000 system. I haven't investigated the rate of convergence of either system to be able to make an accurate prediction, and at any rate, it would take a supercomputer to solve a 3000x3000 system, or a regular desktop several weeks using fast hard disk arrays in place of RAM.
Below I show the first few terms of the power series of this conjugate-logarithm pair, then the slog solution, then the "residue" after removing the singularities. Notice that I intentionally omit the constant term, as I don't feel it's well-defined yet. We can at best say that it's a real number.
\(
\begin{array}{c|ccc} \text{Term} & \log_{a_0}\left(z-a_0\right) + \log_{\overline{a_0}}\left(z-\overline{a_0}\right) & \text{slog}\left(z\right) & \text{slog}\left(z\right)\ -\ {\normalsize \left(\log_{a_0}\left(z-a_0\right) + \log_{\overline{a_0}}\left(z-\overline{a_0}\right)\right)} \\
\hline
1 & 0.945130773 & 0.915946056 & -0.029184717 \\
2 & 0.248253691 & 0.249354599 & 0.001100908 \\
3 & -0.111008639 & -0.11046476 & 0.00054388 \\
4 & -0.093733042 & -0.093936255 & -0.000203213 \\
5 & 0.01000001 & 0.010003233 & 3.22281E-06 \\
6 & 0.035879455 & 0.035897922 & 1.84669E-05 \\
7 & 0.006575953 & 0.006573401 & -2.55239E-06 \\
8 & -0.012304686 & -0.01230686 & -2.17352E-06 \\
9 & -0.006390236 & -0.006389803 & 4.33349E-07 \\
10 & 0.003273231 & 0.00327359 & 3.59009E-07 \\
11 & 0.003769267 & 0.003769203 & -6.43927E-08 \\
12 & -0.000280141 & -0.000280217 & -7.58188E-08 \\
13 & -0.001775114 & -0.001775107 & 7.30188E-09 \\
14 & -0.000427988 & -0.00042797 & 1.83129E-08 \\
15 & 0.000679723 & 0.000679723 & 4.01473E-10 \\
16 & 0.000412797 & 0.000412793 & -4.67886E-09 \\
17 & -0.000186597 & -0.000186598 & -7.82734E-10 \\
18 & -0.00025355 & -0.000253549 & 1.1938E-09 \\
\end{array}
\)
The residue is not due to numerical inaccuracies. Rather, it's due to the fact that the two singularities I removed are only one component of the slog. These two singularities are the primary affectors, when a power series is derived at the origin. I've already predicted singularities at 2k*pi*i offsets of these primary singularities, though those logarithms are so far from the origin that they hardly affect more than the first half dozen terms. The root test of the "residue" seems to indicate another pair of singularities only slightly further from the origin than the first two. The root test for a 640x640 solution climbs as high as 0.67, but studying the progression for smaller solutions indicates that the root test will climb higher, probably to about 0.69, give or take. If I multiply the residue by a few thousand, the root test seems to flatten out, so this seems to indicate that the singularity, whereever it is, is in an extremely narrow "well".
It's also possible that multiple singularities lying on the same line of sight from the origin are adding together and appearing to inflate the root test, such that if I had several thousand coefficients, I'd see the root test start to go back down. I have too little information at this point, so I have to rely on good old fashioned though experiments to try to figure this out.
One thought I've had is that, because the "worlds" on either side of the singularity are completely different from each other, I'm getting interference as the derivatives "refract" around the singularity I thought I'd removed. I may need to evaluate the slog function at other points closer to the primary singularities and perhaps at various rotations around them, to see if I can locate this next closest singularity, identify it, remove it, and extract even more precision in the power series constructed at the origin.
~ Jay Daniel Fox
