08/12/2009, 02:15 AM

(08/12/2009, 01:31 AM)sheldonison Wrote: Oh, I get it, you're using the well defined function base change function at the real axis only, for sexp_e(z). And then your using those real values to generate the taylor series???? And graphing that taylor series in the complex plane?Yes, this version of sexp is for base e (hence the lack of a subscript), and is found using the change of base formula with the cheta function. I'm working in the double-logarithmic scale, so the cheta function turns out to be the continuous iteration of e^x-1 (which is why I don't have a specific reference to eta).

So I calculate sexp(m*2^p), for m an integer between -64 and 64, for example, and p equal to -108. Because I'm only using reals, convergence is assured. I'm using sexp(0)=0, because Andrew's slog is centered at 0, so I can have slog(sexp(0)) and expect an exact answer.

Anyway, I can then use those 129 points to calculate the various derivatives (actually, I use a matrix inversion that directly gives me the coefficients of the approximated Taylor series). Once I have a Taylor series, I can then move away from the real line.

See the following wikipedia article for the general idea of how the grid of points can be used to get derivatives:

http://en.wikipedia.orirg/wiki/Five-point_stencil

Instead of a 5-point stencil, I use a 129-point stencil. I don't yet have a formula for the coefficients for the stencil, so I have to use a matrix inversion to derive them "the hard way". For an n-point stencil, the matrix inversion is the primary limiting factor for this method, because it uses O(n^3) memory, whereas a vector method would use O(n^2). (The matrix has n^2 terms, each using a multiple of n bits, so the space is order n^3).

~ Jay Daniel Fox