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 regular vs intuitive (formerly: natural) bo198214 Administrator Posts: 1,528 Threads: 97 Joined: Aug 2007 04/27/2008, 10:00 AM (This post was last modified: 06/26/2010, 11:41 AM by bo198214.) I computed $\delta:=\text{rsexp}_{\sqrt{2}}(\text{nslog}_{\sqrt{2}}(0.3))-0.3$ with a precision of rsexp of 20 digits and varying matrix sizes for nslog from 10 to 100. The following graphs show $\delta$ in dependence on the matrix size (x-Axis).         This looks quite as if regular and natural method are equal. Yippee! Guys, I think we are on a great trail of recognizing natural, regular and diagonal tetration being equal. GFR Member Posts: 174 Threads: 4 Joined: Aug 2007 04/27/2008, 02:21 PM Congratulations !!! bo198214 Administrator Posts: 1,528 Threads: 97 Joined: Aug 2007 04/27/2008, 02:54 PM GFR Wrote:Congratulations !!! There is not yet much to congratulate, everything is in the state of conjectures and proofs are reluctant to reveal so far. GFR Member Posts: 174 Threads: 4 Joined: Aug 2007 04/27/2008, 10:14 PM Hey, Henryk! I just wished to sincerely congratulate you for the good possibility of acceptable coincident results of theese different strategies. The problems that we are trying to solve are really difficult. I wonder why they are so difficult. GFR bo198214 Administrator Posts: 1,528 Threads: 97 Joined: Aug 2007 06/21/2010, 01:34 PM (04/27/2008, 10:00 AM)bo198214 Wrote: I computed $\delta:=\text{rsexp}_{\sqrt{2}}(\text{nslog}_{\sqrt{2}}(0.3))-0.3$ with a precision of rsexp of 20 digits and varying matrix sizes for nslog from 10 to 100. The following graphs show $\delta$ in dependence on the matrix size (x-Axis). ... This looks quite as if regular and natural method are equal. Yippee! And here we see again how numerics can be misleading. Actually I increased the matrix size to 300. And plotted the difference $\text{rslog}-\text{islog}$, the result is the expected dilated oscillation for two different Abel functions:     In other words we can be quite sure that regular and intuitive tetration is different. mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 06/24/2010, 02:10 AM (This post was last modified: 06/24/2010, 06:00 AM by mike3.) I'm curious: What programs and methods are you using to compute this? Esp. how do you compute the solution in reasonable time? Also, can you try to expand, say, the $\exp_{\sqrt{2}}^{1/2}(z)$ generated by this method about the fixed point 2? If it's not the same as the regular iteration then it should fail to be holomorphic there, right? (as that's the thing that characterizes the regular iteration -- that it's differentiable at the fixed point, no?) If not, then what happens if you try comparing coefficients to those obtained from the regular iteration and looking for the difference? bo198214 Administrator Posts: 1,528 Threads: 97 Joined: Aug 2007 06/24/2010, 09:15 AM (06/24/2010, 02:10 AM)mike3 Wrote: I'm curious: What programs and methods are you using to compute this? Esp. how do you compute the solution in reasonable time? You mean the intuitive Abel function via the Abel matrix? Well I use some self written algorithm in Sage. It takes some time to compute the coefficients for a 300 matrix. I get 100x100 in less than a minute. But regardless how long it takes you only need to compute it once (even if it takes 5 days ), save it, and later you can use it anytime. If you need code, just tell me. But J.D. Fox development much more sophisticated algorithms, giving better precision. I will have a look at it later and improve my own algorithms accordingly. Quote:Also, can you try to expand, say, the $\exp_{\sqrt{2}}^{1/2}(z)$ generated by this method about the fixed point 2? If it's not the same as the regular iteration then it should fail to be holomorphic there, right? (as that's the thing that characterizes the regular iteration -- that it's differentiable at the fixed point, no?) yes. I will have a go at it (when I implemented Fox's improvements). Gottfried Ultimate Fellow Posts: 853 Threads: 126 Joined: Aug 2007 06/24/2010, 11:37 AM (06/24/2010, 09:15 AM)bo198214 Wrote: (06/24/2010, 02:10 AM)mike3 Wrote: I'm curious: What programs and methods are you using to compute this? Esp. how do you compute the solution in reasonable time? You mean the intuitive Abel function via the Abel matrix? Well I use some self written algorithm in Sage. It takes some time to compute the coefficients for a 300 matrix. I get 100x100 in less than a minute. But regardless how long it takes you only need to compute it once (even if it takes 5 days ), save it, and later you can use it anytime. If you need code, just tell me. But J.D. Fox development much more sophisticated algorithms, giving better precision. I will have a look at it later and improve my own algorithms accordingly. Also you could look at my computation at http://math.eretrandre.org/tetrationforu...27#pid2827 Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

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