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Taylor polynomial. System of equations for the coefficients.
#5
(05/01/2015, 01:57 AM)marraco Wrote: This is a numerical example for base a=e

I had shown how to get 2 sides of a system of equations for solving the coefficients. One side is blue (the linear one), and the other is red (nonlinear, and I still don't know the pattern to generate the complete system of equations. I only got up to x^7, and I can get more, but of course, what I need is a more general expression.
....
You might want to compare this to the intuitive slog method. Jay took that farther than anyone else, getting results accurate to 21 decimal digits using his accelerated technique. http://math.eretrandre.org/tetrationforu...php?tid=63 With 8 terms, accuracy is limited to magnitude of the eighth term, and your eighth term is actually larger than Kneser's. If you were able to generate a 20 term approximation, then theoretically, one could have a around 14 decimal digits of accuracy ... assuming the terms scale based on the nearest logarithmic singularity at sexp(-2). For base e, the Taylor's series coefficients of Kneser's solution approach log(x+2) as n gets larger.
- Sheldon
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