andydude Wrote:@Gottfried: Your graph confuses me, why is the 1-tick for b=... and the 1-tick for f=... not aligned? and why do you have X's for points, do you have any options for boxes? diamonds? circles? no dots? That would make the graph much nicer to look at. But anyway, I got the general idea of the graph, I just think it could use some work. I'm not sure what the benefit of such a change-of-variables would be, unless it would simplify the series for b^b^x.Hi Andrew -
the line for b is just for completeness. It could have had the same y-scale as the other lines, but I wanted to prevent overlay of the curves, so the y-scale for the b-curve is at the right side of the graph.
I'm copy&pasting the data computed by Pari/gp into Excel and do the standard graph with some editing. Not always optimal, though... ;-(
Quote:@Gottfried: As for your series expansion, I really, really like your presentation of the series for (b^x-1), it shows patterns in the coefficients, where I never saw patterns beforeWell It's quite a time ago, early eighties, when I sat night after night in the campus' computing center and hacked the PL/I-runtime-system by analyzing the pointers and data-structures from memory-dumps. The greatest success was to implement a routine, which was able to modify the file-parameters at runtime so that I could open-and reopen any file and assign record-length and block-size-parameters dynamically. I could then provide *very* versatile programs for file-backup, file-restructuring and transformation in dialog-programs; usually with PL/I you needed to define fixed file-types in the source-code of an application.Good job.
So sometimes I get nice associations to this inspired time of pattern-detection...
For the given coefficients a_k you may observe, that in each of that the numerical coefficients of the last column contains the Stirling numbers 1'st kind and the first column that of 2'nd kind, scaled by factorials. The condition, that with integer h the a_k-polynomials produce multiples of the denominator seem to make the other coefficients unique, so I manually solved for the numerical coefficients in a_4, to get an example of the path for a general solution This is in principle possible for all other a_k, so this description allows to compute each single term a_k independently of the matrix-eigensystem-computation. I think, I don't need to explain, why this is a remarkable achievement - in principle. I say "in principle" because the computation-requirements seem again to be huge, and I don't have a good recursive algorithm yet. But the possibility to do the computation for as many terms of the powerseries of x as wished sequentially and independently from each other, frees us principally from the impossible memory management for matrices of sizes for even some dozen powerseries-terms only and of the computation of their symbolical eigensystem-decomposition, and allows, for instance, to compute term a_96 this week, term a_97 next week and so on, if that computation needs any such amount of time.
---
... and I love series, too :-) ("Euler - the master of us all...")
Gottfried
Ps. your explanation of the role of x0 and x1 are good, I understand now.
Gottfried Helms, Kassel