• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 [Update] Comparision of 5 methods of interpolation to continuous tetration Gottfried Ultimate Fellow Posts: 816 Threads: 123 Joined: Aug 2007 10/13/2013, 02:20 PM (This post was last modified: 10/15/2013, 10:25 PM by Gottfried.) I've just updated my discussion from 2010 where I provided pictures and short commentars for the basic introduction into different interpolation proposals for the tetration. I included now also the comaprision with the Kneser-method, where I used one of the Pari/GP-scripts which Sheldon has kindly provided here. Here is the link: http://go.helms-net.de/math/tetdocs/Comp...ations.pdf I'll attach it also here for the possibility that some website might drop down... [updates]: included comparisions of the Kneser with the polynomial 32x32, polynomial 48x48 and polynomial 64x64 - interpolations. Impression/conclusion:The bigger the matrix-size, the better the Kneser solution is approximated. [/end update] Ahh, ps: I would like it much to include more material of someone else, who has some other practical procedure and can provide data for the same environment (of base b=4, and the 1/20 to 1/40-step iterations with the given initial values) such that I can include them in my Excel-tables for plotting. Have fun - Gottfried see for more material: http://go.helms-net.de/math/tetdocs/ Attached Files   ComparisionOfInterpolations.pdf (Size: 352.81 KB / Downloads: 710) Gottfried Helms, Kassel MikeSmith Junior Fellow Posts: 40 Threads: 16 Joined: Nov 2011 10/14/2013, 10:42 AM Gottfried, your math looks difficult to understand for an amateur in continuous functional iteration, but has very interesting pictures nonetheless. I never was good at complex analysis, otherwise I might have a chance to understand. The presentation in your papers is very nice Gottfried Ultimate Fellow Posts: 816 Threads: 123 Joined: Aug 2007 10/14/2013, 01:19 PM (This post was last modified: 10/14/2013, 01:21 PM by Gottfried.) (10/14/2013, 10:42 AM)MikeSmith Wrote: Gottfried, your math looks difficult to understand for an amateur in continuous functional iteration, but has very interesting pictures nonetheless. I never was good at complex analysis, otherwise I might have a chance to understand. The presentation in your papers is very nice Hi Mike - thank you for your kind response. Well, it sounds strange for me to hear, that my math is too complex. You know, I've started all this as a complete amateur, just accidentally wenn I looked at properties of the Pascalmatrix, and power of the Pascalmatrix, and... Then by a very natural and naive interest in the question, what happens if I apply the same procedures to the matrix of Stirlingnumbers I fell in... yes, the area of powertowers. And from there, by considering: how in the world could fractional powers of such matrices be constructed?, to the powertowers with fractional heights, aka tetration. Perhaps after so many years of intense work with it my language became more sophisticated and/or overly jargon-ned ... I'd always liked to make that steps into the matter transparent for other interested ones and as joyful/exciting as they were for me - if you (or someone else here) is interested in a conversation here to make my approach better understandable for an amateur then let me know. Possibly it needs only rewriting of my first articles, or a collaborative rewriting of them? Well, Mike, just share your thoughts - Gottfried Gottfried Helms, Kassel MikeSmith Junior Fellow Posts: 40 Threads: 16 Joined: Nov 2011 10/14/2013, 08:22 PM (10/14/2013, 01:19 PM)Gottfried Wrote: (10/14/2013, 10:42 AM)MikeSmith Wrote: Gottfried, your math looks difficult to understand for an amateur in continuous functional iteration, but has very interesting pictures nonetheless. I never was good at complex analysis, otherwise I might have a chance to understand. The presentation in your papers is very nice Hi Mike - thank you for your kind response. Well, it sounds strange for me to hear, that my math is too complex. You know, I've started all this as a complete amateur, just accidentally wenn I looked at properties of the Pascalmatrix, and power of the Pascalmatrix, and... Then by a very natural and naive interest in the question, what happens if I apply the same procedures to the matrix of Stirlingnumbers I fell in... yes, the area of powertowers. And from there, by considering: how in the world could fractional powers of such matrices be constructed?, to the powertowers with fractional heights, aka tetration. Perhaps after so many years of intense work with it my language became more sophisticated and/or overly jargon-ned ... I'd always liked to make that steps into the matter transparent for other interested ones and as joyful/exciting as they were for me - if you (or someone else here) is interested in a conversation here to make my approach better understandable for an amateur then let me know. Possibly it needs only rewriting of my first articles, or a collaborative rewriting of them? Well, Mike, just share your thoughts - Gottfried thanks for explaining some of the prerequisite knowledge. I should spend some time on this Wikipedia page http://en.wikipedia.org/wiki/List_of_matrices to get more familiar with ideas about matrices sheldonison Long Time Fellow Posts: 684 Threads: 24 Joined: Oct 2008 10/14/2013, 10:07 PM (10/13/2013, 02:20 PM)Gottfried Wrote: I've just updated my discussion from 2010 where I provided pictures and short commentars for the basic introduction into different interpolation proposals for the tetration. I included now also the comaprision with the Kneser-method, where I used one of the Pari/GP-scripts which Sheldon has kindly provided here. Here is the link: http://go.helms-net.de/math/tetdocs/Comp...ations.pdf Gottfried, I was able to reproduce your graph on page 9, by plotting sexp(z+k), where $k=1+\text{slog}(0.1i) \approx -0.002+0.107i$, and z varied from -7 to 1. You might try other values of k. As imag(k) increases, the Kneser technique behaves more and more like the Schroeder function. For example, try k=0.5i, graphing sexp(z+k) from z=-7 to z=1, and you see a very nice well defined spiral towards to fixed point, just like the Schroeder function solution. Here interpolation works very well, since interpolation naturally approaches the Schroeder function solution. Closer to the real axis, the singularities at integer values <=-2 become more and more pronounced, which causes problems for interpolation. - Sheldon Gottfried Ultimate Fellow Posts: 816 Threads: 123 Joined: Aug 2007 10/15/2013, 12:00 AM (This post was last modified: 10/15/2013, 12:02 AM by Gottfried.) Hi Sheldon - thanks for your comment. What remains for me is, whether my impression, that the Kneser-method and the polynomial-method ("quick&dirty-eigendecomposition") approximate if I increase the size of the truncation of the Carlemanmatrix. That would be a really remarkable statement! It would raise the subsequent question, whether the triangular Carleman-matrix, which results from the fixpoint-shift and gives the basis for the regular tetration, or its triangular eigenmatrices, simply need some completion factor, perhaps something like the integral in the Ramanujan-summation for the divergent series to agree with the two other results... I am in search of such a thing for long but with no avail so far - perhaps from the current observation one can get a hint where to dig? Gottfried Gottfried Helms, Kassel Gottfried Ultimate Fellow Posts: 816 Threads: 123 Joined: Aug 2007 10/15/2013, 12:59 AM (This post was last modified: 10/15/2013, 09:02 AM by Gottfried.) (10/14/2013, 08:22 PM)MikeSmith Wrote: http://en.wikipedia.org/wiki/List_of_matrices to get more familiar with ideas about matricesHi Mike - wow, the wikipedia-list is long... Well, for me/for the Carlemanmatrix-approach the following are relevant: * diagonal, * subdiagonal, * triangular and * square shape ("which parts of the matrix are not 'systematically' zero?") Matrices with constant entries: * Triangular with ones filled in, * Pascalmatrix (=lower or upper triangular with binomial entries), * Stirlingmatrices S1,S2 = first and second kind * Standard Vandermondematrix VZ Matrices/vectors with variable entries: * Vandermonde vector of argument x: V(x) = [1,x,x^2,x^3,...], its diagonal ("dV(x)") or row or column form * Vandermonde matrix - some collection of Vandermondevectors, for instance VZ = [V(0),V(1),V(2),V(3),...] * Z-vector of exponent argument Z(w) = [1,1/2^w, 1/3^w, 1/4^w, ...] for work with dirichletseries and derivatives * Factorial vector of exponent argument w: F(w)= [0!^w,1!^w,2!^w,...] for work with exponential series In principle, this is it.