08/23/2007, 04:43 PM
Andrew, I got you a present:
That's the first 60 odd derivatives of your slog, base e. They're scaled, of course.
To create this graph, first I solved the 400x400 matrix. Then I created a RealField with 5120 bits of precision (I probably only needed about 3000 bits or so...). Then, for each x point I wanted to graph, I calculated samples spaced 1/32678 apart. (I think next time I'll put them closer together, since I had the precision available.) Actually, I picked y points, then found x with slog, rather than try to solve the inverse. Because my x points weren't spaced evenly, I used divided differences to approximate the derivatives, and rather than multiply by k factorial, I multiplied by 2.46^k. That would explain the scaling.
Anyway, just a few derivatives later, I lost concavity. By the 67th odd derivative, I hit the x-axis, and it went haywire from there. I'm not sure if the derivatives started to go haywire because my sample points are spaced too far apart, or if it's because I only used a 400x400 matrix.
That's the first 60 odd derivatives of your slog, base e. They're scaled, of course.
To create this graph, first I solved the 400x400 matrix. Then I created a RealField with 5120 bits of precision (I probably only needed about 3000 bits or so...). Then, for each x point I wanted to graph, I calculated samples spaced 1/32678 apart. (I think next time I'll put them closer together, since I had the precision available.) Actually, I picked y points, then found x with slog, rather than try to solve the inverse. Because my x points weren't spaced evenly, I used divided differences to approximate the derivatives, and rather than multiply by k factorial, I multiplied by 2.46^k. That would explain the scaling.
Anyway, just a few derivatives later, I lost concavity. By the 67th odd derivative, I hit the x-axis, and it went haywire from there. I'm not sure if the derivatives started to go haywire because my sample points are spaced too far apart, or if it's because I only used a 400x400 matrix.
~ Jay Daniel Fox
